Precalculus Examples

$x^{2} - 4 y^{2} - 20 x - 64 y - 172 = 0$ , $16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1

Solve for $x$ in $x^{2} - 4 y^{2} - 20 x - 64 y - 172 = 0$ .

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Step 1.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.2

Substitute the values $a = 1$ , $b = - 20$ , and $c = - 4 y^{2} - 64 y - 172$ into the quadratic formula and solve for $x$ .

$\frac{20 \pm \sqrt{{(- 20)}^{2} - 4 \cdot (1 \cdot (- 4 y^{2} - 64 y - 172))}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.3

Simplify.

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Step 1.3.1

Simplify the numerator.

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Step 1.3.1.1

Raise $- 20$ to the power of $2$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot 1 \cdot (- 4 y^{2} - 64 y - 172)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.3.1.2

Multiply $- 4$ by $1$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot (- 4 y^{2} - 64 y - 172)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.3.1.3

Apply the distributive property.

$x = \frac{20 \pm \sqrt{400 - 4 (- 4 y^{2}) - 4 (- 64 y) - 4 \cdot - 172}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.3.1.4

Simplify.

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Step 1.3.1.4.1

Multiply $- 4$ by $- 4$ .

$x = \frac{20 \pm \sqrt{400 + 16 y^{2} - 4 (- 64 y) - 4 \cdot - 172}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.3.1.4.2

Multiply $- 64$ by $- 4$ .

$x = \frac{20 \pm \sqrt{400 + 16 y^{2} + 256 y - 4 \cdot - 172}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.3.1.4.3

Multiply $- 4$ by $- 172$ .

$x = \frac{20 \pm \sqrt{400 + 16 y^{2} + 256 y + 688}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

$x = \frac{20 \pm \sqrt{400 + 16 y^{2} + 256 y + 688}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.3.1.5

Add $400$ and $688$ .

$x = \frac{20 \pm \sqrt{16 y^{2} + 256 y + 1088}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.3.1.6

Factor $16$ out of $16 y^{2} + 256 y + 1088$ .

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Step 1.3.1.6.1

Factor $16$ out of $16 y^{2}$ .

$x = \frac{20 \pm \sqrt{16 (y^{2}) + 256 y + 1088}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.3.1.6.2

Factor $16$ out of $256 y$ .

$x = \frac{20 \pm \sqrt{16 (y^{2}) + 16 (16 y) + 1088}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.3.1.6.3

Factor $16$ out of $1088$ .

$x = \frac{20 \pm \sqrt{16 y^{2} + 16 (16 y) + 16 \cdot 68}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.3.1.6.4

Factor $16$ out of $16 y^{2} + 16 (16 y)$ .

$x = \frac{20 \pm \sqrt{16 (y^{2} + 16 y) + 16 \cdot 68}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.3.1.6.5

Factor $16$ out of $16 (y^{2} + 16 y) + 16 \cdot 68$ .

$x = \frac{20 \pm \sqrt{16 (y^{2} + 16 y + 68)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

$x = \frac{20 \pm \sqrt{16 (y^{2} + 16 y + 68)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.3.1.7

Rewrite $16 (y^{2} + 16 y + 68)$ as ${(2^{2})}^{2} (y^{2} + 16 y + 68)$ .

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Step 1.3.1.7.1

Rewrite $16$ as $4^{2}$ .

$x = \frac{20 \pm \sqrt{4^{2} (y^{2} + 16 y + 68)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.3.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{20 \pm \sqrt{{(2^{2})}^{2} (y^{2} + 16 y + 68)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

$x = \frac{20 \pm \sqrt{{(2^{2})}^{2} (y^{2} + 16 y + 68)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.3.1.8

Pull terms out from under the radical.

$x = \frac{20 \pm 2^{2} \sqrt{y^{2} + 16 y + 68}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.3.1.9

Raise $2$ to the power of $2$ .

$x = \frac{20 \pm 4 \sqrt{y^{2} + 16 y + 68}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

$x = \frac{20 \pm 4 \sqrt{y^{2} + 16 y + 68}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.3.2

Multiply $2$ by $1$ .

$x = \frac{20 \pm 4 \sqrt{y^{2} + 16 y + 68}}{2}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.3.3

Simplify $\frac{20 \pm 4 \sqrt{y^{2} + 16 y + 68}}{2}$ .

$x = 10 \pm 2 \sqrt{y^{2} + 16 y + 68}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

$x = 10 \pm 2 \sqrt{y^{2} + 16 y + 68}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Raise $- 20$ to the power of $2$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot 1 \cdot (- 4 y^{2} - 64 y - 172)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.4.1.2

Multiply $- 4$ by $1$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot (- 4 y^{2} - 64 y - 172)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.4.1.3

Apply the distributive property.

$x = \frac{20 \pm \sqrt{400 - 4 (- 4 y^{2}) - 4 (- 64 y) - 4 \cdot - 172}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.4.1.4

Simplify.

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Step 1.4.1.4.1

Multiply $- 4$ by $- 4$ .

$x = \frac{20 \pm \sqrt{400 + 16 y^{2} - 4 (- 64 y) - 4 \cdot - 172}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.4.1.4.2

Multiply $- 64$ by $- 4$ .

$x = \frac{20 \pm \sqrt{400 + 16 y^{2} + 256 y - 4 \cdot - 172}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.4.1.4.3

Multiply $- 4$ by $- 172$ .

$x = \frac{20 \pm \sqrt{400 + 16 y^{2} + 256 y + 688}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

$x = \frac{20 \pm \sqrt{400 + 16 y^{2} + 256 y + 688}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.4.1.5

Add $400$ and $688$ .

$x = \frac{20 \pm \sqrt{16 y^{2} + 256 y + 1088}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.4.1.6

Factor $16$ out of $16 y^{2} + 256 y + 1088$ .

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Step 1.4.1.6.1

Factor $16$ out of $16 y^{2}$ .

$x = \frac{20 \pm \sqrt{16 (y^{2}) + 256 y + 1088}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.4.1.6.2

Factor $16$ out of $256 y$ .

$x = \frac{20 \pm \sqrt{16 (y^{2}) + 16 (16 y) + 1088}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.4.1.6.3

Factor $16$ out of $1088$ .

$x = \frac{20 \pm \sqrt{16 y^{2} + 16 (16 y) + 16 \cdot 68}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.4.1.6.4

Factor $16$ out of $16 y^{2} + 16 (16 y)$ .

$x = \frac{20 \pm \sqrt{16 (y^{2} + 16 y) + 16 \cdot 68}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.4.1.6.5

Factor $16$ out of $16 (y^{2} + 16 y) + 16 \cdot 68$ .

$x = \frac{20 \pm \sqrt{16 (y^{2} + 16 y + 68)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

$x = \frac{20 \pm \sqrt{16 (y^{2} + 16 y + 68)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.4.1.7

Rewrite $16 (y^{2} + 16 y + 68)$ as ${(2^{2})}^{2} (y^{2} + 16 y + 68)$ .

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Step 1.4.1.7.1

Rewrite $16$ as $4^{2}$ .

$x = \frac{20 \pm \sqrt{4^{2} (y^{2} + 16 y + 68)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.4.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{20 \pm \sqrt{{(2^{2})}^{2} (y^{2} + 16 y + 68)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

$x = \frac{20 \pm \sqrt{{(2^{2})}^{2} (y^{2} + 16 y + 68)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.4.1.8

Pull terms out from under the radical.

$x = \frac{20 \pm 2^{2} \sqrt{y^{2} + 16 y + 68}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.4.1.9

Raise $2$ to the power of $2$ .

$x = \frac{20 \pm 4 \sqrt{y^{2} + 16 y + 68}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

$x = \frac{20 \pm 4 \sqrt{y^{2} + 16 y + 68}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.4.2

Multiply $2$ by $1$ .

$x = \frac{20 \pm 4 \sqrt{y^{2} + 16 y + 68}}{2}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.4.3

Simplify $\frac{20 \pm 4 \sqrt{y^{2} + 16 y + 68}}{2}$ .

$x = 10 \pm 2 \sqrt{y^{2} + 16 y + 68}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.4.4

Change the $\pm$ to $+$ .

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Raise $- 20$ to the power of $2$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot 1 \cdot (- 4 y^{2} - 64 y - 172)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.5.1.2

Multiply $- 4$ by $1$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot (- 4 y^{2} - 64 y - 172)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.5.1.3

Apply the distributive property.

$x = \frac{20 \pm \sqrt{400 - 4 (- 4 y^{2}) - 4 (- 64 y) - 4 \cdot - 172}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.5.1.4

Simplify.

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Step 1.5.1.4.1

Multiply $- 4$ by $- 4$ .

$x = \frac{20 \pm \sqrt{400 + 16 y^{2} - 4 (- 64 y) - 4 \cdot - 172}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.5.1.4.2

Multiply $- 64$ by $- 4$ .

$x = \frac{20 \pm \sqrt{400 + 16 y^{2} + 256 y - 4 \cdot - 172}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.5.1.4.3

Multiply $- 4$ by $- 172$ .

$x = \frac{20 \pm \sqrt{400 + 16 y^{2} + 256 y + 688}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

$x = \frac{20 \pm \sqrt{400 + 16 y^{2} + 256 y + 688}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.5.1.5

Add $400$ and $688$ .

$x = \frac{20 \pm \sqrt{16 y^{2} + 256 y + 1088}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.5.1.6

Factor $16$ out of $16 y^{2} + 256 y + 1088$ .

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Step 1.5.1.6.1

Factor $16$ out of $16 y^{2}$ .

$x = \frac{20 \pm \sqrt{16 (y^{2}) + 256 y + 1088}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.5.1.6.2

Factor $16$ out of $256 y$ .

$x = \frac{20 \pm \sqrt{16 (y^{2}) + 16 (16 y) + 1088}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.5.1.6.3

Factor $16$ out of $1088$ .

$x = \frac{20 \pm \sqrt{16 y^{2} + 16 (16 y) + 16 \cdot 68}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.5.1.6.4

Factor $16$ out of $16 y^{2} + 16 (16 y)$ .

$x = \frac{20 \pm \sqrt{16 (y^{2} + 16 y) + 16 \cdot 68}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.5.1.6.5

Factor $16$ out of $16 (y^{2} + 16 y) + 16 \cdot 68$ .

$x = \frac{20 \pm \sqrt{16 (y^{2} + 16 y + 68)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

$x = \frac{20 \pm \sqrt{16 (y^{2} + 16 y + 68)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.5.1.7

Rewrite $16 (y^{2} + 16 y + 68)$ as ${(2^{2})}^{2} (y^{2} + 16 y + 68)$ .

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Step 1.5.1.7.1

Rewrite $16$ as $4^{2}$ .

$x = \frac{20 \pm \sqrt{4^{2} (y^{2} + 16 y + 68)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.5.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{20 \pm \sqrt{{(2^{2})}^{2} (y^{2} + 16 y + 68)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

$x = \frac{20 \pm \sqrt{{(2^{2})}^{2} (y^{2} + 16 y + 68)}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.5.1.8

Pull terms out from under the radical.

$x = \frac{20 \pm 2^{2} \sqrt{y^{2} + 16 y + 68}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.5.1.9

Raise $2$ to the power of $2$ .

$x = \frac{20 \pm 4 \sqrt{y^{2} + 16 y + 68}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

$x = \frac{20 \pm 4 \sqrt{y^{2} + 16 y + 68}}{2 \cdot 1}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.5.2

Multiply $2$ by $1$ .

$x = \frac{20 \pm 4 \sqrt{y^{2} + 16 y + 68}}{2}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.5.3

Simplify $\frac{20 \pm 4 \sqrt{y^{2} + 16 y + 68}}{2}$ .

$x = 10 \pm 2 \sqrt{y^{2} + 16 y + 68}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.5.4

Change the $\pm$ to $-$ .

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 1.6

The final answer is the combination of both solutions.

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$

Step 2

Solve the system $x = 10 + 2 \sqrt{y^{2} + 16 y + 68}, 16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$ .

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Step 2.1

Replace all occurrences of $x$ with $10 + 2 \sqrt{y^{2} + 16 y + 68}$ in each equation.

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Step 2.1.1

Replace all occurrences of $x$ in $16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$ with $10 + 2 \sqrt{y^{2} + 16 y + 68}$ .

$16 {(10 + 2 \sqrt{y^{2} + 16 y + 68})}^{2} + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify $16 {(10 + 2 \sqrt{y^{2} + 16 y + 68})}^{2} + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600$ .

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Step 2.1.2.1.1

Simplify each term.

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Step 2.1.2.1.1.1

Rewrite ${(10 + 2 \sqrt{y^{2} + 16 y + 68})}^{2}$ as $(10 + 2 \sqrt{y^{2} + 16 y + 68}) (10 + 2 \sqrt{y^{2} + 16 y + 68})$ .

$16 ((10 + 2 \sqrt{y^{2} + 16 y + 68}) (10 + 2 \sqrt{y^{2} + 16 y + 68})) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.2

Expand $(10 + 2 \sqrt{y^{2} + 16 y + 68}) (10 + 2 \sqrt{y^{2} + 16 y + 68})$ using the FOIL Method.

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Step 2.1.2.1.1.2.1

Apply the distributive property.

$16 (10 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 2 \sqrt{y^{2} + 16 y + 68} (10 + 2 \sqrt{y^{2} + 16 y + 68})) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.2.2

Apply the distributive property.

$16 (10 \cdot 10 + 10 (2 \sqrt{y^{2} + 16 y + 68}) + 2 \sqrt{y^{2} + 16 y + 68} (10 + 2 \sqrt{y^{2} + 16 y + 68})) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.2.3

Apply the distributive property.

$16 (10 \cdot 10 + 10 (2 \sqrt{y^{2} + 16 y + 68}) + 2 \sqrt{y^{2} + 16 y + 68} \cdot 10 + 2 \sqrt{y^{2} + 16 y + 68} (2 \sqrt{y^{2} + 16 y + 68})) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.3

Simplify and combine like terms.

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Step 2.1.2.1.1.3.1

Simplify each term.

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Step 2.1.2.1.1.3.1.1

Multiply $10$ by $10$ .

$16 (100 + 10 (2 \sqrt{y^{2} + 16 y + 68}) + 2 \sqrt{y^{2} + 16 y + 68} \cdot 10 + 2 \sqrt{y^{2} + 16 y + 68} (2 \sqrt{y^{2} + 16 y + 68})) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.3.1.2

Multiply $2$ by $10$ .

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 2 \sqrt{y^{2} + 16 y + 68} \cdot 10 + 2 \sqrt{y^{2} + 16 y + 68} (2 \sqrt{y^{2} + 16 y + 68})) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.3.1.3

Multiply $10$ by $2$ .

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 2 \sqrt{y^{2} + 16 y + 68} (2 \sqrt{y^{2} + 16 y + 68})) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.3.1.4

Multiply $2 \sqrt{y^{2} + 16 y + 68} (2 \sqrt{y^{2} + 16 y + 68})$ .

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Step 2.1.2.1.1.3.1.4.1

Multiply $2$ by $2$ .

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 4 \sqrt{y^{2} + 16 y + 68} \sqrt{y^{2} + 16 y + 68}) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.3.1.4.2

Raise $\sqrt{y^{2} + 16 y + 68}$ to the power of $1$ .

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 4 (\sqrt{y^{2} + 16 y + 68} \sqrt{y^{2} + 16 y + 68})) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.3.1.4.3

Raise $\sqrt{y^{2} + 16 y + 68}$ to the power of $1$ .

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 4 (\sqrt{y^{2} + 16 y + 68} \sqrt{y^{2} + 16 y + 68})) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.3.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 4 {\sqrt{y^{2} + 16 y + 68}}^{1 + 1}) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.3.1.4.5

Add $1$ and $1$ .

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 4 {\sqrt{y^{2} + 16 y + 68}}^{2}) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 4 {\sqrt{y^{2} + 16 y + 68}}^{2}) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.3.1.5

Rewrite ${\sqrt{y^{2} + 16 y + 68}}^{2}$ as $y^{2} + 16 y + 68$ .

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Step 2.1.2.1.1.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y^{2} + 16 y + 68}$ as ${(y^{2} + 16 y + 68)}^{\frac{1}{2}}$ .

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 4 {({(y^{2} + 16 y + 68)}^{\frac{1}{2}})}^{2}) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.3.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 4 {(y^{2} + 16 y + 68)}^{\frac{1}{2} \cdot 2}) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.3.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 4 {(y^{2} + 16 y + 68)}^{\frac{2}{2}}) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.3.1.5.4.1

Cancel the common factor.

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 4 {(y^{2} + 16 y + 68)}^{\frac{2}{2}}) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.3.1.5.4.2

Rewrite the expression.

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 4 (y^{2} + 16 y + 68)) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 4 (y^{2} + 16 y + 68)) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.3.1.5.5

Simplify.

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 4 (y^{2} + 16 y + 68)) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 4 (y^{2} + 16 y + 68)) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.3.1.6

Apply the distributive property.

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 4 y^{2} + 4 (16 y) + 4 \cdot 68) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.3.1.7

Simplify.

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Step 2.1.2.1.1.3.1.7.1

Multiply $16$ by $4$ .

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 4 y^{2} + 64 y + 4 \cdot 68) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.3.1.7.2

Multiply $4$ by $68$ .

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 4 y^{2} + 64 y + 272) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 4 y^{2} + 64 y + 272) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$16 (100 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 4 y^{2} + 64 y + 272) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.3.2

Add $100$ and $272$ .

$16 (372 + 20 \sqrt{y^{2} + 16 y + 68} + 20 \sqrt{y^{2} + 16 y + 68} + 4 y^{2} + 64 y) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.3.3

Add $20 \sqrt{y^{2} + 16 y + 68}$ and $20 \sqrt{y^{2} + 16 y + 68}$ .

$16 (372 + 40 \sqrt{y^{2} + 16 y + 68} + 4 y^{2} + 64 y) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$16 (372 + 40 \sqrt{y^{2} + 16 y + 68} + 4 y^{2} + 64 y) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.4

Apply the distributive property.

$16 \cdot 372 + 16 (40 \sqrt{y^{2} + 16 y + 68}) + 16 (4 y^{2}) + 16 (64 y) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.5

Simplify.

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Step 2.1.2.1.1.5.1

Multiply $16$ by $372$ .

$5952 + 16 (40 \sqrt{y^{2} + 16 y + 68}) + 16 (4 y^{2}) + 16 (64 y) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.5.2

Multiply $40$ by $16$ .

$5952 + 640 \sqrt{y^{2} + 16 y + 68} + 16 (4 y^{2}) + 16 (64 y) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.5.3

Multiply $4$ by $16$ .

$5952 + 640 \sqrt{y^{2} + 16 y + 68} + 64 y^{2} + 16 (64 y) + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.5.4

Multiply $64$ by $16$ .

$5952 + 640 \sqrt{y^{2} + 16 y + 68} + 64 y^{2} + 1024 y + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$5952 + 640 \sqrt{y^{2} + 16 y + 68} + 64 y^{2} + 1024 y + 4 y^{2} - 320 (10 + 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.6

Apply the distributive property.

$5952 + 640 \sqrt{y^{2} + 16 y + 68} + 64 y^{2} + 1024 y + 4 y^{2} - 320 \cdot 10 - 320 (2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.7

Multiply $- 320$ by $10$ .

$5952 + 640 \sqrt{y^{2} + 16 y + 68} + 64 y^{2} + 1024 y + 4 y^{2} - 3200 - 320 (2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.1.8

Multiply $2$ by $- 320$ .

$5952 + 640 \sqrt{y^{2} + 16 y + 68} + 64 y^{2} + 1024 y + 4 y^{2} - 3200 - 640 \sqrt{y^{2} + 16 y + 68} + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$5952 + 640 \sqrt{y^{2} + 16 y + 68} + 64 y^{2} + 1024 y + 4 y^{2} - 3200 - 640 \sqrt{y^{2} + 16 y + 68} + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.2

Simplify by adding terms.

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Step 2.1.2.1.2.1

Combine the opposite terms in $5952 + 640 \sqrt{y^{2} + 16 y + 68} + 64 y^{2} + 1024 y + 4 y^{2} - 3200 - 640 \sqrt{y^{2} + 16 y + 68} + 64 y + 1600$ .

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Step 2.1.2.1.2.1.1

Subtract $640 \sqrt{y^{2} + 16 y + 68}$ from $640 \sqrt{y^{2} + 16 y + 68}$ .

$5952 + 64 y^{2} + 1024 y + 4 y^{2} - 3200 + 0 + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.2.1.2

Add $5952 + 64 y^{2} + 1024 y + 4 y^{2} - 3200$ and $0$ .

$5952 + 64 y^{2} + 1024 y + 4 y^{2} - 3200 + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$5952 + 64 y^{2} + 1024 y + 4 y^{2} - 3200 + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.2.2

Subtract $3200$ from $5952$ .

$64 y^{2} + 1024 y + 4 y^{2} + 2752 + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.2.3

Add $64 y^{2}$ and $4 y^{2}$ .

$68 y^{2} + 1024 y + 2752 + 64 y + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.2.4

Add $1024 y$ and $64 y$ .

$68 y^{2} + 1088 y + 2752 + 1600 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.1.2.1.2.5

Add $2752$ and $1600$ .

$68 y^{2} + 1088 y + 4352 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$68 y^{2} + 1088 y + 4352 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$68 y^{2} + 1088 y + 4352 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$68 y^{2} + 1088 y + 4352 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$68 y^{2} + 1088 y + 4352 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.2

Solve for $y$ in $68 y^{2} + 1088 y + 4352 = 0$ .

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Step 2.2.1

Factor the left side of the equation.

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Step 2.2.1.1

Factor $68$ out of $68 y^{2} + 1088 y + 4352$ .

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Step 2.2.1.1.1

Factor $68$ out of $68 y^{2}$ .

$68 (y^{2}) + 1088 y + 4352 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.2.1.1.2

Factor $68$ out of $1088 y$ .

$68 (y^{2}) + 68 (16 y) + 4352 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.2.1.1.3

Factor $68$ out of $4352$ .

$68 y^{2} + 68 (16 y) + 68 \cdot 64 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.2.1.1.4

Factor $68$ out of $68 y^{2} + 68 (16 y)$ .

$68 (y^{2} + 16 y) + 68 \cdot 64 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.2.1.1.5

Factor $68$ out of $68 (y^{2} + 16 y) + 68 \cdot 64$ .

$68 (y^{2} + 16 y + 64) = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$68 (y^{2} + 16 y + 64) = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.2.1.2

Factor using the perfect square rule.

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Step 2.2.1.2.1

Rewrite $64$ as $8^{2}$ .

$68 (y^{2} + 16 y + 8^{2}) = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.2.1.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$16 y = 2 \cdot y \cdot 8$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.2.1.2.3

Rewrite the polynomial.

$68 (y^{2} + 2 \cdot y \cdot 8 + 8^{2}) = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.2.1.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = y$ and $b = 8$ .

$68 {(y + 8)}^{2} = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$68 {(y + 8)}^{2} = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$68 {(y + 8)}^{2} = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.2.2

Divide each term in $68 {(y + 8)}^{2} = 0$ by $68$ and simplify.

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Step 2.2.2.1

Divide each term in $68 {(y + 8)}^{2} = 0$ by $68$ .

$\frac{68 {(y + 8)}^{2}}{68} = \frac{0}{68}$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $68$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{68 {(y + 8)}^{2}}{68} = \frac{0}{68}$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.2.2.2.1.2

Divide ${(y + 8)}^{2}$ by $1$ .

${(y + 8)}^{2} = \frac{0}{68}$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

${(y + 8)}^{2} = \frac{0}{68}$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

${(y + 8)}^{2} = \frac{0}{68}$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Divide $0$ by $68$ .

${(y + 8)}^{2} = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

${(y + 8)}^{2} = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

${(y + 8)}^{2} = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.2.3

Set the $y + 8$ equal to $0$ .

$y + 8 = 0$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.2.4

Subtract $8$ from both sides of the equation.

$y = - 8$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

$y = - 8$

$x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$

Step 2.3

Replace all occurrences of $y$ with $- 8$ in each equation.

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Step 2.3.1

Replace all occurrences of $y$ in $x = 10 + 2 \sqrt{y^{2} + 16 y + 68}$ with $- 8$ .

$x = 10 + 2 \sqrt{{(- 8)}^{2} + 16 (- 8) + 68}$

$y = - 8$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $10 + 2 \sqrt{{(- 8)}^{2} + 16 (- 8) + 68}$ .

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Step 2.3.2.1.1

Simplify each term.

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Step 2.3.2.1.1.1

Raise $- 8$ to the power of $2$ .

$x = 10 + 2 \sqrt{64 + 16 (- 8) + 68}$

$y = - 8$

Step 2.3.2.1.1.2

Multiply $16$ by $- 8$ .

$x = 10 + 2 \sqrt{64 - 128 + 68}$

$y = - 8$

Step 2.3.2.1.1.3

Subtract $128$ from $64$ .

$x = 10 + 2 \sqrt{- 64 + 68}$

$y = - 8$

Step 2.3.2.1.1.4

Add $- 64$ and $68$ .

$x = 10 + 2 \sqrt{4}$

$y = - 8$

Step 2.3.2.1.1.5

Rewrite $4$ as $2^{2}$ .

$x = 10 + 2 \sqrt{2^{2}}$

$y = - 8$

Step 2.3.2.1.1.6

Pull terms out from under the radical, assuming positive real numbers.

$x = 10 + 2 \cdot 2$

$y = - 8$

Step 2.3.2.1.1.7

Multiply $2$ by $2$ .

$x = 10 + 4$

$y = - 8$

$x = 10 + 4$

$y = - 8$

Step 2.3.2.1.2

Add $10$ and $4$ .

$x = 14$

$y = - 8$

$x = 14$

$y = - 8$

$x = 14$

$y = - 8$

$x = 14$

$y = - 8$

$x = 14$

$y = - 8$

Step 3

Solve the system $x = 10 - 2 \sqrt{y^{2} + 16 y + 68}, 16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$ .

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Step 3.1

Replace all occurrences of $x$ with $10 - 2 \sqrt{y^{2} + 16 y + 68}$ in each equation.

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Step 3.1.1

Replace all occurrences of $x$ in $16 x^{2} + 4 y^{2} - 320 x + 64 y + 1600 = 0$ with $10 - 2 \sqrt{y^{2} + 16 y + 68}$ .

$16 {(10 - 2 \sqrt{y^{2} + 16 y + 68})}^{2} + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify $16 {(10 - 2 \sqrt{y^{2} + 16 y + 68})}^{2} + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600$ .

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Step 3.1.2.1.1

Simplify each term.

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Step 3.1.2.1.1.1

Rewrite ${(10 - 2 \sqrt{y^{2} + 16 y + 68})}^{2}$ as $(10 - 2 \sqrt{y^{2} + 16 y + 68}) (10 - 2 \sqrt{y^{2} + 16 y + 68})$ .

$16 ((10 - 2 \sqrt{y^{2} + 16 y + 68}) (10 - 2 \sqrt{y^{2} + 16 y + 68})) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.2

Expand $(10 - 2 \sqrt{y^{2} + 16 y + 68}) (10 - 2 \sqrt{y^{2} + 16 y + 68})$ using the FOIL Method.

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Step 3.1.2.1.1.2.1

Apply the distributive property.

$16 (10 (10 - 2 \sqrt{y^{2} + 16 y + 68}) - 2 \sqrt{y^{2} + 16 y + 68} (10 - 2 \sqrt{y^{2} + 16 y + 68})) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.2.2

Apply the distributive property.

$16 (10 \cdot 10 + 10 (- 2 \sqrt{y^{2} + 16 y + 68}) - 2 \sqrt{y^{2} + 16 y + 68} (10 - 2 \sqrt{y^{2} + 16 y + 68})) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.2.3

Apply the distributive property.

$16 (10 \cdot 10 + 10 (- 2 \sqrt{y^{2} + 16 y + 68}) - 2 \sqrt{y^{2} + 16 y + 68} \cdot 10 - 2 \sqrt{y^{2} + 16 y + 68} (- 2 \sqrt{y^{2} + 16 y + 68})) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.3

Simplify and combine like terms.

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Step 3.1.2.1.1.3.1

Simplify each term.

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Step 3.1.2.1.1.3.1.1

Multiply $10$ by $10$ .

$16 (100 + 10 (- 2 \sqrt{y^{2} + 16 y + 68}) - 2 \sqrt{y^{2} + 16 y + 68} \cdot 10 - 2 \sqrt{y^{2} + 16 y + 68} (- 2 \sqrt{y^{2} + 16 y + 68})) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.3.1.2

Multiply $- 2$ by $10$ .

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 2 \sqrt{y^{2} + 16 y + 68} \cdot 10 - 2 \sqrt{y^{2} + 16 y + 68} (- 2 \sqrt{y^{2} + 16 y + 68})) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.3.1.3

Multiply $10$ by $- 2$ .

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} - 2 \sqrt{y^{2} + 16 y + 68} (- 2 \sqrt{y^{2} + 16 y + 68})) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.3.1.4

Multiply $- 2 \sqrt{y^{2} + 16 y + 68} (- 2 \sqrt{y^{2} + 16 y + 68})$ .

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Step 3.1.2.1.1.3.1.4.1

Multiply $- 2$ by $- 2$ .

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} + 4 \sqrt{y^{2} + 16 y + 68} \sqrt{y^{2} + 16 y + 68}) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.3.1.4.2

Raise $\sqrt{y^{2} + 16 y + 68}$ to the power of $1$ .

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} + 4 (\sqrt{y^{2} + 16 y + 68} \sqrt{y^{2} + 16 y + 68})) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.3.1.4.3

Raise $\sqrt{y^{2} + 16 y + 68}$ to the power of $1$ .

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} + 4 (\sqrt{y^{2} + 16 y + 68} \sqrt{y^{2} + 16 y + 68})) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.3.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} + 4 {\sqrt{y^{2} + 16 y + 68}}^{1 + 1}) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.3.1.4.5

Add $1$ and $1$ .

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} + 4 {\sqrt{y^{2} + 16 y + 68}}^{2}) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} + 4 {\sqrt{y^{2} + 16 y + 68}}^{2}) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.3.1.5

Rewrite ${\sqrt{y^{2} + 16 y + 68}}^{2}$ as $y^{2} + 16 y + 68$ .

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Step 3.1.2.1.1.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y^{2} + 16 y + 68}$ as ${(y^{2} + 16 y + 68)}^{\frac{1}{2}}$ .

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} + 4 {({(y^{2} + 16 y + 68)}^{\frac{1}{2}})}^{2}) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.3.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} + 4 {(y^{2} + 16 y + 68)}^{\frac{1}{2} \cdot 2}) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.3.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} + 4 {(y^{2} + 16 y + 68)}^{\frac{2}{2}}) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.1.3.1.5.4.1

Cancel the common factor.

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} + 4 {(y^{2} + 16 y + 68)}^{\frac{2}{2}}) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.3.1.5.4.2

Rewrite the expression.

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} + 4 (y^{2} + 16 y + 68)) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} + 4 (y^{2} + 16 y + 68)) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.3.1.5.5

Simplify.

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} + 4 (y^{2} + 16 y + 68)) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} + 4 (y^{2} + 16 y + 68)) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.3.1.6

Apply the distributive property.

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} + 4 y^{2} + 4 (16 y) + 4 \cdot 68) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.3.1.7

Simplify.

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Step 3.1.2.1.1.3.1.7.1

Multiply $16$ by $4$ .

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} + 4 y^{2} + 64 y + 4 \cdot 68) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.3.1.7.2

Multiply $4$ by $68$ .

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} + 4 y^{2} + 64 y + 272) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} + 4 y^{2} + 64 y + 272) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$16 (100 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} + 4 y^{2} + 64 y + 272) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.3.2

Add $100$ and $272$ .

$16 (372 - 20 \sqrt{y^{2} + 16 y + 68} - 20 \sqrt{y^{2} + 16 y + 68} + 4 y^{2} + 64 y) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.3.3

Subtract $20 \sqrt{y^{2} + 16 y + 68}$ from $- 20 \sqrt{y^{2} + 16 y + 68}$ .

$16 (372 - 40 \sqrt{y^{2} + 16 y + 68} + 4 y^{2} + 64 y) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$16 (372 - 40 \sqrt{y^{2} + 16 y + 68} + 4 y^{2} + 64 y) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.4

Apply the distributive property.

$16 \cdot 372 + 16 (- 40 \sqrt{y^{2} + 16 y + 68}) + 16 (4 y^{2}) + 16 (64 y) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.5

Simplify.

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Step 3.1.2.1.1.5.1

Multiply $16$ by $372$ .

$5952 + 16 (- 40 \sqrt{y^{2} + 16 y + 68}) + 16 (4 y^{2}) + 16 (64 y) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.5.2

Multiply $- 40$ by $16$ .

$5952 - 640 \sqrt{y^{2} + 16 y + 68} + 16 (4 y^{2}) + 16 (64 y) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.5.3

Multiply $4$ by $16$ .

$5952 - 640 \sqrt{y^{2} + 16 y + 68} + 64 y^{2} + 16 (64 y) + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.5.4

Multiply $64$ by $16$ .

$5952 - 640 \sqrt{y^{2} + 16 y + 68} + 64 y^{2} + 1024 y + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$5952 - 640 \sqrt{y^{2} + 16 y + 68} + 64 y^{2} + 1024 y + 4 y^{2} - 320 (10 - 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.6

Apply the distributive property.

$5952 - 640 \sqrt{y^{2} + 16 y + 68} + 64 y^{2} + 1024 y + 4 y^{2} - 320 \cdot 10 - 320 (- 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.7

Multiply $- 320$ by $10$ .

$5952 - 640 \sqrt{y^{2} + 16 y + 68} + 64 y^{2} + 1024 y + 4 y^{2} - 3200 - 320 (- 2 \sqrt{y^{2} + 16 y + 68}) + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.1.8

Multiply $- 2$ by $- 320$ .

$5952 - 640 \sqrt{y^{2} + 16 y + 68} + 64 y^{2} + 1024 y + 4 y^{2} - 3200 + 640 \sqrt{y^{2} + 16 y + 68} + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$5952 - 640 \sqrt{y^{2} + 16 y + 68} + 64 y^{2} + 1024 y + 4 y^{2} - 3200 + 640 \sqrt{y^{2} + 16 y + 68} + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.2

Simplify by adding terms.

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Step 3.1.2.1.2.1

Combine the opposite terms in $5952 - 640 \sqrt{y^{2} + 16 y + 68} + 64 y^{2} + 1024 y + 4 y^{2} - 3200 + 640 \sqrt{y^{2} + 16 y + 68} + 64 y + 1600$ .

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Step 3.1.2.1.2.1.1

Add $- 640 \sqrt{y^{2} + 16 y + 68}$ and $640 \sqrt{y^{2} + 16 y + 68}$ .

$5952 + 64 y^{2} + 1024 y + 4 y^{2} - 3200 + 0 + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.2.1.2

Add $5952 + 64 y^{2} + 1024 y + 4 y^{2} - 3200$ and $0$ .

$5952 + 64 y^{2} + 1024 y + 4 y^{2} - 3200 + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$5952 + 64 y^{2} + 1024 y + 4 y^{2} - 3200 + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.2.2

Subtract $3200$ from $5952$ .

$64 y^{2} + 1024 y + 4 y^{2} + 2752 + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.2.3

Add $64 y^{2}$ and $4 y^{2}$ .

$68 y^{2} + 1024 y + 2752 + 64 y + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.2.4

Add $1024 y$ and $64 y$ .

$68 y^{2} + 1088 y + 2752 + 1600 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.1.2.1.2.5

Add $2752$ and $1600$ .

$68 y^{2} + 1088 y + 4352 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$68 y^{2} + 1088 y + 4352 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$68 y^{2} + 1088 y + 4352 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$68 y^{2} + 1088 y + 4352 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$68 y^{2} + 1088 y + 4352 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.2

Solve for $y$ in $68 y^{2} + 1088 y + 4352 = 0$ .

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Step 3.2.1

Factor the left side of the equation.

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Step 3.2.1.1

Factor $68$ out of $68 y^{2} + 1088 y + 4352$ .

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Step 3.2.1.1.1

Factor $68$ out of $68 y^{2}$ .

$68 (y^{2}) + 1088 y + 4352 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.2.1.1.2

Factor $68$ out of $1088 y$ .

$68 (y^{2}) + 68 (16 y) + 4352 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.2.1.1.3

Factor $68$ out of $4352$ .

$68 y^{2} + 68 (16 y) + 68 \cdot 64 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.2.1.1.4

Factor $68$ out of $68 y^{2} + 68 (16 y)$ .

$68 (y^{2} + 16 y) + 68 \cdot 64 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.2.1.1.5

Factor $68$ out of $68 (y^{2} + 16 y) + 68 \cdot 64$ .

$68 (y^{2} + 16 y + 64) = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$68 (y^{2} + 16 y + 64) = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.2.1.2

Factor using the perfect square rule.

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Step 3.2.1.2.1

Rewrite $64$ as $8^{2}$ .

$68 (y^{2} + 16 y + 8^{2}) = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.2.1.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$16 y = 2 \cdot y \cdot 8$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.2.1.2.3

Rewrite the polynomial.

$68 (y^{2} + 2 \cdot y \cdot 8 + 8^{2}) = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.2.1.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = y$ and $b = 8$ .

$68 {(y + 8)}^{2} = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$68 {(y + 8)}^{2} = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$68 {(y + 8)}^{2} = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.2.2

Divide each term in $68 {(y + 8)}^{2} = 0$ by $68$ and simplify.

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Step 3.2.2.1

Divide each term in $68 {(y + 8)}^{2} = 0$ by $68$ .

$\frac{68 {(y + 8)}^{2}}{68} = \frac{0}{68}$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $68$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{68 {(y + 8)}^{2}}{68} = \frac{0}{68}$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.2.2.2.1.2

Divide ${(y + 8)}^{2}$ by $1$ .

${(y + 8)}^{2} = \frac{0}{68}$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

${(y + 8)}^{2} = \frac{0}{68}$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

${(y + 8)}^{2} = \frac{0}{68}$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Divide $0$ by $68$ .

${(y + 8)}^{2} = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

${(y + 8)}^{2} = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

${(y + 8)}^{2} = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.2.3

Set the $y + 8$ equal to $0$ .

$y + 8 = 0$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.2.4

Subtract $8$ from both sides of the equation.

$y = - 8$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

$y = - 8$

$x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$

Step 3.3

Replace all occurrences of $y$ with $- 8$ in each equation.

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Step 3.3.1

Replace all occurrences of $y$ in $x = 10 - 2 \sqrt{y^{2} + 16 y + 68}$ with $- 8$ .

$x = 10 - 2 \sqrt{{(- 8)}^{2} + 16 (- 8) + 68}$

$y = - 8$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $10 - 2 \sqrt{{(- 8)}^{2} + 16 (- 8) + 68}$ .

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Step 3.3.2.1.1

Simplify each term.

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Step 3.3.2.1.1.1

Raise $- 8$ to the power of $2$ .

$x = 10 - 2 \sqrt{64 + 16 (- 8) + 68}$

$y = - 8$

Step 3.3.2.1.1.2

Multiply $16$ by $- 8$ .

$x = 10 - 2 \sqrt{64 - 128 + 68}$

$y = - 8$

Step 3.3.2.1.1.3

Subtract $128$ from $64$ .

$x = 10 - 2 \sqrt{- 64 + 68}$

$y = - 8$

Step 3.3.2.1.1.4

Add $- 64$ and $68$ .

$x = 10 - 2 \sqrt{4}$

$y = - 8$

Step 3.3.2.1.1.5

Rewrite $4$ as $2^{2}$ .

$x = 10 - 2 \sqrt{2^{2}}$

$y = - 8$

Step 3.3.2.1.1.6

Pull terms out from under the radical, assuming positive real numbers.

$x = 10 - 2 \cdot 2$

$y = - 8$

Step 3.3.2.1.1.7

Multiply $- 2$ by $2$ .

$x = 10 - 4$

$y = - 8$

$x = 10 - 4$

$y = - 8$

Step 3.3.2.1.2

Subtract $4$ from $10$ .

$x = 6$

$y = - 8$

$x = 6$

$y = - 8$

$x = 6$

$y = - 8$

$x = 6$

$y = - 8$

$x = 6$

$y = - 8$

Step 4

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(14, - 8)$

$(6, - 8)$

Step 5

The result can be shown in multiple forms.

Point Form:

$(14, - 8), (6, - 8)$

Equation Form:

$x = 14, y = - 8$

$x = 6, y = - 8$

Step 6