Trigonometry Examples

$x^{2} + y^{2} - 20 x - 16 y + 100 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 1$ , $b = - 16$ , and $c = x^{2} - 20 x + 100$ into the quadratic formula and solve for $y$ .

$\frac{16 \pm \sqrt{{(- 16)}^{2} - 4 \cdot (1 \cdot (x^{2} - 20 x + 100))}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Rewrite $4 \cdot 1 \cdot (x^{2} - 20 x + 100)$ as ${(2 \cdot 1 \cdot (x - 10))}^{2}$ .

$y = \frac{16 \pm \sqrt{{(- 16)}^{2} - {(2 \cdot 1 \cdot (x - 10))}^{2}}}{2 \cdot 1}$

Step 3.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = - 16$ and $b = 2 \cdot 1 \cdot (x - 10)$ .

$y = \frac{16 \pm \sqrt{(- 16 + 2 \cdot 1 \cdot (x - 10)) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 3.1.3

Simplify.

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Step 3.1.3.1

Multiply $2$ by $1$ .

$y = \frac{16 \pm \sqrt{(- 16 + 2 \cdot (x - 10)) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 3.1.3.2

Apply the distributive property.

$y = \frac{16 \pm \sqrt{(- 16 + 2 x + 2 \cdot - 10) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 3.1.3.3

Multiply $2$ by $- 10$ .

$y = \frac{16 \pm \sqrt{(- 16 + 2 x - 20) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 3.1.3.4

Subtract $20$ from $- 16$ .

$y = \frac{16 \pm \sqrt{(2 x - 36) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 3.1.3.5

Factor $2$ out of $2 x - 36$ .

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Step 3.1.3.5.1

Factor $2$ out of $2 x$ .

$y = \frac{16 \pm \sqrt{(2 (x) - 36) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 3.1.3.5.2

Factor $2$ out of $- 36$ .

$y = \frac{16 \pm \sqrt{(2 x + 2 \cdot - 18) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 3.1.3.5.3

Factor $2$ out of $2 x + 2 \cdot - 18$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 3.1.3.6

Combine exponents.

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Step 3.1.3.6.1

Multiply $2$ by $1$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (- 16 - (2 \cdot (x - 10)))}}{2 \cdot 1}$

Step 3.1.3.6.2

Multiply $2$ by $- 1$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (- 16 - 2 (x - 10))}}{2 \cdot 1}$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

Apply the distributive property.

$y = \frac{16 \pm \sqrt{2 (x - 18) (- 16 - 2 x - 2 \cdot - 10)}}{2 \cdot 1}$

Step 3.1.4.2

Multiply $- 2$ by $- 10$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (- 16 - 2 x + 20)}}{2 \cdot 1}$

Step 3.1.5

Add $- 16$ and $20$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (- 2 x + 4)}}{2 \cdot 1}$

Step 3.1.6

Factor $2$ out of $- 2 x + 4$ .

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Step 3.1.6.1

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

$y = \frac{16 \pm \sqrt{2 (x - 18) (2 (- x) + 4)}}{2 \cdot 1}$

Step 3.1.6.2

Factor $2$ out of $4$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (2 (- x) + 2 (2))}}{2 \cdot 1}$

Step 3.1.6.3

Factor $2$ out of $2 (- x) + 2 (2)$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (2 (- x + 2))}}{2 \cdot 1}$

$y = \frac{16 \pm \sqrt{2 (x - 18) \cdot (2 (- x + 2))}}{2 \cdot 1}$

Step 3.1.7

Multiply $2$ by $2$ .

$y = \frac{16 \pm \sqrt{4 (x - 18) (- x + 2)}}{2 \cdot 1}$

Step 3.1.8

Rewrite $4 (x - 18) (- x + 2)$ as $2^{2} ((x - 18) (- x + 2))$ .

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Step 3.1.8.1

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

$y = \frac{16 \pm \sqrt{2^{2} (x - 18) (- x + 2)}}{2 \cdot 1}$

Step 3.1.8.2

Add parentheses.

$y = \frac{16 \pm \sqrt{2^{2} ((x - 18) (- x + 2))}}{2 \cdot 1}$

Step 3.1.9

Pull terms out from under the radical.

$y = \frac{16 \pm 2 \sqrt{(x - 18) (- x + 2)}}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

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

Step 3.3

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

$y = 8 \pm \sqrt{(x - 18) (- x + 2)}$

Step 4

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

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Rewrite $4 \cdot 1 \cdot (x^{2} - 20 x + 100)$ as ${(2 \cdot 1 \cdot (x - 10))}^{2}$ .

$y = \frac{16 \pm \sqrt{{(- 16)}^{2} - {(2 \cdot 1 \cdot (x - 10))}^{2}}}{2 \cdot 1}$

Step 4.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = - 16$ and $b = 2 \cdot 1 \cdot (x - 10)$ .

$y = \frac{16 \pm \sqrt{(- 16 + 2 \cdot 1 \cdot (x - 10)) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 4.1.3

Simplify.

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Step 4.1.3.1

Multiply $2$ by $1$ .

$y = \frac{16 \pm \sqrt{(- 16 + 2 \cdot (x - 10)) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 4.1.3.2

Apply the distributive property.

$y = \frac{16 \pm \sqrt{(- 16 + 2 x + 2 \cdot - 10) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 4.1.3.3

Multiply $2$ by $- 10$ .

$y = \frac{16 \pm \sqrt{(- 16 + 2 x - 20) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 4.1.3.4

Subtract $20$ from $- 16$ .

$y = \frac{16 \pm \sqrt{(2 x - 36) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 4.1.3.5

Factor $2$ out of $2 x - 36$ .

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Step 4.1.3.5.1

Factor $2$ out of $2 x$ .

$y = \frac{16 \pm \sqrt{(2 (x) - 36) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 4.1.3.5.2

Factor $2$ out of $- 36$ .

$y = \frac{16 \pm \sqrt{(2 x + 2 \cdot - 18) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 4.1.3.5.3

Factor $2$ out of $2 x + 2 \cdot - 18$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 4.1.3.6

Combine exponents.

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Step 4.1.3.6.1

Multiply $2$ by $1$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (- 16 - (2 \cdot (x - 10)))}}{2 \cdot 1}$

Step 4.1.3.6.2

Multiply $2$ by $- 1$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (- 16 - 2 (x - 10))}}{2 \cdot 1}$

Step 4.1.4

Simplify each term.

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Step 4.1.4.1

Apply the distributive property.

$y = \frac{16 \pm \sqrt{2 (x - 18) (- 16 - 2 x - 2 \cdot - 10)}}{2 \cdot 1}$

Step 4.1.4.2

Multiply $- 2$ by $- 10$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (- 16 - 2 x + 20)}}{2 \cdot 1}$

Step 4.1.5

Add $- 16$ and $20$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (- 2 x + 4)}}{2 \cdot 1}$

Step 4.1.6

Factor $2$ out of $- 2 x + 4$ .

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Step 4.1.6.1

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

$y = \frac{16 \pm \sqrt{2 (x - 18) (2 (- x) + 4)}}{2 \cdot 1}$

Step 4.1.6.2

Factor $2$ out of $4$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (2 (- x) + 2 (2))}}{2 \cdot 1}$

Step 4.1.6.3

Factor $2$ out of $2 (- x) + 2 (2)$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (2 (- x + 2))}}{2 \cdot 1}$

$y = \frac{16 \pm \sqrt{2 (x - 18) \cdot (2 (- x + 2))}}{2 \cdot 1}$

Step 4.1.7

Multiply $2$ by $2$ .

$y = \frac{16 \pm \sqrt{4 (x - 18) (- x + 2)}}{2 \cdot 1}$

Step 4.1.8

Rewrite $4 (x - 18) (- x + 2)$ as $2^{2} ((x - 18) (- x + 2))$ .

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Step 4.1.8.1

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

$y = \frac{16 \pm \sqrt{2^{2} (x - 18) (- x + 2)}}{2 \cdot 1}$

Step 4.1.8.2

Add parentheses.

$y = \frac{16 \pm \sqrt{2^{2} ((x - 18) (- x + 2))}}{2 \cdot 1}$

Step 4.1.9

Pull terms out from under the radical.

$y = \frac{16 \pm 2 \sqrt{(x - 18) (- x + 2)}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

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

Step 4.3

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

$y = 8 \pm \sqrt{(x - 18) (- x + 2)}$

Step 4.4

Change the $\pm$ to $+$ .

$y = 8 + \sqrt{(x - 18) (- x + 2)}$

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Rewrite $4 \cdot 1 \cdot (x^{2} - 20 x + 100)$ as ${(2 \cdot 1 \cdot (x - 10))}^{2}$ .

$y = \frac{16 \pm \sqrt{{(- 16)}^{2} - {(2 \cdot 1 \cdot (x - 10))}^{2}}}{2 \cdot 1}$

Step 5.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = - 16$ and $b = 2 \cdot 1 \cdot (x - 10)$ .

$y = \frac{16 \pm \sqrt{(- 16 + 2 \cdot 1 \cdot (x - 10)) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 5.1.3

Simplify.

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Step 5.1.3.1

Multiply $2$ by $1$ .

$y = \frac{16 \pm \sqrt{(- 16 + 2 \cdot (x - 10)) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 5.1.3.2

Apply the distributive property.

$y = \frac{16 \pm \sqrt{(- 16 + 2 x + 2 \cdot - 10) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 5.1.3.3

Multiply $2$ by $- 10$ .

$y = \frac{16 \pm \sqrt{(- 16 + 2 x - 20) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 5.1.3.4

Subtract $20$ from $- 16$ .

$y = \frac{16 \pm \sqrt{(2 x - 36) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 5.1.3.5

Factor $2$ out of $2 x - 36$ .

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Step 5.1.3.5.1

Factor $2$ out of $2 x$ .

$y = \frac{16 \pm \sqrt{(2 (x) - 36) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 5.1.3.5.2

Factor $2$ out of $- 36$ .

$y = \frac{16 \pm \sqrt{(2 x + 2 \cdot - 18) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 5.1.3.5.3

Factor $2$ out of $2 x + 2 \cdot - 18$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (- 16 - (2 \cdot 1 \cdot (x - 10)))}}{2 \cdot 1}$

Step 5.1.3.6

Combine exponents.

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Step 5.1.3.6.1

Multiply $2$ by $1$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (- 16 - (2 \cdot (x - 10)))}}{2 \cdot 1}$

Step 5.1.3.6.2

Multiply $2$ by $- 1$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (- 16 - 2 (x - 10))}}{2 \cdot 1}$

Step 5.1.4

Simplify each term.

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Step 5.1.4.1

Apply the distributive property.

$y = \frac{16 \pm \sqrt{2 (x - 18) (- 16 - 2 x - 2 \cdot - 10)}}{2 \cdot 1}$

Step 5.1.4.2

Multiply $- 2$ by $- 10$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (- 16 - 2 x + 20)}}{2 \cdot 1}$

Step 5.1.5

Add $- 16$ and $20$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (- 2 x + 4)}}{2 \cdot 1}$

Step 5.1.6

Factor $2$ out of $- 2 x + 4$ .

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Step 5.1.6.1

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

$y = \frac{16 \pm \sqrt{2 (x - 18) (2 (- x) + 4)}}{2 \cdot 1}$

Step 5.1.6.2

Factor $2$ out of $4$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (2 (- x) + 2 (2))}}{2 \cdot 1}$

Step 5.1.6.3

Factor $2$ out of $2 (- x) + 2 (2)$ .

$y = \frac{16 \pm \sqrt{2 (x - 18) (2 (- x + 2))}}{2 \cdot 1}$

$y = \frac{16 \pm \sqrt{2 (x - 18) \cdot (2 (- x + 2))}}{2 \cdot 1}$

Step 5.1.7

Multiply $2$ by $2$ .

$y = \frac{16 \pm \sqrt{4 (x - 18) (- x + 2)}}{2 \cdot 1}$

Step 5.1.8

Rewrite $4 (x - 18) (- x + 2)$ as $2^{2} ((x - 18) (- x + 2))$ .

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Step 5.1.8.1

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

$y = \frac{16 \pm \sqrt{2^{2} (x - 18) (- x + 2)}}{2 \cdot 1}$

Step 5.1.8.2

Add parentheses.

$y = \frac{16 \pm \sqrt{2^{2} ((x - 18) (- x + 2))}}{2 \cdot 1}$

Step 5.1.9

Pull terms out from under the radical.

$y = \frac{16 \pm 2 \sqrt{(x - 18) (- x + 2)}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

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

Step 5.3

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

$y = 8 \pm \sqrt{(x - 18) (- x + 2)}$

Step 5.4

Change the $\pm$ to $-$ .

$y = 8 - \sqrt{(x - 18) (- x + 2)}$

Step 6

The final answer is the combination of both solutions.

$y = 8 + \sqrt{(x - 18) (- x + 2)}$

$y = 8 - \sqrt{(x - 18) (- x + 2)}$

Step 7

Interchange the variables. Create an equation for each expression.

$x = 8 + \sqrt{(y - 18) (- y + 2)}, x = 8 - \sqrt{(y - 18) (- y + 2)}$

Step 8

Solve for $y$ .

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Step 8.1

Rewrite the equation as $8 + \sqrt{(y - 18) (- y + 2)} = x$ .

$8 + \sqrt{(y - 18) (- y + 2)} = x$

Step 8.2

Subtract $8$ from both sides of the equation.

$\sqrt{(y - 18) (- y + 2)} = x - 8$

Step 8.3

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

${\sqrt{(y - 18) (- y + 2)}}^{2} = {(x - 8)}^{2}$

Step 8.4

Simplify each side of the equation.

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Step 8.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(y - 18) (- y + 2)}$ as ${((y - 18) (- y + 2))}^{\frac{1}{2}}$ .

${({((y - 18) (- y + 2))}^{\frac{1}{2}})}^{2} = {(x - 8)}^{2}$

Step 8.4.2

Simplify the left side.

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Step 8.4.2.1

Simplify ${({((y - 18) (- y + 2))}^{\frac{1}{2}})}^{2}$ .

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Step 8.4.2.1.1

Multiply the exponents in ${({((y - 18) (- y + 2))}^{\frac{1}{2}})}^{2}$ .

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Step 8.4.2.1.1.1

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

${((y - 18) (- y + 2))}^{\frac{1}{2} \cdot 2} = {(x - 8)}^{2}$

Step 8.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 8.4.2.1.1.2.1

Cancel the common factor.

${((y - 18) (- y + 2))}^{\frac{1}{2} \cdot 2} = {(x - 8)}^{2}$

Step 8.4.2.1.1.2.2

Rewrite the expression.

${((y - 18) (- y + 2))}^{1} = {(x - 8)}^{2}$

Step 8.4.2.1.2

Expand $(y - 18) (- y + 2)$ using the FOIL Method.

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Step 8.4.2.1.2.1

Apply the distributive property.

${(y (- y + 2) - 18 (- y + 2))}^{1} = {(x - 8)}^{2}$

Step 8.4.2.1.2.2

Apply the distributive property.

${(y (- y) + y \cdot 2 - 18 (- y + 2))}^{1} = {(x - 8)}^{2}$

Step 8.4.2.1.2.3

Apply the distributive property.

${(y (- y) + y \cdot 2 - 18 (- y) - 18 \cdot 2)}^{1} = {(x - 8)}^{2}$

Step 8.4.2.1.3

Simplify and combine like terms.

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Step 8.4.2.1.3.1

Simplify each term.

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Step 8.4.2.1.3.1.1

Rewrite using the commutative property of multiplication.

${(- y \cdot y + y \cdot 2 - 18 (- y) - 18 \cdot 2)}^{1} = {(x - 8)}^{2}$

Step 8.4.2.1.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 8.4.2.1.3.1.2.1

Move $y$ .

${(- (y \cdot y) + y \cdot 2 - 18 (- y) - 18 \cdot 2)}^{1} = {(x - 8)}^{2}$

Step 8.4.2.1.3.1.2.2

Multiply $y$ by $y$ .

${(- y^{2} + y \cdot 2 - 18 (- y) - 18 \cdot 2)}^{1} = {(x - 8)}^{2}$

Step 8.4.2.1.3.1.3

Move $2$ to the left of $y$ .

${(- y^{2} + 2 \cdot y - 18 (- y) - 18 \cdot 2)}^{1} = {(x - 8)}^{2}$

Step 8.4.2.1.3.1.4

Multiply $- 1$ by $- 18$ .

${(- y^{2} + 2 y + 18 y - 18 \cdot 2)}^{1} = {(x - 8)}^{2}$

Step 8.4.2.1.3.1.5

Multiply $- 18$ by $2$ .

${(- y^{2} + 2 y + 18 y - 36)}^{1} = {(x - 8)}^{2}$

Step 8.4.2.1.3.2

Add $2 y$ and $18 y$ .

${(- y^{2} + 20 y - 36)}^{1} = {(x - 8)}^{2}$

Step 8.4.2.1.4

Simplify.

$- y^{2} + 20 y - 36 = {(x - 8)}^{2}$

Step 8.4.3

Simplify the right side.

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Step 8.4.3.1

Simplify ${(x - 8)}^{2}$ .

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Step 8.4.3.1.1

Rewrite ${(x - 8)}^{2}$ as $(x - 8) (x - 8)$ .

$- y^{2} + 20 y - 36 = (x - 8) (x - 8)$

Step 8.4.3.1.2

Expand $(x - 8) (x - 8)$ using the FOIL Method.

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Step 8.4.3.1.2.1

Apply the distributive property.

$- y^{2} + 20 y - 36 = x (x - 8) - 8 (x - 8)$

Step 8.4.3.1.2.2

Apply the distributive property.

$- y^{2} + 20 y - 36 = x \cdot x + x \cdot - 8 - 8 (x - 8)$

Step 8.4.3.1.2.3

Apply the distributive property.

$- y^{2} + 20 y - 36 = x \cdot x + x \cdot - 8 - 8 x - 8 \cdot - 8$

Step 8.4.3.1.3

Simplify and combine like terms.

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Step 8.4.3.1.3.1

Simplify each term.

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Step 8.4.3.1.3.1.1

Multiply $x$ by $x$ .

$- y^{2} + 20 y - 36 = x^{2} + x \cdot - 8 - 8 x - 8 \cdot - 8$

Step 8.4.3.1.3.1.2

Move $- 8$ to the left of $x$ .

$- y^{2} + 20 y - 36 = x^{2} - 8 \cdot x - 8 x - 8 \cdot - 8$

Step 8.4.3.1.3.1.3

Multiply $- 8$ by $- 8$ .

$- y^{2} + 20 y - 36 = x^{2} - 8 x - 8 x + 64$

Step 8.4.3.1.3.2

Subtract $8 x$ from $- 8 x$ .

$- y^{2} + 20 y - 36 = x^{2} - 16 x + 64$

Step 8.5

Solve for $y$ .

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Step 8.5.1

Move all terms to the left side of the equation and simplify.

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Step 8.5.1.1

Move all the expressions to the left side of the equation.

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Step 8.5.1.1.1

Subtract $x^{2}$ from both sides of the equation.

$- y^{2} + 20 y - 36 - x^{2} = - 16 x + 64$

Step 8.5.1.1.2

Add $16 x$ to both sides of the equation.

$- y^{2} + 20 y - 36 - x^{2} + 16 x = 64$

Step 8.5.1.1.3

Subtract $64$ from both sides of the equation.

$- y^{2} + 20 y - 36 - x^{2} + 16 x - 64 = 0$

Step 8.5.1.2

Subtract $64$ from $- 36$ .

$- y^{2} + 20 y - x^{2} + 16 x - 100 = 0$

Step 8.5.2

Use the quadratic formula to find the solutions.

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

Step 8.5.3

Substitute the values $a = - 1$ , $b = 20$ , and $c = - x^{2} + 16 x - 100$ into the quadratic formula and solve for $y$ .

$\frac{- 20 \pm \sqrt{20^{2} - 4 \cdot (- 1 \cdot (- x^{2} + 16 x - 100))}}{2 \cdot - 1}$

Step 8.5.4

Simplify.

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Step 8.5.4.1

Simplify the numerator.

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Step 8.5.4.1.1

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

$y = \frac{- 20 \pm \sqrt{400 - 4 \cdot - 1 \cdot (- x^{2} + 16 x - 100)}}{2 \cdot - 1}$

Step 8.5.4.1.2

Multiply $- 4$ by $- 1$ .

$y = \frac{- 20 \pm \sqrt{400 + 4 \cdot (- x^{2} + 16 x - 100)}}{2 \cdot - 1}$

Step 8.5.4.1.3

Apply the distributive property.

$y = \frac{- 20 \pm \sqrt{400 + 4 (- x^{2}) + 4 (16 x) + 4 \cdot - 100}}{2 \cdot - 1}$

Step 8.5.4.1.4

Simplify.

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Step 8.5.4.1.4.1

Multiply $- 1$ by $4$ .

$y = \frac{- 20 \pm \sqrt{400 - 4 x^{2} + 4 (16 x) + 4 \cdot - 100}}{2 \cdot - 1}$

Step 8.5.4.1.4.2

Multiply $16$ by $4$ .

$y = \frac{- 20 \pm \sqrt{400 - 4 x^{2} + 64 x + 4 \cdot - 100}}{2 \cdot - 1}$

Step 8.5.4.1.4.3

Multiply $4$ by $- 100$ .

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

Step 8.5.4.1.5

Subtract $400$ from $400$ .

$y = \frac{- 20 \pm \sqrt{- 4 x^{2} + 64 x + 0}}{2 \cdot - 1}$

Step 8.5.4.1.6

Add $- 4 x^{2} + 64 x$ and $0$ .

$y = \frac{- 20 \pm \sqrt{- 4 x^{2} + 64 x}}{2 \cdot - 1}$

Step 8.5.4.1.7

Factor $4 x$ out of $- 4 x^{2} + 64 x$ .

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Step 8.5.4.1.7.1

Factor $4 x$ out of $- 4 x^{2}$ .

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

Step 8.5.4.1.7.2

Factor $4 x$ out of $64 x$ .

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

Step 8.5.4.1.7.3

Factor $4 x$ out of $4 x (- x) + 4 x (16)$ .

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

Step 8.5.4.1.8

Rewrite $4 x (- x + 16)$ as $2^{2} (x (- x + 4^{2}))$ .

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Step 8.5.4.1.8.1

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

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

Step 8.5.4.1.8.2

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

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

Step 8.5.4.1.8.3

Add parentheses.

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

Step 8.5.4.1.9

Pull terms out from under the radical.

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

Step 8.5.4.1.10

Raise $4$ to the power of $2$ .

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

Step 8.5.4.2

Multiply $2$ by $- 1$ .

$y = \frac{- 20 \pm 2 \sqrt{x (- x + 16)}}{- 2}$

Step 8.5.4.3

Simplify $\frac{- 20 \pm 2 \sqrt{x (- x + 16)}}{- 2}$ .

$y = 10 \pm \sqrt{x (- x + 16)}$

Step 8.5.5

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

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Step 8.5.5.1

Simplify the numerator.

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Step 8.5.5.1.1

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

$y = \frac{- 20 \pm \sqrt{400 - 4 \cdot - 1 \cdot (- x^{2} + 16 x - 100)}}{2 \cdot - 1}$

Step 8.5.5.1.2

Multiply $- 4$ by $- 1$ .

$y = \frac{- 20 \pm \sqrt{400 + 4 \cdot (- x^{2} + 16 x - 100)}}{2 \cdot - 1}$

Step 8.5.5.1.3

Apply the distributive property.

$y = \frac{- 20 \pm \sqrt{400 + 4 (- x^{2}) + 4 (16 x) + 4 \cdot - 100}}{2 \cdot - 1}$

Step 8.5.5.1.4

Simplify.

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Step 8.5.5.1.4.1

Multiply $- 1$ by $4$ .

$y = \frac{- 20 \pm \sqrt{400 - 4 x^{2} + 4 (16 x) + 4 \cdot - 100}}{2 \cdot - 1}$

Step 8.5.5.1.4.2

Multiply $16$ by $4$ .

$y = \frac{- 20 \pm \sqrt{400 - 4 x^{2} + 64 x + 4 \cdot - 100}}{2 \cdot - 1}$

Step 8.5.5.1.4.3

Multiply $4$ by $- 100$ .

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

Step 8.5.5.1.5

Subtract $400$ from $400$ .

$y = \frac{- 20 \pm \sqrt{- 4 x^{2} + 64 x + 0}}{2 \cdot - 1}$

Step 8.5.5.1.6

Add $- 4 x^{2} + 64 x$ and $0$ .

$y = \frac{- 20 \pm \sqrt{- 4 x^{2} + 64 x}}{2 \cdot - 1}$

Step 8.5.5.1.7

Factor $4 x$ out of $- 4 x^{2} + 64 x$ .

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Step 8.5.5.1.7.1

Factor $4 x$ out of $- 4 x^{2}$ .

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

Step 8.5.5.1.7.2

Factor $4 x$ out of $64 x$ .

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

Step 8.5.5.1.7.3

Factor $4 x$ out of $4 x (- x) + 4 x (16)$ .

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

Step 8.5.5.1.8

Rewrite $4 x (- x + 16)$ as $2^{2} (x (- x + 4^{2}))$ .

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Step 8.5.5.1.8.1

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

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

Step 8.5.5.1.8.2

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

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

Step 8.5.5.1.8.3

Add parentheses.

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

Step 8.5.5.1.9

Pull terms out from under the radical.

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

Step 8.5.5.1.10

Raise $4$ to the power of $2$ .

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

Step 8.5.5.2

Multiply $2$ by $- 1$ .

$y = \frac{- 20 \pm 2 \sqrt{x (- x + 16)}}{- 2}$

Step 8.5.5.3

Simplify $\frac{- 20 \pm 2 \sqrt{x (- x + 16)}}{- 2}$ .

$y = 10 \pm \sqrt{x (- x + 16)}$

Step 8.5.5.4

Change the $\pm$ to $+$ .

$y = 10 + \sqrt{x (- x + 16)}$

Step 8.5.6

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

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Step 8.5.6.1

Simplify the numerator.

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Step 8.5.6.1.1

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

$y = \frac{- 20 \pm \sqrt{400 - 4 \cdot - 1 \cdot (- x^{2} + 16 x - 100)}}{2 \cdot - 1}$

Step 8.5.6.1.2

Multiply $- 4$ by $- 1$ .

$y = \frac{- 20 \pm \sqrt{400 + 4 \cdot (- x^{2} + 16 x - 100)}}{2 \cdot - 1}$

Step 8.5.6.1.3

Apply the distributive property.

$y = \frac{- 20 \pm \sqrt{400 + 4 (- x^{2}) + 4 (16 x) + 4 \cdot - 100}}{2 \cdot - 1}$

Step 8.5.6.1.4

Simplify.

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Step 8.5.6.1.4.1

Multiply $- 1$ by $4$ .

$y = \frac{- 20 \pm \sqrt{400 - 4 x^{2} + 4 (16 x) + 4 \cdot - 100}}{2 \cdot - 1}$

Step 8.5.6.1.4.2

Multiply $16$ by $4$ .

$y = \frac{- 20 \pm \sqrt{400 - 4 x^{2} + 64 x + 4 \cdot - 100}}{2 \cdot - 1}$

Step 8.5.6.1.4.3

Multiply $4$ by $- 100$ .

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

Step 8.5.6.1.5

Subtract $400$ from $400$ .

$y = \frac{- 20 \pm \sqrt{- 4 x^{2} + 64 x + 0}}{2 \cdot - 1}$

Step 8.5.6.1.6

Add $- 4 x^{2} + 64 x$ and $0$ .

$y = \frac{- 20 \pm \sqrt{- 4 x^{2} + 64 x}}{2 \cdot - 1}$

Step 8.5.6.1.7

Factor $4 x$ out of $- 4 x^{2} + 64 x$ .

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Step 8.5.6.1.7.1

Factor $4 x$ out of $- 4 x^{2}$ .

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

Step 8.5.6.1.7.2

Factor $4 x$ out of $64 x$ .

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

Step 8.5.6.1.7.3

Factor $4 x$ out of $4 x (- x) + 4 x (16)$ .

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

Step 8.5.6.1.8

Rewrite $4 x (- x + 16)$ as $2^{2} (x (- x + 4^{2}))$ .

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Step 8.5.6.1.8.1

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

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

Step 8.5.6.1.8.2

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

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

Step 8.5.6.1.8.3

Add parentheses.

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

Step 8.5.6.1.9

Pull terms out from under the radical.

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

Step 8.5.6.1.10

Raise $4$ to the power of $2$ .

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

Step 8.5.6.2

Multiply $2$ by $- 1$ .

$y = \frac{- 20 \pm 2 \sqrt{x (- x + 16)}}{- 2}$

Step 8.5.6.3

Simplify $\frac{- 20 \pm 2 \sqrt{x (- x + 16)}}{- 2}$ .

$y = 10 \pm \sqrt{x (- x + 16)}$

Step 8.5.6.4

Change the $\pm$ to $-$ .

$y = 10 - \sqrt{x (- x + 16)}$

Step 8.5.7

The final answer is the combination of both solutions.

$y = 10 + \sqrt{x (- x + 16)}$

$y = 10 - \sqrt{x (- x + 16)}$

$y = 10 + \sqrt{x (- x + 16)}$

$y = 10 - \sqrt{x (- x + 16)}$

$y = 10 + \sqrt{x (- x + 16)}$

$y = 10 - \sqrt{x (- x + 16)}$

Step 9

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = 10 + \sqrt{x (- x + 16)}, 10 - \sqrt{x (- x + 16)}$

Step 10

Verify if $f^{- 1} (x) = 10 + \sqrt{x (- x + 16)}, 10 - \sqrt{x (- x + 16)}$ is the inverse of $f (x) = 8 + \sqrt{(x - 18) (- x + 2)}, 8 - \sqrt{(x - 18) (- x + 2)}$ .

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Step 10.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = 8 + \sqrt{(x - 18) (- x + 2)}, 8 - \sqrt{(x - 18) (- x + 2)}$ and $f^{- 1} (x) = 10 + \sqrt{x (- x + 16)}, 10 - \sqrt{x (- x + 16)}$ and compare them.

Step 10.2

Find the range of $8 + \sqrt{(x - 18) (- x + 2)}$ .

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Step 10.2.1

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[8, 8]$

Step 10.3

Find the domain of $10 + \sqrt{x (- x + 16)}$ .

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Step 10.3.1

Set the radicand in $\sqrt{x (- x + 16)}$ greater than or equal to $0$ to find where the expression is defined.

$x (- x + 16) \geq 0$

Step 10.3.2

Solve for $x$ .

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Step 10.3.2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$- x + 16 = 0$

Step 10.3.2.2

Set $x$ equal to $0$ .

$x = 0$

Step 10.3.2.3

Set $- x + 16$ equal to $0$ and solve for $x$ .

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Step 10.3.2.3.1

Set $- x + 16$ equal to $0$ .

$- x + 16 = 0$

Step 10.3.2.3.2

Solve $- x + 16 = 0$ for $x$ .

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Step 10.3.2.3.2.1

Subtract $16$ from both sides of the equation.

$- x = - 16$

Step 10.3.2.3.2.2

Divide each term in $- x = - 16$ by $- 1$ and simplify.

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Step 10.3.2.3.2.2.1

Divide each term in $- x = - 16$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 16}{- 1}$

Step 10.3.2.3.2.2.2

Simplify the left side.

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Step 10.3.2.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 16}{- 1}$

Step 10.3.2.3.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 16}{- 1}$

Step 10.3.2.3.2.2.3

Simplify the right side.

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Step 10.3.2.3.2.2.3.1

Divide $- 16$ by $- 1$ .

$x = 16$

Step 10.3.2.4

The final solution is all the values that make $x (- x + 16) \geq 0$ true.

$x = 0, 16$

Step 10.3.2.5

Use each root to create test intervals.

$x < 0$

$0 < x < 16$

$x > 16$

Step 10.3.2.6

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 10.3.2.6.1

Test a value on the interval $x < 0$ to see if it makes the inequality true.

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Step 10.3.2.6.1.1

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 10.3.2.6.1.2

Replace $x$ with $- 2$ in the original inequality.

$(- 2) (- (- 2) + 16) \geq 0$

Step 10.3.2.6.1.3

The left side $- 36$ is less than the right side $0$ , which means that the given statement is false.

False

Step 10.3.2.6.2

Test a value on the interval $0 < x < 16$ to see if it makes the inequality true.

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Step 10.3.2.6.2.1

Choose a value on the interval $0 < x < 16$ and see if this value makes the original inequality true.

$x = 2$

Step 10.3.2.6.2.2

Replace $x$ with $2$ in the original inequality.

$(2) (- (2) + 16) \geq 0$

Step 10.3.2.6.2.3

The left side $28$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 10.3.2.6.3

Test a value on the interval $x > 16$ to see if it makes the inequality true.

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Step 10.3.2.6.3.1

Choose a value on the interval $x > 16$ and see if this value makes the original inequality true.

$x = 18$

Step 10.3.2.6.3.2

Replace $x$ with $18$ in the original inequality.

$(18) (- (18) + 16) \geq 0$

Step 10.3.2.6.3.3

The left side $- 36$ is less than the right side $0$ , which means that the given statement is false.

False

Step 10.3.2.6.4

Compare the intervals to determine which ones satisfy the original inequality.

$x < 0$ False

$0 < x < 16$ True

$x > 16$ False

$x < 0$ False

$0 < x < 16$ True

$x > 16$ False

Step 10.3.2.7

The solution consists of all of the true intervals.

$0 \leq x \leq 16$

Step 10.3.3

The domain is all values of $x$ that make the expression defined.

$[0, 16]$

Step 10.4

Since the domain of $f^{- 1} (x) = 10 + \sqrt{x (- x + 16)}, 10 - \sqrt{x (- x + 16)}$ is not equal to the range of $f (x) = 8 + \sqrt{(x - 18) (- x + 2)}, 8 - \sqrt{(x - 18) (- x + 2)}$ , then $f^{- 1} (x) = 10 + \sqrt{x (- x + 16)}, 10 - \sqrt{x (- x + 16)}$ is not an inverse of $f (x) = 8 + \sqrt{(x - 18) (- x + 2)}, 8 - \sqrt{(x - 18) (- x + 2)}$ .

There is no inverse

Step 11