Trigonometry Examples

$x^{2} + y^{2} + 4 x - 6 y - 3 = 0$

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 $a = 1$ , $b = - 6$ , and $c = x^{2} + 4 x - 3$ into the quadratic formula and solve for $y$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot (x^{2} + 4 x - 3))}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $- 6$ to the power of $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot (x^{2} + 4 x - 3)}}{2 \cdot 1}$

Step 3.1.2

Multiply $- 4$ by $1$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot (x^{2} + 4 x - 3)}}{2 \cdot 1}$

Step 3.1.3

Apply the distributive property.

$y = \frac{6 \pm \sqrt{36 - 4 x^{2} - 4 (4 x) - 4 \cdot - 3}}{2 \cdot 1}$

Step 3.1.4

Simplify.

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Step 3.1.4.1

Multiply $4$ by $- 4$ .

$y = \frac{6 \pm \sqrt{36 - 4 x^{2} - 16 x - 4 \cdot - 3}}{2 \cdot 1}$

Step 3.1.4.2

Multiply $- 4$ by $- 3$ .

$y = \frac{6 \pm \sqrt{36 - 4 x^{2} - 16 x + 12}}{2 \cdot 1}$

Step 3.1.5

Add $36$ and $12$ .

$y = \frac{6 \pm \sqrt{- 4 x^{2} - 16 x + 48}}{2 \cdot 1}$

Step 3.1.6

Rewrite $- 4 x^{2} - 16 x + 48$ in a factored form.

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Step 3.1.6.1

Factor $4$ out of $- 4 x^{2} - 16 x + 48$ .

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Step 3.1.6.1.1

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

$y = \frac{6 \pm \sqrt{4 (- x^{2}) - 16 x + 48}}{2 \cdot 1}$

Step 3.1.6.1.2

Factor $4$ out of $- 16 x$ .

$y = \frac{6 \pm \sqrt{4 (- x^{2}) + 4 (- 4 x) + 48}}{2 \cdot 1}$

Step 3.1.6.1.3

Factor $4$ out of $48$ .

$y = \frac{6 \pm \sqrt{4 (- x^{2}) + 4 (- 4 x) + 4 (12)}}{2 \cdot 1}$

Step 3.1.6.1.4

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

$y = \frac{6 \pm \sqrt{4 (- x^{2} - 4 x) + 4 (12)}}{2 \cdot 1}$

Step 3.1.6.1.5

Factor $4$ out of $4 (- x^{2} - 4 x) + 4 (12)$ .

$y = \frac{6 \pm \sqrt{4 (- x^{2} - 4 x + 12)}}{2 \cdot 1}$

Step 3.1.6.2

Factor by grouping.

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Step 3.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 12 = - 12$ and whose sum is $b = - 4$ .

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Step 3.1.6.2.1.1

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

$y = \frac{6 \pm \sqrt{4 (- x^{2} - 4 x + 12)}}{2 \cdot 1}$

Step 3.1.6.2.1.2

Rewrite $- 4$ as $2$ plus $- 6$

$y = \frac{6 \pm \sqrt{4 (- x^{2} + (2 - 6) x + 12)}}{2 \cdot 1}$

Step 3.1.6.2.1.3

Apply the distributive property.

$y = \frac{6 \pm \sqrt{4 (- x^{2} + 2 x - 6 x + 12)}}{2 \cdot 1}$

Step 3.1.6.2.2

Factor out the greatest common factor from each group.

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Step 3.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{6 \pm \sqrt{4 ((- x^{2} + 2 x) - 6 x + 12)}}{2 \cdot 1}$

Step 3.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{6 \pm \sqrt{4 (x (- x + 2) + 6 (- x + 2))}}{2 \cdot 1}$

Step 3.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 2$ .

$y = \frac{6 \pm \sqrt{4 ((- x + 2) (x + 6))}}{2 \cdot 1}$

$y = \frac{6 \pm \sqrt{4 (- x + 2) (x + 6)}}{2 \cdot 1}$

Step 3.1.7

Rewrite $4 (- x + 2) (x + 6)$ as $2^{2} ((- x + 2) (x + 6))$ .

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Step 3.1.7.1

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

$y = \frac{6 \pm \sqrt{2^{2} (- x + 2) (x + 6)}}{2 \cdot 1}$

Step 3.1.7.2

Add parentheses.

$y = \frac{6 \pm \sqrt{2^{2} ((- x + 2) (x + 6))}}{2 \cdot 1}$

Step 3.1.8

Pull terms out from under the radical.

$y = \frac{6 \pm 2 \sqrt{(- x + 2) (x + 6)}}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

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

Step 3.3

Simplify $\frac{6 \pm 2 \sqrt{(- x + 2) (x + 6)}}{2}$ .

$y = 3 \pm \sqrt{(- x + 2) (x + 6)}$

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

Raise $- 6$ to the power of $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot (x^{2} + 4 x - 3)}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4$ by $1$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot (x^{2} + 4 x - 3)}}{2 \cdot 1}$

Step 4.1.3

Apply the distributive property.

$y = \frac{6 \pm \sqrt{36 - 4 x^{2} - 4 (4 x) - 4 \cdot - 3}}{2 \cdot 1}$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Multiply $4$ by $- 4$ .

$y = \frac{6 \pm \sqrt{36 - 4 x^{2} - 16 x - 4 \cdot - 3}}{2 \cdot 1}$

Step 4.1.4.2

Multiply $- 4$ by $- 3$ .

$y = \frac{6 \pm \sqrt{36 - 4 x^{2} - 16 x + 12}}{2 \cdot 1}$

Step 4.1.5

Add $36$ and $12$ .

$y = \frac{6 \pm \sqrt{- 4 x^{2} - 16 x + 48}}{2 \cdot 1}$

Step 4.1.6

Rewrite $- 4 x^{2} - 16 x + 48$ in a factored form.

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Step 4.1.6.1

Factor $4$ out of $- 4 x^{2} - 16 x + 48$ .

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Step 4.1.6.1.1

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

$y = \frac{6 \pm \sqrt{4 (- x^{2}) - 16 x + 48}}{2 \cdot 1}$

Step 4.1.6.1.2

Factor $4$ out of $- 16 x$ .

$y = \frac{6 \pm \sqrt{4 (- x^{2}) + 4 (- 4 x) + 48}}{2 \cdot 1}$

Step 4.1.6.1.3

Factor $4$ out of $48$ .

$y = \frac{6 \pm \sqrt{4 (- x^{2}) + 4 (- 4 x) + 4 (12)}}{2 \cdot 1}$

Step 4.1.6.1.4

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

$y = \frac{6 \pm \sqrt{4 (- x^{2} - 4 x) + 4 (12)}}{2 \cdot 1}$

Step 4.1.6.1.5

Factor $4$ out of $4 (- x^{2} - 4 x) + 4 (12)$ .

$y = \frac{6 \pm \sqrt{4 (- x^{2} - 4 x + 12)}}{2 \cdot 1}$

Step 4.1.6.2

Factor by grouping.

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Step 4.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 12 = - 12$ and whose sum is $b = - 4$ .

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Step 4.1.6.2.1.1

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

$y = \frac{6 \pm \sqrt{4 (- x^{2} - 4 x + 12)}}{2 \cdot 1}$

Step 4.1.6.2.1.2

Rewrite $- 4$ as $2$ plus $- 6$

$y = \frac{6 \pm \sqrt{4 (- x^{2} + (2 - 6) x + 12)}}{2 \cdot 1}$

Step 4.1.6.2.1.3

Apply the distributive property.

$y = \frac{6 \pm \sqrt{4 (- x^{2} + 2 x - 6 x + 12)}}{2 \cdot 1}$

Step 4.1.6.2.2

Factor out the greatest common factor from each group.

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Step 4.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{6 \pm \sqrt{4 ((- x^{2} + 2 x) - 6 x + 12)}}{2 \cdot 1}$

Step 4.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{6 \pm \sqrt{4 (x (- x + 2) + 6 (- x + 2))}}{2 \cdot 1}$

Step 4.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 2$ .

$y = \frac{6 \pm \sqrt{4 ((- x + 2) (x + 6))}}{2 \cdot 1}$

$y = \frac{6 \pm \sqrt{4 (- x + 2) (x + 6)}}{2 \cdot 1}$

Step 4.1.7

Rewrite $4 (- x + 2) (x + 6)$ as $2^{2} ((- x + 2) (x + 6))$ .

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Step 4.1.7.1

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

$y = \frac{6 \pm \sqrt{2^{2} (- x + 2) (x + 6)}}{2 \cdot 1}$

Step 4.1.7.2

Add parentheses.

$y = \frac{6 \pm \sqrt{2^{2} ((- x + 2) (x + 6))}}{2 \cdot 1}$

Step 4.1.8

Pull terms out from under the radical.

$y = \frac{6 \pm 2 \sqrt{(- x + 2) (x + 6)}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

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

Step 4.3

Simplify $\frac{6 \pm 2 \sqrt{(- x + 2) (x + 6)}}{2}$ .

$y = 3 \pm \sqrt{(- x + 2) (x + 6)}$

Step 4.4

Change the $\pm$ to $+$ .

$y = 3 + \sqrt{(- x + 2) (x + 6)}$

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

Raise $- 6$ to the power of $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot (x^{2} + 4 x - 3)}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4$ by $1$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot (x^{2} + 4 x - 3)}}{2 \cdot 1}$

Step 5.1.3

Apply the distributive property.

$y = \frac{6 \pm \sqrt{36 - 4 x^{2} - 4 (4 x) - 4 \cdot - 3}}{2 \cdot 1}$

Step 5.1.4

Simplify.

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Step 5.1.4.1

Multiply $4$ by $- 4$ .

$y = \frac{6 \pm \sqrt{36 - 4 x^{2} - 16 x - 4 \cdot - 3}}{2 \cdot 1}$

Step 5.1.4.2

Multiply $- 4$ by $- 3$ .

$y = \frac{6 \pm \sqrt{36 - 4 x^{2} - 16 x + 12}}{2 \cdot 1}$

Step 5.1.5

Add $36$ and $12$ .

$y = \frac{6 \pm \sqrt{- 4 x^{2} - 16 x + 48}}{2 \cdot 1}$

Step 5.1.6

Rewrite $- 4 x^{2} - 16 x + 48$ in a factored form.

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Step 5.1.6.1

Factor $4$ out of $- 4 x^{2} - 16 x + 48$ .

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Step 5.1.6.1.1

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

$y = \frac{6 \pm \sqrt{4 (- x^{2}) - 16 x + 48}}{2 \cdot 1}$

Step 5.1.6.1.2

Factor $4$ out of $- 16 x$ .

$y = \frac{6 \pm \sqrt{4 (- x^{2}) + 4 (- 4 x) + 48}}{2 \cdot 1}$

Step 5.1.6.1.3

Factor $4$ out of $48$ .

$y = \frac{6 \pm \sqrt{4 (- x^{2}) + 4 (- 4 x) + 4 (12)}}{2 \cdot 1}$

Step 5.1.6.1.4

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

$y = \frac{6 \pm \sqrt{4 (- x^{2} - 4 x) + 4 (12)}}{2 \cdot 1}$

Step 5.1.6.1.5

Factor $4$ out of $4 (- x^{2} - 4 x) + 4 (12)$ .

$y = \frac{6 \pm \sqrt{4 (- x^{2} - 4 x + 12)}}{2 \cdot 1}$

Step 5.1.6.2

Factor by grouping.

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Step 5.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 12 = - 12$ and whose sum is $b = - 4$ .

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Step 5.1.6.2.1.1

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

$y = \frac{6 \pm \sqrt{4 (- x^{2} - 4 x + 12)}}{2 \cdot 1}$

Step 5.1.6.2.1.2

Rewrite $- 4$ as $2$ plus $- 6$

$y = \frac{6 \pm \sqrt{4 (- x^{2} + (2 - 6) x + 12)}}{2 \cdot 1}$

Step 5.1.6.2.1.3

Apply the distributive property.

$y = \frac{6 \pm \sqrt{4 (- x^{2} + 2 x - 6 x + 12)}}{2 \cdot 1}$

Step 5.1.6.2.2

Factor out the greatest common factor from each group.

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Step 5.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{6 \pm \sqrt{4 ((- x^{2} + 2 x) - 6 x + 12)}}{2 \cdot 1}$

Step 5.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{6 \pm \sqrt{4 (x (- x + 2) + 6 (- x + 2))}}{2 \cdot 1}$

Step 5.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 2$ .

$y = \frac{6 \pm \sqrt{4 ((- x + 2) (x + 6))}}{2 \cdot 1}$

$y = \frac{6 \pm \sqrt{4 (- x + 2) (x + 6)}}{2 \cdot 1}$

Step 5.1.7

Rewrite $4 (- x + 2) (x + 6)$ as $2^{2} ((- x + 2) (x + 6))$ .

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Step 5.1.7.1

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

$y = \frac{6 \pm \sqrt{2^{2} (- x + 2) (x + 6)}}{2 \cdot 1}$

Step 5.1.7.2

Add parentheses.

$y = \frac{6 \pm \sqrt{2^{2} ((- x + 2) (x + 6))}}{2 \cdot 1}$

Step 5.1.8

Pull terms out from under the radical.

$y = \frac{6 \pm 2 \sqrt{(- x + 2) (x + 6)}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

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

Step 5.3

Simplify $\frac{6 \pm 2 \sqrt{(- x + 2) (x + 6)}}{2}$ .

$y = 3 \pm \sqrt{(- x + 2) (x + 6)}$

Step 5.4

Change the $\pm$ to $-$ .

$y = 3 - \sqrt{(- x + 2) (x + 6)}$

Step 6

The final answer is the combination of both solutions.

$y = 3 + \sqrt{(- x + 2) (x + 6)}$

$y = 3 - \sqrt{(- x + 2) (x + 6)}$

Step 7

Interchange the variables. Create an equation for each expression.

$x = 3 + \sqrt{(- y + 2) (y + 6)}, x = 3 - \sqrt{(- y + 2) (y + 6)}$

Step 8

Solve for $y$ .

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Step 8.1

Rewrite the equation as $3 + \sqrt{(- y + 2) (y + 6)} = x$ .

$3 + \sqrt{(- y + 2) (y + 6)} = x$

Step 8.2

Subtract $3$ from both sides of the equation.

$\sqrt{(- y + 2) (y + 6)} = x - 3$

Step 8.3

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

${\sqrt{(- y + 2) (y + 6)}}^{2} = {(x - 3)}^{2}$

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 $\sqrt{(- y + 2) (y + 6)}$ as ${((- y + 2) (y + 6))}^{\frac{1}{2}}$ .

${({((- y + 2) (y + 6))}^{\frac{1}{2}})}^{2} = {(x - 3)}^{2}$

Step 8.4.2

Simplify the left side.

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Step 8.4.2.1

Simplify ${({((- y + 2) (y + 6))}^{\frac{1}{2}})}^{2}$ .

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Step 8.4.2.1.1

Multiply the exponents in ${({((- y + 2) (y + 6))}^{\frac{1}{2}})}^{2}$ .

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Step 8.4.2.1.1.1

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

${((- y + 2) (y + 6))}^{\frac{1}{2} \cdot 2} = {(x - 3)}^{2}$

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.

${((- y + 2) (y + 6))}^{\frac{1}{2} \cdot 2} = {(x - 3)}^{2}$

Step 8.4.2.1.1.2.2

Rewrite the expression.

${((- y + 2) (y + 6))}^{1} = {(x - 3)}^{2}$

Step 8.4.2.1.2

Expand $(- y + 2) (y + 6)$ using the FOIL Method.

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Step 8.4.2.1.2.1

Apply the distributive property.

${(- y (y + 6) + 2 (y + 6))}^{1} = {(x - 3)}^{2}$

Step 8.4.2.1.2.2

Apply the distributive property.

${(- y \cdot y - y \cdot 6 + 2 (y + 6))}^{1} = {(x - 3)}^{2}$

Step 8.4.2.1.2.3

Apply the distributive property.

${(- y \cdot y - y \cdot 6 + 2 y + 2 \cdot 6)}^{1} = {(x - 3)}^{2}$

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

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

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Step 8.4.2.1.3.1.1.1

Move $y$ .

${(- (y \cdot y) - y \cdot 6 + 2 y + 2 \cdot 6)}^{1} = {(x - 3)}^{2}$

Step 8.4.2.1.3.1.1.2

Multiply $y$ by $y$ .

${(- y^{2} - y \cdot 6 + 2 y + 2 \cdot 6)}^{1} = {(x - 3)}^{2}$

Step 8.4.2.1.3.1.2

Multiply $6$ by $- 1$ .

${(- y^{2} - 6 y + 2 y + 2 \cdot 6)}^{1} = {(x - 3)}^{2}$

Step 8.4.2.1.3.1.3

Multiply $2$ by $6$ .

${(- y^{2} - 6 y + 2 y + 12)}^{1} = {(x - 3)}^{2}$

Step 8.4.2.1.3.2

Add $- 6 y$ and $2 y$ .

${(- y^{2} - 4 y + 12)}^{1} = {(x - 3)}^{2}$

Step 8.4.2.1.4

Simplify.

$- y^{2} - 4 y + 12 = {(x - 3)}^{2}$

Step 8.4.3

Simplify the right side.

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Step 8.4.3.1

Simplify ${(x - 3)}^{2}$ .

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Step 8.4.3.1.1

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$- y^{2} - 4 y + 12 = (x - 3) (x - 3)$

Step 8.4.3.1.2

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 8.4.3.1.2.1

Apply the distributive property.

$- y^{2} - 4 y + 12 = x (x - 3) - 3 (x - 3)$

Step 8.4.3.1.2.2

Apply the distributive property.

$- y^{2} - 4 y + 12 = x \cdot x + x \cdot - 3 - 3 (x - 3)$

Step 8.4.3.1.2.3

Apply the distributive property.

$- y^{2} - 4 y + 12 = x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3$

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 $x$ .

$- y^{2} - 4 y + 12 = x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3$

Step 8.4.3.1.3.1.2

Move $- 3$ to the left of $x$ .

$- y^{2} - 4 y + 12 = x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3$

Step 8.4.3.1.3.1.3

Multiply $- 3$ by $- 3$ .

$- y^{2} - 4 y + 12 = x^{2} - 3 x - 3 x + 9$

Step 8.4.3.1.3.2

Subtract $3 x$ from $- 3 x$ .

$- y^{2} - 4 y + 12 = x^{2} - 6 x + 9$

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.

$- y^{2} - 4 y + 12 - x^{2} = - 6 x + 9$

Step 8.5.1.1.2

Add $6 x$ to both sides of the equation.

$- y^{2} - 4 y + 12 - x^{2} + 6 x = 9$

Step 8.5.1.1.3

Subtract $9$ from both sides of the equation.

$- y^{2} - 4 y + 12 - x^{2} + 6 x - 9 = 0$

Step 8.5.1.2

Subtract $9$ from $12$ .

$- y^{2} - 4 y - x^{2} + 6 x + 3 = 0$

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 $a = - 1$ , $b = - 4$ , and $c = - x^{2} + 6 x + 3$ into the quadratic formula and solve for $y$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (- 1 \cdot (- x^{2} + 6 x + 3))}}{2 \cdot - 1}$

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

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot - 1 \cdot (- x^{2} + 6 x + 3)$ .

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Step 8.5.4.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot - 1 \cdot (- x^{2} + 6 x + 3)}}{2 \cdot - 1}$

Step 8.5.4.1.1.2

Factor $- 4$ out of $- 4 \cdot - 1 \cdot (- x^{2} + 6 x + 3)$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (- 1 \cdot (- x^{2} + 6 x + 3))}}{2 \cdot - 1}$

Step 8.5.4.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (- 1 \cdot (- x^{2} + 6 x + 3))$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 - 1 \cdot (- x^{2} + 6 x + 3))}}{2 \cdot - 1}$

Step 8.5.4.1.2

Factor $- 1$ out of $- 4 - 1 \cdot (- x^{2} + 6 x + 3)$ .

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Step 8.5.4.1.2.1

Reorder $- 4$ and $- 1 \cdot (- x^{2} + 6 x + 3)$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (- x^{2} + 6 x + 3) - 4)}}{2 \cdot - 1}$

Step 8.5.4.1.2.2

Rewrite $- 4$ as $- 1 (4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (- x^{2} + 6 x + 3) - 1 \cdot 4)}}{2 \cdot - 1}$

Step 8.5.4.1.2.3

Factor $- 1$ out of $- 1 (- x^{2} + 6 x + 3) - 1 (4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (- x^{2} + 6 x + 3 + 4))}}{2 \cdot - 1}$

Step 8.5.4.1.2.4

Rewrite $- 1 (- x^{2} + 6 x + 3 + 4)$ as $- (- x^{2} + 6 x + 3 + 4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- (- x^{2} + 6 x + 3 + 4))}}{2 \cdot - 1}$

Step 8.5.4.1.3

Add $3$ and $4$ .

$y = \frac{4 \pm \sqrt{- 4 (- (- x^{2} + 6 x + 7))}}{2 \cdot - 1}$

Step 8.5.4.1.4

Factor.

$y = \frac{4 \pm \sqrt{- 4 \cdot (- 1 (- x - 1) (x - 7))}}{2 \cdot - 1}$

Step 8.5.4.1.5

Combine exponents.

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Step 8.5.4.1.5.1

Factor out negative.

$y = \frac{4 \pm \sqrt{- ((- 4 (- x - 1)) (x - 7))}}{2 \cdot - 1}$

Step 8.5.4.1.5.2

Multiply $- 4$ by $- 1$ .

$y = \frac{4 \pm \sqrt{4 ((- x - 1) (x - 7))}}{2 \cdot - 1}$

$y = \frac{4 \pm \sqrt{4 (- x - 1) (x - 7)}}{2 \cdot - 1}$

Step 8.5.4.1.6

Rewrite $4 (- x - 1) (x - 7)$ as $2^{2} ((- x - 1) (x - 7))$ .

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Step 8.5.4.1.6.1

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

$y = \frac{4 \pm \sqrt{2^{2} (- x - 1) (x - 7)}}{2 \cdot - 1}$

Step 8.5.4.1.6.2

Add parentheses.

$y = \frac{4 \pm \sqrt{2^{2} ((- x - 1) (x - 7))}}{2 \cdot - 1}$

Step 8.5.4.1.7

Pull terms out from under the radical.

$y = \frac{4 \pm 2 \sqrt{(- x - 1) (x - 7)}}{2 \cdot - 1}$

Step 8.5.4.2

Multiply $2$ by $- 1$ .

$y = \frac{4 \pm 2 \sqrt{(- x - 1) (x - 7)}}{- 2}$

Step 8.5.4.3

Simplify $\frac{4 \pm 2 \sqrt{(- x - 1) (x - 7)}}{- 2}$ .

$y = \frac{2 \pm \sqrt{(- x - 1) (x - 7)}}{- 1}$

Step 8.5.4.4

Move the negative one from the denominator of $\frac{2 \pm \sqrt{(- x - 1) (x - 7)}}{- 1}$ .

$y = - 1 \cdot (2 \pm \sqrt{(- x - 1) (x - 7)})$

Step 8.5.4.5

Rewrite $- 1 \cdot (2 \pm \sqrt{(- x - 1) (x - 7)})$ as $- (2 \pm \sqrt{(- x - 1) (x - 7)})$ .

$y = - (2 \pm \sqrt{(- x - 1) (x - 7)})$

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

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot - 1 \cdot (- x^{2} + 6 x + 3)$ .

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Step 8.5.5.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot - 1 \cdot (- x^{2} + 6 x + 3)}}{2 \cdot - 1}$

Step 8.5.5.1.1.2

Factor $- 4$ out of $- 4 \cdot - 1 \cdot (- x^{2} + 6 x + 3)$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (- 1 \cdot (- x^{2} + 6 x + 3))}}{2 \cdot - 1}$

Step 8.5.5.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (- 1 \cdot (- x^{2} + 6 x + 3))$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 - 1 \cdot (- x^{2} + 6 x + 3))}}{2 \cdot - 1}$

Step 8.5.5.1.2

Factor $- 1$ out of $- 4 - 1 \cdot (- x^{2} + 6 x + 3)$ .

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Step 8.5.5.1.2.1

Reorder $- 4$ and $- 1 \cdot (- x^{2} + 6 x + 3)$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (- x^{2} + 6 x + 3) - 4)}}{2 \cdot - 1}$

Step 8.5.5.1.2.2

Rewrite $- 4$ as $- 1 (4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (- x^{2} + 6 x + 3) - 1 \cdot 4)}}{2 \cdot - 1}$

Step 8.5.5.1.2.3

Factor $- 1$ out of $- 1 (- x^{2} + 6 x + 3) - 1 (4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (- x^{2} + 6 x + 3 + 4))}}{2 \cdot - 1}$

Step 8.5.5.1.2.4

Rewrite $- 1 (- x^{2} + 6 x + 3 + 4)$ as $- (- x^{2} + 6 x + 3 + 4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- (- x^{2} + 6 x + 3 + 4))}}{2 \cdot - 1}$

Step 8.5.5.1.3

Add $3$ and $4$ .

$y = \frac{4 \pm \sqrt{- 4 (- (- x^{2} + 6 x + 7))}}{2 \cdot - 1}$

Step 8.5.5.1.4

Factor.

$y = \frac{4 \pm \sqrt{- 4 \cdot (- 1 (- x - 1) (x - 7))}}{2 \cdot - 1}$

Step 8.5.5.1.5

Combine exponents.

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Step 8.5.5.1.5.1

Factor out negative.

$y = \frac{4 \pm \sqrt{- ((- 4 (- x - 1)) (x - 7))}}{2 \cdot - 1}$

Step 8.5.5.1.5.2

Multiply $- 4$ by $- 1$ .

$y = \frac{4 \pm \sqrt{4 ((- x - 1) (x - 7))}}{2 \cdot - 1}$

$y = \frac{4 \pm \sqrt{4 (- x - 1) (x - 7)}}{2 \cdot - 1}$

Step 8.5.5.1.6

Rewrite $4 (- x - 1) (x - 7)$ as $2^{2} ((- x - 1) (x - 7))$ .

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Step 8.5.5.1.6.1

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

$y = \frac{4 \pm \sqrt{2^{2} (- x - 1) (x - 7)}}{2 \cdot - 1}$

Step 8.5.5.1.6.2

Add parentheses.

$y = \frac{4 \pm \sqrt{2^{2} ((- x - 1) (x - 7))}}{2 \cdot - 1}$

Step 8.5.5.1.7

Pull terms out from under the radical.

$y = \frac{4 \pm 2 \sqrt{(- x - 1) (x - 7)}}{2 \cdot - 1}$

Step 8.5.5.2

Multiply $2$ by $- 1$ .

$y = \frac{4 \pm 2 \sqrt{(- x - 1) (x - 7)}}{- 2}$

Step 8.5.5.3

Simplify $\frac{4 \pm 2 \sqrt{(- x - 1) (x - 7)}}{- 2}$ .

$y = \frac{2 \pm \sqrt{(- x - 1) (x - 7)}}{- 1}$

Step 8.5.5.4

Move the negative one from the denominator of $\frac{2 \pm \sqrt{(- x - 1) (x - 7)}}{- 1}$ .

$y = - 1 \cdot (2 \pm \sqrt{(- x - 1) (x - 7)})$

Step 8.5.5.5

Rewrite $- 1 \cdot (2 \pm \sqrt{(- x - 1) (x - 7)})$ as $- (2 \pm \sqrt{(- x - 1) (x - 7)})$ .

$y = - (2 \pm \sqrt{(- x - 1) (x - 7)})$

Step 8.5.5.6

Change the $\pm$ to $+$ .

$y = - (2 + \sqrt{(- x - 1) (x - 7)})$

Step 8.5.5.7

Apply the distributive property.

$y = - 1 \cdot 2 - \sqrt{(- x - 1) (x - 7)}$

Step 8.5.5.8

Multiply $- 1$ by $2$ .

$y = - 2 - \sqrt{(- x - 1) (x - 7)}$

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

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot - 1 \cdot (- x^{2} + 6 x + 3)$ .

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Step 8.5.6.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot - 1 \cdot (- x^{2} + 6 x + 3)}}{2 \cdot - 1}$

Step 8.5.6.1.1.2

Factor $- 4$ out of $- 4 \cdot - 1 \cdot (- x^{2} + 6 x + 3)$ .

$y = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (- 1 \cdot (- x^{2} + 6 x + 3))}}{2 \cdot - 1}$

Step 8.5.6.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (- 1 \cdot (- x^{2} + 6 x + 3))$ .

$y = \frac{4 \pm \sqrt{- 4 (- 4 - 1 \cdot (- x^{2} + 6 x + 3))}}{2 \cdot - 1}$

Step 8.5.6.1.2

Factor $- 1$ out of $- 4 - 1 \cdot (- x^{2} + 6 x + 3)$ .

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Step 8.5.6.1.2.1

Reorder $- 4$ and $- 1 \cdot (- x^{2} + 6 x + 3)$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (- x^{2} + 6 x + 3) - 4)}}{2 \cdot - 1}$

Step 8.5.6.1.2.2

Rewrite $- 4$ as $- 1 (4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (- x^{2} + 6 x + 3) - 1 \cdot 4)}}{2 \cdot - 1}$

Step 8.5.6.1.2.3

Factor $- 1$ out of $- 1 (- x^{2} + 6 x + 3) - 1 (4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- 1 (- x^{2} + 6 x + 3 + 4))}}{2 \cdot - 1}$

Step 8.5.6.1.2.4

Rewrite $- 1 (- x^{2} + 6 x + 3 + 4)$ as $- (- x^{2} + 6 x + 3 + 4)$ .

$y = \frac{4 \pm \sqrt{- 4 (- (- x^{2} + 6 x + 3 + 4))}}{2 \cdot - 1}$

Step 8.5.6.1.3

Add $3$ and $4$ .

$y = \frac{4 \pm \sqrt{- 4 (- (- x^{2} + 6 x + 7))}}{2 \cdot - 1}$

Step 8.5.6.1.4

Factor.

$y = \frac{4 \pm \sqrt{- 4 \cdot (- 1 (- x - 1) (x - 7))}}{2 \cdot - 1}$

Step 8.5.6.1.5

Combine exponents.

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Step 8.5.6.1.5.1

Factor out negative.

$y = \frac{4 \pm \sqrt{- ((- 4 (- x - 1)) (x - 7))}}{2 \cdot - 1}$

Step 8.5.6.1.5.2

Multiply $- 4$ by $- 1$ .

$y = \frac{4 \pm \sqrt{4 ((- x - 1) (x - 7))}}{2 \cdot - 1}$

$y = \frac{4 \pm \sqrt{4 (- x - 1) (x - 7)}}{2 \cdot - 1}$

Step 8.5.6.1.6

Rewrite $4 (- x - 1) (x - 7)$ as $2^{2} ((- x - 1) (x - 7))$ .

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Step 8.5.6.1.6.1

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

$y = \frac{4 \pm \sqrt{2^{2} (- x - 1) (x - 7)}}{2 \cdot - 1}$

Step 8.5.6.1.6.2

Add parentheses.

$y = \frac{4 \pm \sqrt{2^{2} ((- x - 1) (x - 7))}}{2 \cdot - 1}$

Step 8.5.6.1.7

Pull terms out from under the radical.

$y = \frac{4 \pm 2 \sqrt{(- x - 1) (x - 7)}}{2 \cdot - 1}$

Step 8.5.6.2

Multiply $2$ by $- 1$ .

$y = \frac{4 \pm 2 \sqrt{(- x - 1) (x - 7)}}{- 2}$

Step 8.5.6.3

Simplify $\frac{4 \pm 2 \sqrt{(- x - 1) (x - 7)}}{- 2}$ .

$y = \frac{2 \pm \sqrt{(- x - 1) (x - 7)}}{- 1}$

Step 8.5.6.4

Move the negative one from the denominator of $\frac{2 \pm \sqrt{(- x - 1) (x - 7)}}{- 1}$ .

$y = - 1 \cdot (2 \pm \sqrt{(- x - 1) (x - 7)})$

Step 8.5.6.5

Rewrite $- 1 \cdot (2 \pm \sqrt{(- x - 1) (x - 7)})$ as $- (2 \pm \sqrt{(- x - 1) (x - 7)})$ .

$y = - (2 \pm \sqrt{(- x - 1) (x - 7)})$

Step 8.5.6.6

Change the $\pm$ to $-$ .

$y = - (2 - \sqrt{(- x - 1) (x - 7)})$

Step 8.5.6.7

Apply the distributive property.

$y = - 1 \cdot 2 + \sqrt{(- x - 1) (x - 7)}$

Step 8.5.6.8

Multiply $- 1$ by $2$ .

$y = - 2 + \sqrt{(- x - 1) (x - 7)}$

Step 8.5.6.9

Multiply $- - \sqrt{(- x - 1) (x - 7)}$ .

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Step 8.5.6.9.1

Multiply $- 1$ by $- 1$ .

$y = - 2 + 1 \sqrt{(- x - 1) (x - 7)}$

Step 8.5.6.9.2

Multiply $\sqrt{(- x - 1) (x - 7)}$ by $1$ .

$y = - 2 + \sqrt{(- x - 1) (x - 7)}$

Step 8.5.7

The final answer is the combination of both solutions.

$y = - 2 - \sqrt{(- x - 1) (x - 7)}$

$y = - 2 + \sqrt{(- x - 1) (x - 7)}$

$y = - 2 - \sqrt{(- x - 1) (x - 7)}$

$y = - 2 + \sqrt{(- x - 1) (x - 7)}$

$y = - 2 - \sqrt{(- x - 1) (x - 7)}$

$y = - 2 + \sqrt{(- x - 1) (x - 7)}$

Step 9

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

$f^{- 1} (x) = - 2 - \sqrt{(- x - 1) (x - 7)}, - 2 + \sqrt{(- x - 1) (x - 7)}$

Step 10

Verify if $f^{- 1} (x) = - 2 - \sqrt{(- x - 1) (x - 7)}, - 2 + \sqrt{(- x - 1) (x - 7)}$ is the inverse of $f (x) = 3 + \sqrt{(- x + 2) (x + 6)}, 3 - \sqrt{(- x + 2) (x + 6)}$ .

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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 $f (x) = 3 + \sqrt{(- x + 2) (x + 6)}, 3 - \sqrt{(- x + 2) (x + 6)}$ and $f^{- 1} (x) = - 2 - \sqrt{(- x - 1) (x - 7)}, - 2 + \sqrt{(- x - 1) (x - 7)}$ and compare them.

Step 10.2

Find the range of $f (x) = 3 + \sqrt{(- x + 2) (x + 6)}, 3 - \sqrt{(- x + 2) (x + 6)}$ .

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Step 10.2.1

Find the range of $3 + \sqrt{(- x + 2) (x + 6)}$ .

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Step 10.2.1.1

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

Interval Notation:

$[3, 7]$

Step 10.2.2

Find the range of $3 - \sqrt{(- x + 2) (x + 6)}$ .

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Step 10.2.2.1

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

Interval Notation:

$[- 1, 3]$

Step 10.2.3

Find the union of $[3, 7], [- 1, 3]$ .

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Step 10.2.3.1

The union consists of all of the elements that are contained in each interval.

$[- 1, 7]$

Step 10.3

Find the domain of $- 2 - \sqrt{(- x - 1) (x - 7)}$ .

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Step 10.3.1

Set the radicand in $\sqrt{(- x - 1) (x - 7)}$ greater than or equal to $0$ to find where the expression is defined.

$(- x - 1) (x - 7) \geq 0$

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 $0$ .

$- x - 1 = 0$

$x - 7 = 0$

Step 10.3.2.2

Set $- x - 1$ equal to $0$ and solve for $x$ .

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Step 10.3.2.2.1

Set $- x - 1$ equal to $0$ .

$- x - 1 = 0$

Step 10.3.2.2.2

Solve $- x - 1 = 0$ for $x$ .

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Step 10.3.2.2.2.1

Add $1$ to both sides of the equation.

$- x = 1$

Step 10.3.2.2.2.2

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

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Step 10.3.2.2.2.2.1

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

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

Step 10.3.2.2.2.2.2

Simplify the left side.

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Step 10.3.2.2.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 10.3.2.2.2.2.2.2

Divide $x$ by $1$ .

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

Step 10.3.2.2.2.2.3

Simplify the right side.

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Step 10.3.2.2.2.2.3.1

Divide $1$ by $- 1$ .

$x = - 1$

Step 10.3.2.3

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 10.3.2.3.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 10.3.2.3.2

Add $7$ to both sides of the equation.

$x = 7$

Step 10.3.2.4

The final solution is all the values that make $(- x - 1) (x - 7) \geq 0$ true.

$x = - 1, 7$

Step 10.3.2.5

Use each root to create test intervals.

$x < - 1$

$- 1 < x < 7$

$x > 7$

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 $x < - 1$ 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 $x < - 1$ and see if this value makes the original inequality true.

$x = - 4$

Step 10.3.2.6.1.2

Replace $x$ with $- 4$ in the original inequality.

$(- (- 4) - 1) ((- 4) - 7) \geq 0$

Step 10.3.2.6.1.3

The left side $- 33$ 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 $- 1 < x < 7$ 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 $- 1 < x < 7$ and see if this value makes the original inequality true.

$x = 0$

Step 10.3.2.6.2.2

Replace $x$ with $0$ in the original inequality.

$(- (0) - 1) ((0) - 7) \geq 0$

Step 10.3.2.6.2.3

The left side $7$ 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 $x > 7$ 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 $x > 7$ and see if this value makes the original inequality true.

$x = 10$

Step 10.3.2.6.3.2

Replace $x$ with $10$ in the original inequality.

$(- (10) - 1) ((10) - 7) \geq 0$

Step 10.3.2.6.3.3

The left side $- 33$ 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.

$x < - 1$ False

$- 1 < x < 7$ True

$x > 7$ False

$x < - 1$ False

$- 1 < x < 7$ True

$x > 7$ False

Step 10.3.2.7

The solution consists of all of the true intervals.

$- 1 \leq x \leq 7$

Step 10.3.3

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

$[- 1, 7]$

Step 10.4

Find the domain of $3 + \sqrt{(- x + 2) (x + 6)}$ .

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Step 10.4.1

Set the radicand in $\sqrt{(- x + 2) (x + 6)}$ greater than or equal to $0$ to find where the expression is defined.

$(- x + 2) (x + 6) \geq 0$

Step 10.4.2

Solve for $x$ .

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Step 10.4.2.1

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

$- x + 2 = 0$

$x + 6 = 0$

Step 10.4.2.2

Set $- x + 2$ equal to $0$ and solve for $x$ .

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Step 10.4.2.2.1

Set $- x + 2$ equal to $0$ .

$- x + 2 = 0$

Step 10.4.2.2.2

Solve $- x + 2 = 0$ for $x$ .

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Step 10.4.2.2.2.1

Subtract $2$ from both sides of the equation.

$- x = - 2$

Step 10.4.2.2.2.2

Divide each term in $- x = - 2$ by $- 1$ and simplify.

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Step 10.4.2.2.2.2.1

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

Step 10.4.2.2.2.2.2

Simplify the left side.

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Step 10.4.2.2.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 2}{- 1}$

Step 10.4.2.2.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Step 10.4.2.2.2.2.3

Simplify the right side.

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Step 10.4.2.2.2.2.3.1

Divide $- 2$ by $- 1$ .

$x = 2$

Step 10.4.2.3

Set $x + 6$ equal to $0$ and solve for $x$ .

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Step 10.4.2.3.1

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Step 10.4.2.3.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 10.4.2.4

The final solution is all the values that make $(- x + 2) (x + 6) \geq 0$ true.

$x = 2, - 6$

Step 10.4.2.5

Use each root to create test intervals.

$x < - 6$

$- 6 < x < 2$

$x > 2$

Step 10.4.2.6

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 10.4.2.6.1

Test a value on the interval $x < - 6$ to see if it makes the inequality true.

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Step 10.4.2.6.1.1

Choose a value on the interval $x < - 6$ and see if this value makes the original inequality true.

$x = - 8$

Step 10.4.2.6.1.2

Replace $x$ with $- 8$ in the original inequality.

$(- (- 8) + 2) ((- 8) + 6) \geq 0$

Step 10.4.2.6.1.3

The left side $- 20$ is less than the right side $0$ , which means that the given statement is false.

False

Step 10.4.2.6.2

Test a value on the interval $- 6 < x < 2$ to see if it makes the inequality true.

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Step 10.4.2.6.2.1

Choose a value on the interval $- 6 < x < 2$ and see if this value makes the original inequality true.

$x = 0$

Step 10.4.2.6.2.2

Replace $x$ with $0$ in the original inequality.

$(- (0) + 2) ((0) + 6) \geq 0$

Step 10.4.2.6.2.3

The left side $12$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 10.4.2.6.3

Test a value on the interval $x > 2$ to see if it makes the inequality true.

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Step 10.4.2.6.3.1

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

Step 10.4.2.6.3.2

Replace $x$ with $4$ in the original inequality.

$(- (4) + 2) ((4) + 6) \geq 0$

Step 10.4.2.6.3.3

The left side $- 20$ is less than the right side $0$ , which means that the given statement is false.

False

Step 10.4.2.6.4

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

$x < - 6$ False

$- 6 < x < 2$ True

$x > 2$ False

$x < - 6$ False

$- 6 < x < 2$ True

$x > 2$ False

Step 10.4.2.7

The solution consists of all of the true intervals.

$- 6 \leq x \leq 2$

Step 10.4.3

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

$[- 6, 2]$

Step 10.5

Find the range of $- 2 - \sqrt{(- x - 1) (x - 7)}$ .

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Step 10.5.1

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

Interval Notation:

$[- 2, - 2]$

Step 10.6

Since the range of $f^{- 1} (x) = - 2 - \sqrt{(- x - 1) (x - 7)}, - 2 + \sqrt{(- x - 1) (x - 7)}$ is not equal to the domain of $f (x) = 3 + \sqrt{(- x + 2) (x + 6)}, 3 - \sqrt{(- x + 2) (x + 6)}$ , then $f^{- 1} (x) = - 2 - \sqrt{(- x - 1) (x - 7)}, - 2 + \sqrt{(- x - 1) (x - 7)}$ is not an inverse of $f (x) = 3 + \sqrt{(- x + 2) (x + 6)}, 3 - \sqrt{(- x + 2) (x + 6)}$ .

There is no inverse

Step 11