Precalculus Examples

$y = \sqrt{144 - x^{2}}$ , $y = 16 - x$

Step 1

Eliminate the equal sides of each equation and combine.

$\sqrt{144 - x^{2}} = 16 - x$

Step 2

Solve $\sqrt{144 - x^{2}} = 16 - x$ for $x$ .

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Step 2.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{144 - x^{2}}}^{2} = {(16 - x)}^{2}$

Step 2.2

Simplify each side of the equation.

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{144 - x^{2}}$ as ${(144 - x^{2})}^{\frac{1}{2}}$ .

${({(144 - x^{2})}^{\frac{1}{2}})}^{2} = {(16 - x)}^{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify ${({(144 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1

Multiply the exponents in ${({(144 - x^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1.1

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

${(144 - x^{2})}^{\frac{1}{2} \cdot 2} = {(16 - x)}^{2}$

Step 2.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.2.1.1.2.1

Cancel the common factor.

${(144 - x^{2})}^{\frac{1}{2} \cdot 2} = {(16 - x)}^{2}$

Step 2.2.2.1.1.2.2

Rewrite the expression.

${(144 - x^{2})}^{1} = {(16 - x)}^{2}$

Step 2.2.2.1.2

Simplify.

$144 - x^{2} = {(16 - x)}^{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Simplify ${(16 - x)}^{2}$ .

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Step 2.2.3.1.1

Rewrite ${(16 - x)}^{2}$ as $(16 - x) (16 - x)$ .

$144 - x^{2} = (16 - x) (16 - x)$

Step 2.2.3.1.2

Expand $(16 - x) (16 - x)$ using the FOIL Method.

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Step 2.2.3.1.2.1

Apply the distributive property.

$144 - x^{2} = 16 (16 - x) - x (16 - x)$

Step 2.2.3.1.2.2

Apply the distributive property.

$144 - x^{2} = 16 \cdot 16 + 16 (- x) - x (16 - x)$

Step 2.2.3.1.2.3

Apply the distributive property.

$144 - x^{2} = 16 \cdot 16 + 16 (- x) - x \cdot 16 - x (- x)$

Step 2.2.3.1.3

Simplify and combine like terms.

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Step 2.2.3.1.3.1

Simplify each term.

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Step 2.2.3.1.3.1.1

Multiply $16$ by $16$ .

$144 - x^{2} = 256 + 16 (- x) - x \cdot 16 - x (- x)$

Step 2.2.3.1.3.1.2

Multiply $- 1$ by $16$ .

$144 - x^{2} = 256 - 16 x - x \cdot 16 - x (- x)$

Step 2.2.3.1.3.1.3

Multiply $16$ by $- 1$ .

$144 - x^{2} = 256 - 16 x - 16 x - x (- x)$

Step 2.2.3.1.3.1.4

Rewrite using the commutative property of multiplication.

$144 - x^{2} = 256 - 16 x - 16 x - 1 \cdot - 1 x \cdot x$

Step 2.2.3.1.3.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 2.2.3.1.3.1.5.1

Move $x$ .

$144 - x^{2} = 256 - 16 x - 16 x - 1 \cdot - 1 (x \cdot x)$

Step 2.2.3.1.3.1.5.2

Multiply $x$ by $x$ .

$144 - x^{2} = 256 - 16 x - 16 x - 1 \cdot - 1 x^{2}$

Step 2.2.3.1.3.1.6

Multiply $- 1$ by $- 1$ .

$144 - x^{2} = 256 - 16 x - 16 x + 1 x^{2}$

Step 2.2.3.1.3.1.7

Multiply $x^{2}$ by $1$ .

$144 - x^{2} = 256 - 16 x - 16 x + x^{2}$

Step 2.2.3.1.3.2

Subtract $16 x$ from $- 16 x$ .

$144 - x^{2} = 256 - 32 x + x^{2}$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$256 - 32 x + x^{2} = 144 - x^{2}$

Step 2.3.2

Move all terms containing $x$ to the left side of the equation.

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Step 2.3.2.1

Add $x^{2}$ to both sides of the equation.

$256 - 32 x + x^{2} + x^{2} = 144$

Step 2.3.2.2

Add $x^{2}$ and $x^{2}$ .

$256 - 32 x + 2 x^{2} = 144$

Step 2.3.3

Subtract $144$ from both sides of the equation.

$256 - 32 x + 2 x^{2} - 144 = 0$

Step 2.3.4

Subtract $144$ from $256$ .

$- 32 x + 2 x^{2} + 112 = 0$

Step 2.3.5

Factor the left side of the equation.

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Step 2.3.5.1

Factor $2$ out of $- 32 x + 2 x^{2} + 112$ .

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Step 2.3.5.1.1

Factor $2$ out of $- 32 x$ .

$2 (- 16 x) + 2 x^{2} + 112 = 0$

Step 2.3.5.1.2

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

$2 (- 16 x) + 2 (x^{2}) + 112 = 0$

Step 2.3.5.1.3

Factor $2$ out of $112$ .

$2 (- 16 x) + 2 x^{2} + 2 \cdot 56 = 0$

Step 2.3.5.1.4

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

$2 (- 16 x + x^{2}) + 2 \cdot 56 = 0$

Step 2.3.5.1.5

Factor $2$ out of $2 (- 16 x + x^{2}) + 2 \cdot 56$ .

$2 (- 16 x + x^{2} + 56) = 0$

Step 2.3.5.2

Reorder terms.

$2 (x^{2} - 16 x + 56) = 0$

Step 2.3.6

Divide each term in $2 (x^{2} - 16 x + 56) = 0$ by $2$ and simplify.

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Step 2.3.6.1

Divide each term in $2 (x^{2} - 16 x + 56) = 0$ by $2$ .

$\frac{2 (x^{2} - 16 x + 56)}{2} = \frac{0}{2}$

Step 2.3.6.2

Simplify the left side.

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Step 2.3.6.2.1

Cancel the common factor of $2$ .

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Step 2.3.6.2.1.1

Cancel the common factor.

$\frac{2 (x^{2} - 16 x + 56)}{2} = \frac{0}{2}$

Step 2.3.6.2.1.2

Divide $x^{2} - 16 x + 56$ by $1$ .

$x^{2} - 16 x + 56 = \frac{0}{2}$

Step 2.3.6.3

Simplify the right side.

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Step 2.3.6.3.1

Divide $0$ by $2$ .

$x^{2} - 16 x + 56 = 0$

Step 2.3.7

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + y = 16 - x$

Step 2.3.8

Substitute the values $a = 1$ , $b = - 16$ , and $c = 56$ into the quadratic formula and solve for $x$ .

$\frac{16 \pm \sqrt{{(- 16)}^{2} - 4 \cdot (1 \cdot 56)}}{2 \cdot 1} + y = 16 - x$

Step 2.3.9

Simplify.

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Step 2.3.9.1

Simplify the numerator.

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Step 2.3.9.1.1

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

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 1 \cdot 56}}{2 \cdot 1}$

Step 2.3.9.1.2

Multiply $- 4 \cdot 1 \cdot 56$ .

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Step 2.3.9.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 56}}{2 \cdot 1}$

Step 2.3.9.1.2.2

Multiply $- 4$ by $56$ .

$x = \frac{16 \pm \sqrt{256 - 224}}{2 \cdot 1}$

Step 2.3.9.1.3

Subtract $224$ from $256$ .

$x = \frac{16 \pm \sqrt{32}}{2 \cdot 1}$

Step 2.3.9.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 2.3.9.1.4.1

Factor $16$ out of $32$ .

$x = \frac{16 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 2.3.9.1.4.2

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

$x = \frac{16 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Step 2.3.9.1.5

Pull terms out from under the radical.

$x = \frac{16 \pm 4 \sqrt{2}}{2 \cdot 1}$

Step 2.3.9.2

Multiply $2$ by $1$ .

$x = \frac{16 \pm 4 \sqrt{2}}{2}$

Step 2.3.9.3

Simplify $\frac{16 \pm 4 \sqrt{2}}{2}$ .

$x = 8 \pm 2 \sqrt{2}$

Step 2.3.10

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

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Step 2.3.10.1

Simplify the numerator.

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Step 2.3.10.1.1

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

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 1 \cdot 56}}{2 \cdot 1}$

Step 2.3.10.1.2

Multiply $- 4 \cdot 1 \cdot 56$ .

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Step 2.3.10.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 56}}{2 \cdot 1}$

Step 2.3.10.1.2.2

Multiply $- 4$ by $56$ .

$x = \frac{16 \pm \sqrt{256 - 224}}{2 \cdot 1}$

Step 2.3.10.1.3

Subtract $224$ from $256$ .

$x = \frac{16 \pm \sqrt{32}}{2 \cdot 1}$

Step 2.3.10.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 2.3.10.1.4.1

Factor $16$ out of $32$ .

$x = \frac{16 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 2.3.10.1.4.2

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

$x = \frac{16 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Step 2.3.10.1.5

Pull terms out from under the radical.

$x = \frac{16 \pm 4 \sqrt{2}}{2 \cdot 1}$

Step 2.3.10.2

Multiply $2$ by $1$ .

$x = \frac{16 \pm 4 \sqrt{2}}{2}$

Step 2.3.10.3

Simplify $\frac{16 \pm 4 \sqrt{2}}{2}$ .

$x = 8 \pm 2 \sqrt{2}$

Step 2.3.10.4

Change the $\pm$ to $+$ .

$x = 8 + 2 \sqrt{2}$

Step 2.3.11

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

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Step 2.3.11.1

Simplify the numerator.

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Step 2.3.11.1.1

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

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 1 \cdot 56}}{2 \cdot 1}$

Step 2.3.11.1.2

Multiply $- 4 \cdot 1 \cdot 56$ .

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Step 2.3.11.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 56}}{2 \cdot 1}$

Step 2.3.11.1.2.2

Multiply $- 4$ by $56$ .

$x = \frac{16 \pm \sqrt{256 - 224}}{2 \cdot 1}$

Step 2.3.11.1.3

Subtract $224$ from $256$ .

$x = \frac{16 \pm \sqrt{32}}{2 \cdot 1}$

Step 2.3.11.1.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 2.3.11.1.4.1

Factor $16$ out of $32$ .

$x = \frac{16 \pm \sqrt{16 (2)}}{2 \cdot 1}$

Step 2.3.11.1.4.2

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

$x = \frac{16 \pm \sqrt{4^{2} \cdot 2}}{2 \cdot 1}$

Step 2.3.11.1.5

Pull terms out from under the radical.

$x = \frac{16 \pm 4 \sqrt{2}}{2 \cdot 1}$

Step 2.3.11.2

Multiply $2$ by $1$ .

$x = \frac{16 \pm 4 \sqrt{2}}{2}$

Step 2.3.11.3

Simplify $\frac{16 \pm 4 \sqrt{2}}{2}$ .

$x = 8 \pm 2 \sqrt{2}$

Step 2.3.11.4

Change the $\pm$ to $-$ .

$x = 8 - 2 \sqrt{2}$

Step 2.3.12

The final answer is the combination of both solutions.

$x = 8 + 2 \sqrt{2}, 8 - 2 \sqrt{2}$

Step 3

Evaluate $y$ when $x = 8 + 2 \sqrt{2}$ .

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Step 3.1

Substitute $8 + 2 \sqrt{2}$ for $x$ .

$y = 16 - (8 + 2 \sqrt{2})$

Step 3.2

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

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Apply the distributive property.

$y = 16 - 1 \cdot 8 - (2 \sqrt{2})$

Step 3.2.1.2

Multiply $- 1$ by $8$ .

$y = 16 - 8 - (2 \sqrt{2})$

Step 3.2.1.3

Multiply $2$ by $- 1$ .

$y = 16 - 8 - 2 \sqrt{2}$

Step 3.2.2

Subtract $8$ from $16$ .

$y = 8 - 2 \sqrt{2}$

Step 4

Evaluate $y$ when $x = 8 - 2 \sqrt{2}$ .

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Step 4.1

Substitute $8 - 2 \sqrt{2}$ for $x$ .

$y = 16 - (8 - 2 \sqrt{2})$

Step 4.2

Simplify $16 - (8 - 2 \sqrt{2})$ .

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Apply the distributive property.

$y = 16 - 1 \cdot 8 - (- 2 \sqrt{2})$

Step 4.2.1.2

Multiply $- 1$ by $8$ .

$y = 16 - 8 - (- 2 \sqrt{2})$

Step 4.2.1.3

Multiply $- 2$ by $- 1$ .

$y = 16 - 8 + 2 \sqrt{2}$

Step 4.2.2

Subtract $8$ from $16$ .

$y = 8 + 2 \sqrt{2}$

Step 5

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

$(8 + 2 \sqrt{2}, 8 - 2 \sqrt{2})$

$(8 - 2 \sqrt{2}, 8 + 2 \sqrt{2})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(8 + 2 \sqrt{2}, 8 - 2 \sqrt{2}), (8 - 2 \sqrt{2}, 8 + 2 \sqrt{2})$

Equation Form:

$x = 8 + 2 \sqrt{2}, y = 8 - 2 \sqrt{2}$

$x = 8 - 2 \sqrt{2}, y = 8 + 2 \sqrt{2}$

Step 7