Pre-Algebra Examples

$2 (x^{2} + 6 x) + 4 (y^{2} - 4 y) = 20$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$2 x^{2} + 2 (6 x) + 4 (y^{2} - 4 y) = 20$

Step 1.2

Multiply $6$ by $2$ .

$2 x^{2} + 12 x + 4 (y^{2} - 4 y) = 20$

Step 1.3

Apply the distributive property.

$2 x^{2} + 12 x + 4 y^{2} + 4 (- 4 y) = 20$

Step 1.4

Multiply $- 4$ by $4$ .

$2 x^{2} + 12 x + 4 y^{2} - 16 y = 20$

Step 2

Subtract $20$ from both sides of the equation.

$2 x^{2} + 12 x + 4 y^{2} - 16 y - 20 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 4$ , $b = - 16$ , and $c = 2 x^{2} + 12 x - 20$ into the quadratic formula and solve for $y$ .

$\frac{16 \pm \sqrt{{(- 16)}^{2} - 4 \cdot (4 \cdot (2 x^{2} + 12 x - 20))}}{2 \cdot 4}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 16$ to the power of $2$ .

$y = \frac{16 \pm \sqrt{256 - 4 \cdot 4 \cdot (2 x^{2} + 12 x - 20)}}{2 \cdot 4}$

Step 5.1.2

Multiply $- 4$ by $4$ .

$y = \frac{16 \pm \sqrt{256 - 16 \cdot (2 x^{2} + 12 x - 20)}}{2 \cdot 4}$

Step 5.1.3

Apply the distributive property.

$y = \frac{16 \pm \sqrt{256 - 16 (2 x^{2}) - 16 (12 x) - 16 \cdot - 20}}{2 \cdot 4}$

Step 5.1.4

Simplify.

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Step 5.1.4.1

Multiply $2$ by $- 16$ .

$y = \frac{16 \pm \sqrt{256 - 32 x^{2} - 16 (12 x) - 16 \cdot - 20}}{2 \cdot 4}$

Step 5.1.4.2

Multiply $12$ by $- 16$ .

$y = \frac{16 \pm \sqrt{256 - 32 x^{2} - 192 x - 16 \cdot - 20}}{2 \cdot 4}$

Step 5.1.4.3

Multiply $- 16$ by $- 20$ .

$y = \frac{16 \pm \sqrt{256 - 32 x^{2} - 192 x + 320}}{2 \cdot 4}$

Step 5.1.5

Add $256$ and $320$ .

$y = \frac{16 \pm \sqrt{- 32 x^{2} - 192 x + 576}}{2 \cdot 4}$

Step 5.1.6

Factor $32$ out of $- 32 x^{2} - 192 x + 576$ .

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Step 5.1.6.1

Factor $32$ out of $- 32 x^{2}$ .

$y = \frac{16 \pm \sqrt{32 (- x^{2}) - 192 x + 576}}{2 \cdot 4}$

Step 5.1.6.2

Factor $32$ out of $- 192 x$ .

$y = \frac{16 \pm \sqrt{32 (- x^{2}) + 32 (- 6 x) + 576}}{2 \cdot 4}$

Step 5.1.6.3

Factor $32$ out of $576$ .

$y = \frac{16 \pm \sqrt{32 (- x^{2}) + 32 (- 6 x) + 32 (18)}}{2 \cdot 4}$

Step 5.1.6.4

Factor $32$ out of $32 (- x^{2}) + 32 (- 6 x)$ .

$y = \frac{16 \pm \sqrt{32 (- x^{2} - 6 x) + 32 (18)}}{2 \cdot 4}$

Step 5.1.6.5

Factor $32$ out of $32 (- x^{2} - 6 x) + 32 (18)$ .

$y = \frac{16 \pm \sqrt{32 (- x^{2} - 6 x + 18)}}{2 \cdot 4}$

Step 5.1.7

Rewrite $32 (- x^{2} - 6 x + 18)$ as ${(2^{2})}^{2} \cdot (2 (- x^{2} - 6 x + 18))$ .

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Step 5.1.7.1

Factor $16$ out of $32$ .

$y = \frac{16 \pm \sqrt{16 (2) (- x^{2} - 6 x + 18)}}{2 \cdot 4}$

Step 5.1.7.2

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

$y = \frac{16 \pm \sqrt{4^{2} \cdot (2 (- x^{2} - 6 x + 18))}}{2 \cdot 4}$

Step 5.1.7.3

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

$y = \frac{16 \pm \sqrt{{(2^{2})}^{2} \cdot (2 (- x^{2} - 6 x + 18))}}{2 \cdot 4}$

Step 5.1.7.4

Add parentheses.

$y = \frac{16 \pm \sqrt{{(2^{2})}^{2} \cdot (2 (- x^{2} - 6 x + 18))}}{2 \cdot 4}$

Step 5.1.8

Pull terms out from under the radical.

$y = \frac{16 \pm 2^{2} \sqrt{2 (- x^{2} - 6 x + 18)}}{2 \cdot 4}$

Step 5.1.9

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

$y = \frac{16 \pm 4 \sqrt{2 (- x^{2} - 6 x + 18)}}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$y = \frac{16 \pm 4 \sqrt{2 (- x^{2} - 6 x + 18)}}{8}$

Step 5.3

Simplify $\frac{16 \pm 4 \sqrt{2 (- x^{2} - 6 x + 18)}}{8}$ .

$y = \frac{4 \pm \sqrt{2 (- x^{2} - 6 x + 18)}}{2}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 16$ to the power of $2$ .

$y = \frac{16 \pm \sqrt{256 - 4 \cdot 4 \cdot (2 x^{2} + 12 x - 20)}}{2 \cdot 4}$

Step 6.1.2

Multiply $- 4$ by $4$ .

$y = \frac{16 \pm \sqrt{256 - 16 \cdot (2 x^{2} + 12 x - 20)}}{2 \cdot 4}$

Step 6.1.3

Apply the distributive property.

$y = \frac{16 \pm \sqrt{256 - 16 (2 x^{2}) - 16 (12 x) - 16 \cdot - 20}}{2 \cdot 4}$

Step 6.1.4

Simplify.

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Step 6.1.4.1

Multiply $2$ by $- 16$ .

$y = \frac{16 \pm \sqrt{256 - 32 x^{2} - 16 (12 x) - 16 \cdot - 20}}{2 \cdot 4}$

Step 6.1.4.2

Multiply $12$ by $- 16$ .

$y = \frac{16 \pm \sqrt{256 - 32 x^{2} - 192 x - 16 \cdot - 20}}{2 \cdot 4}$

Step 6.1.4.3

Multiply $- 16$ by $- 20$ .

$y = \frac{16 \pm \sqrt{256 - 32 x^{2} - 192 x + 320}}{2 \cdot 4}$

Step 6.1.5

Add $256$ and $320$ .

$y = \frac{16 \pm \sqrt{- 32 x^{2} - 192 x + 576}}{2 \cdot 4}$

Step 6.1.6

Factor $32$ out of $- 32 x^{2} - 192 x + 576$ .

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Step 6.1.6.1

Factor $32$ out of $- 32 x^{2}$ .

$y = \frac{16 \pm \sqrt{32 (- x^{2}) - 192 x + 576}}{2 \cdot 4}$

Step 6.1.6.2

Factor $32$ out of $- 192 x$ .

$y = \frac{16 \pm \sqrt{32 (- x^{2}) + 32 (- 6 x) + 576}}{2 \cdot 4}$

Step 6.1.6.3

Factor $32$ out of $576$ .

$y = \frac{16 \pm \sqrt{32 (- x^{2}) + 32 (- 6 x) + 32 (18)}}{2 \cdot 4}$

Step 6.1.6.4

Factor $32$ out of $32 (- x^{2}) + 32 (- 6 x)$ .

$y = \frac{16 \pm \sqrt{32 (- x^{2} - 6 x) + 32 (18)}}{2 \cdot 4}$

Step 6.1.6.5

Factor $32$ out of $32 (- x^{2} - 6 x) + 32 (18)$ .

$y = \frac{16 \pm \sqrt{32 (- x^{2} - 6 x + 18)}}{2 \cdot 4}$

Step 6.1.7

Rewrite $32 (- x^{2} - 6 x + 18)$ as ${(2^{2})}^{2} \cdot (2 (- x^{2} - 6 x + 18))$ .

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Step 6.1.7.1

Factor $16$ out of $32$ .

$y = \frac{16 \pm \sqrt{16 (2) (- x^{2} - 6 x + 18)}}{2 \cdot 4}$

Step 6.1.7.2

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

$y = \frac{16 \pm \sqrt{4^{2} \cdot (2 (- x^{2} - 6 x + 18))}}{2 \cdot 4}$

Step 6.1.7.3

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

$y = \frac{16 \pm \sqrt{{(2^{2})}^{2} \cdot (2 (- x^{2} - 6 x + 18))}}{2 \cdot 4}$

Step 6.1.7.4

Add parentheses.

$y = \frac{16 \pm \sqrt{{(2^{2})}^{2} \cdot (2 (- x^{2} - 6 x + 18))}}{2 \cdot 4}$

Step 6.1.8

Pull terms out from under the radical.

$y = \frac{16 \pm 2^{2} \sqrt{2 (- x^{2} - 6 x + 18)}}{2 \cdot 4}$

Step 6.1.9

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

$y = \frac{16 \pm 4 \sqrt{2 (- x^{2} - 6 x + 18)}}{2 \cdot 4}$

Step 6.2

Multiply $2$ by $4$ .

$y = \frac{16 \pm 4 \sqrt{2 (- x^{2} - 6 x + 18)}}{8}$

Step 6.3

Simplify $\frac{16 \pm 4 \sqrt{2 (- x^{2} - 6 x + 18)}}{8}$ .

$y = \frac{4 \pm \sqrt{2 (- x^{2} - 6 x + 18)}}{2}$

Step 6.4

Change the $\pm$ to $+$ .

$y = \frac{4 + \sqrt{2 (- x^{2} - 6 x + 18)}}{2}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 16$ to the power of $2$ .

$y = \frac{16 \pm \sqrt{256 - 4 \cdot 4 \cdot (2 x^{2} + 12 x - 20)}}{2 \cdot 4}$

Step 7.1.2

Multiply $- 4$ by $4$ .

$y = \frac{16 \pm \sqrt{256 - 16 \cdot (2 x^{2} + 12 x - 20)}}{2 \cdot 4}$

Step 7.1.3

Apply the distributive property.

$y = \frac{16 \pm \sqrt{256 - 16 (2 x^{2}) - 16 (12 x) - 16 \cdot - 20}}{2 \cdot 4}$

Step 7.1.4

Simplify.

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Step 7.1.4.1

Multiply $2$ by $- 16$ .

$y = \frac{16 \pm \sqrt{256 - 32 x^{2} - 16 (12 x) - 16 \cdot - 20}}{2 \cdot 4}$

Step 7.1.4.2

Multiply $12$ by $- 16$ .

$y = \frac{16 \pm \sqrt{256 - 32 x^{2} - 192 x - 16 \cdot - 20}}{2 \cdot 4}$

Step 7.1.4.3

Multiply $- 16$ by $- 20$ .

$y = \frac{16 \pm \sqrt{256 - 32 x^{2} - 192 x + 320}}{2 \cdot 4}$

Step 7.1.5

Add $256$ and $320$ .

$y = \frac{16 \pm \sqrt{- 32 x^{2} - 192 x + 576}}{2 \cdot 4}$

Step 7.1.6

Factor $32$ out of $- 32 x^{2} - 192 x + 576$ .

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Step 7.1.6.1

Factor $32$ out of $- 32 x^{2}$ .

$y = \frac{16 \pm \sqrt{32 (- x^{2}) - 192 x + 576}}{2 \cdot 4}$

Step 7.1.6.2

Factor $32$ out of $- 192 x$ .

$y = \frac{16 \pm \sqrt{32 (- x^{2}) + 32 (- 6 x) + 576}}{2 \cdot 4}$

Step 7.1.6.3

Factor $32$ out of $576$ .

$y = \frac{16 \pm \sqrt{32 (- x^{2}) + 32 (- 6 x) + 32 (18)}}{2 \cdot 4}$

Step 7.1.6.4

Factor $32$ out of $32 (- x^{2}) + 32 (- 6 x)$ .

$y = \frac{16 \pm \sqrt{32 (- x^{2} - 6 x) + 32 (18)}}{2 \cdot 4}$

Step 7.1.6.5

Factor $32$ out of $32 (- x^{2} - 6 x) + 32 (18)$ .

$y = \frac{16 \pm \sqrt{32 (- x^{2} - 6 x + 18)}}{2 \cdot 4}$

Step 7.1.7

Rewrite $32 (- x^{2} - 6 x + 18)$ as ${(2^{2})}^{2} \cdot (2 (- x^{2} - 6 x + 18))$ .

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Step 7.1.7.1

Factor $16$ out of $32$ .

$y = \frac{16 \pm \sqrt{16 (2) (- x^{2} - 6 x + 18)}}{2 \cdot 4}$

Step 7.1.7.2

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

$y = \frac{16 \pm \sqrt{4^{2} \cdot (2 (- x^{2} - 6 x + 18))}}{2 \cdot 4}$

Step 7.1.7.3

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

$y = \frac{16 \pm \sqrt{{(2^{2})}^{2} \cdot (2 (- x^{2} - 6 x + 18))}}{2 \cdot 4}$

Step 7.1.7.4

Add parentheses.

$y = \frac{16 \pm \sqrt{{(2^{2})}^{2} \cdot (2 (- x^{2} - 6 x + 18))}}{2 \cdot 4}$

Step 7.1.8

Pull terms out from under the radical.

$y = \frac{16 \pm 2^{2} \sqrt{2 (- x^{2} - 6 x + 18)}}{2 \cdot 4}$

Step 7.1.9

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

$y = \frac{16 \pm 4 \sqrt{2 (- x^{2} - 6 x + 18)}}{2 \cdot 4}$

Step 7.2

Multiply $2$ by $4$ .

$y = \frac{16 \pm 4 \sqrt{2 (- x^{2} - 6 x + 18)}}{8}$

Step 7.3

Simplify $\frac{16 \pm 4 \sqrt{2 (- x^{2} - 6 x + 18)}}{8}$ .

$y = \frac{4 \pm \sqrt{2 (- x^{2} - 6 x + 18)}}{2}$

Step 7.4

Change the $\pm$ to $-$ .

$y = \frac{4 - \sqrt{2 (- x^{2} - 6 x + 18)}}{2}$

Step 8

The final answer is the combination of both solutions.

$y = \frac{4 + \sqrt{2 (- x^{2} - 6 x + 18)}}{2}$

$y = \frac{4 - \sqrt{2 (- x^{2} - 6 x + 18)}}{2}$

Step 9

The given equation $y = \frac{4 + \sqrt{2 (- x^{2} - 6 x + 18)}}{2}, \frac{4 - \sqrt{2 (- x^{2} - 6 x + 18)}}{2}$ can not be written as $y = k x^{2}$ , so $y$ doesn't vary directly with $x^{2}$ .

$y$ doesn't vary directly with $x$