Trigonometry Examples

$4 x^{2} + 3 y^{2} + 8 x - 6 y - 29 = 0$

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 $a = 3$ , $b = - 6$ , and $c = 4 x^{2} + 8 x - 29$ into the quadratic formula and solve for $y$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (3 \cdot (4 x^{2} + 8 x - 29))}}{2 \cdot 3}$

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 $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 3 \cdot (4 x^{2} + 8 x - 29)}}{2 \cdot 3}$

Step 3.1.2

Multiply $- 4$ by $3$ .

$y = \frac{6 \pm \sqrt{36 - 12 \cdot (4 x^{2} + 8 x - 29)}}{2 \cdot 3}$

Step 3.1.3

Apply the distributive property.

$y = \frac{6 \pm \sqrt{36 - 12 (4 x^{2}) - 12 (8 x) - 12 \cdot - 29}}{2 \cdot 3}$

Step 3.1.4

Simplify.

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Step 3.1.4.1

Multiply $4$ by $- 12$ .

$y = \frac{6 \pm \sqrt{36 - 48 x^{2} - 12 (8 x) - 12 \cdot - 29}}{2 \cdot 3}$

Step 3.1.4.2

Multiply $8$ by $- 12$ .

$y = \frac{6 \pm \sqrt{36 - 48 x^{2} - 96 x - 12 \cdot - 29}}{2 \cdot 3}$

Step 3.1.4.3

Multiply $- 12$ by $- 29$ .

$y = \frac{6 \pm \sqrt{36 - 48 x^{2} - 96 x + 348}}{2 \cdot 3}$

Step 3.1.5

Add $36$ and $348$ .

$y = \frac{6 \pm \sqrt{- 48 x^{2} - 96 x + 384}}{2 \cdot 3}$

Step 3.1.6

Rewrite $- 48 x^{2} - 96 x + 384$ in a factored form.

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Step 3.1.6.1

Factor $48$ out of $- 48 x^{2} - 96 x + 384$ .

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Step 3.1.6.1.1

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

$y = \frac{6 \pm \sqrt{48 (- x^{2}) - 96 x + 384}}{2 \cdot 3}$

Step 3.1.6.1.2

Factor $48$ out of $- 96 x$ .

$y = \frac{6 \pm \sqrt{48 (- x^{2}) + 48 (- 2 x) + 384}}{2 \cdot 3}$

Step 3.1.6.1.3

Factor $48$ out of $384$ .

$y = \frac{6 \pm \sqrt{48 (- x^{2}) + 48 (- 2 x) + 48 (8)}}{2 \cdot 3}$

Step 3.1.6.1.4

Factor $48$ out of $48 (- x^{2}) + 48 (- 2 x)$ .

$y = \frac{6 \pm \sqrt{48 (- x^{2} - 2 x) + 48 (8)}}{2 \cdot 3}$

Step 3.1.6.1.5

Factor $48$ out of $48 (- x^{2} - 2 x) + 48 (8)$ .

$y = \frac{6 \pm \sqrt{48 (- x^{2} - 2 x + 8)}}{2 \cdot 3}$

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 $a \cdot c = - 1 \cdot 8 = - 8$ and whose sum is $b = - 2$ .

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Step 3.1.6.2.1.1

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

$y = \frac{6 \pm \sqrt{48 (- x^{2} - 2 x + 8)}}{2 \cdot 3}$

Step 3.1.6.2.1.2

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

$y = \frac{6 \pm \sqrt{48 (- x^{2} + (2 - 4) x + 8)}}{2 \cdot 3}$

Step 3.1.6.2.1.3

Apply the distributive property.

$y = \frac{6 \pm \sqrt{48 (- x^{2} + 2 x - 4 x + 8)}}{2 \cdot 3}$

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.

$y = \frac{6 \pm \sqrt{48 ((- x^{2} + 2 x) - 4 x + 8)}}{2 \cdot 3}$

Step 3.1.6.2.2.2

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

$y = \frac{6 \pm \sqrt{48 (x (- x + 2) + 4 (- x + 2))}}{2 \cdot 3}$

Step 3.1.6.2.3

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

$y = \frac{6 \pm \sqrt{48 ((- x + 2) (x + 4))}}{2 \cdot 3}$

$y = \frac{6 \pm \sqrt{48 (- x + 2) (x + 4)}}{2 \cdot 3}$

Step 3.1.7

Rewrite $48 (- x + 2) (x + 4)$ as ${(2^{2})}^{2} \cdot (3 (- x + 2) (x + 4))$ .

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Step 3.1.7.1

Factor $16$ out of $48$ .

$y = \frac{6 \pm \sqrt{16 (3) (- x + 2) (x + 4)}}{2 \cdot 3}$

Step 3.1.7.2

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

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

Step 3.1.7.3

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

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

Step 3.1.7.4

Add parentheses.

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

Step 3.1.7.5

Add parentheses.

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

Step 3.1.8

Pull terms out from under the radical.

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

Step 3.1.9

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

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

Step 3.2

Multiply $2$ by $3$ .

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

Step 3.3

Simplify $\frac{6 \pm 4 \sqrt{3 (- x + 2) (x + 4)}}{6}$ .

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

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 $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 3 \cdot (4 x^{2} + 8 x - 29)}}{2 \cdot 3}$

Step 4.1.2

Multiply $- 4$ by $3$ .

$y = \frac{6 \pm \sqrt{36 - 12 \cdot (4 x^{2} + 8 x - 29)}}{2 \cdot 3}$

Step 4.1.3

Apply the distributive property.

$y = \frac{6 \pm \sqrt{36 - 12 (4 x^{2}) - 12 (8 x) - 12 \cdot - 29}}{2 \cdot 3}$

Step 4.1.4

Simplify.

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Step 4.1.4.1

Multiply $4$ by $- 12$ .

$y = \frac{6 \pm \sqrt{36 - 48 x^{2} - 12 (8 x) - 12 \cdot - 29}}{2 \cdot 3}$

Step 4.1.4.2

Multiply $8$ by $- 12$ .

$y = \frac{6 \pm \sqrt{36 - 48 x^{2} - 96 x - 12 \cdot - 29}}{2 \cdot 3}$

Step 4.1.4.3

Multiply $- 12$ by $- 29$ .

$y = \frac{6 \pm \sqrt{36 - 48 x^{2} - 96 x + 348}}{2 \cdot 3}$

Step 4.1.5

Add $36$ and $348$ .

$y = \frac{6 \pm \sqrt{- 48 x^{2} - 96 x + 384}}{2 \cdot 3}$

Step 4.1.6

Rewrite $- 48 x^{2} - 96 x + 384$ in a factored form.

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Step 4.1.6.1

Factor $48$ out of $- 48 x^{2} - 96 x + 384$ .

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Step 4.1.6.1.1

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

$y = \frac{6 \pm \sqrt{48 (- x^{2}) - 96 x + 384}}{2 \cdot 3}$

Step 4.1.6.1.2

Factor $48$ out of $- 96 x$ .

$y = \frac{6 \pm \sqrt{48 (- x^{2}) + 48 (- 2 x) + 384}}{2 \cdot 3}$

Step 4.1.6.1.3

Factor $48$ out of $384$ .

$y = \frac{6 \pm \sqrt{48 (- x^{2}) + 48 (- 2 x) + 48 (8)}}{2 \cdot 3}$

Step 4.1.6.1.4

Factor $48$ out of $48 (- x^{2}) + 48 (- 2 x)$ .

$y = \frac{6 \pm \sqrt{48 (- x^{2} - 2 x) + 48 (8)}}{2 \cdot 3}$

Step 4.1.6.1.5

Factor $48$ out of $48 (- x^{2} - 2 x) + 48 (8)$ .

$y = \frac{6 \pm \sqrt{48 (- x^{2} - 2 x + 8)}}{2 \cdot 3}$

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 $a \cdot c = - 1 \cdot 8 = - 8$ and whose sum is $b = - 2$ .

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Step 4.1.6.2.1.1

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

$y = \frac{6 \pm \sqrt{48 (- x^{2} - 2 x + 8)}}{2 \cdot 3}$

Step 4.1.6.2.1.2

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

$y = \frac{6 \pm \sqrt{48 (- x^{2} + (2 - 4) x + 8)}}{2 \cdot 3}$

Step 4.1.6.2.1.3

Apply the distributive property.

$y = \frac{6 \pm \sqrt{48 (- x^{2} + 2 x - 4 x + 8)}}{2 \cdot 3}$

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.

$y = \frac{6 \pm \sqrt{48 ((- x^{2} + 2 x) - 4 x + 8)}}{2 \cdot 3}$

Step 4.1.6.2.2.2

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

$y = \frac{6 \pm \sqrt{48 (x (- x + 2) + 4 (- x + 2))}}{2 \cdot 3}$

Step 4.1.6.2.3

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

$y = \frac{6 \pm \sqrt{48 ((- x + 2) (x + 4))}}{2 \cdot 3}$

$y = \frac{6 \pm \sqrt{48 (- x + 2) (x + 4)}}{2 \cdot 3}$

Step 4.1.7

Rewrite $48 (- x + 2) (x + 4)$ as ${(2^{2})}^{2} \cdot (3 (- x + 2) (x + 4))$ .

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Step 4.1.7.1

Factor $16$ out of $48$ .

$y = \frac{6 \pm \sqrt{16 (3) (- x + 2) (x + 4)}}{2 \cdot 3}$

Step 4.1.7.2

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

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

Step 4.1.7.3

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

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

Step 4.1.7.4

Add parentheses.

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

Step 4.1.7.5

Add parentheses.

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

Step 4.1.8

Pull terms out from under the radical.

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

Step 4.1.9

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

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

Step 4.2

Multiply $2$ by $3$ .

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

Step 4.3

Simplify $\frac{6 \pm 4 \sqrt{3 (- x + 2) (x + 4)}}{6}$ .

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

Step 4.4

Change the $\pm$ to $+$ .

$y = \frac{3 + 2 \sqrt{3 (- x + 2) (x + 4)}}{3}$

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 $2$ .

$y = \frac{6 \pm \sqrt{36 - 4 \cdot 3 \cdot (4 x^{2} + 8 x - 29)}}{2 \cdot 3}$

Step 5.1.2

Multiply $- 4$ by $3$ .

$y = \frac{6 \pm \sqrt{36 - 12 \cdot (4 x^{2} + 8 x - 29)}}{2 \cdot 3}$

Step 5.1.3

Apply the distributive property.

$y = \frac{6 \pm \sqrt{36 - 12 (4 x^{2}) - 12 (8 x) - 12 \cdot - 29}}{2 \cdot 3}$

Step 5.1.4

Simplify.

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Step 5.1.4.1

Multiply $4$ by $- 12$ .

$y = \frac{6 \pm \sqrt{36 - 48 x^{2} - 12 (8 x) - 12 \cdot - 29}}{2 \cdot 3}$

Step 5.1.4.2

Multiply $8$ by $- 12$ .

$y = \frac{6 \pm \sqrt{36 - 48 x^{2} - 96 x - 12 \cdot - 29}}{2 \cdot 3}$

Step 5.1.4.3

Multiply $- 12$ by $- 29$ .

$y = \frac{6 \pm \sqrt{36 - 48 x^{2} - 96 x + 348}}{2 \cdot 3}$

Step 5.1.5

Add $36$ and $348$ .

$y = \frac{6 \pm \sqrt{- 48 x^{2} - 96 x + 384}}{2 \cdot 3}$

Step 5.1.6

Rewrite $- 48 x^{2} - 96 x + 384$ in a factored form.

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Step 5.1.6.1

Factor $48$ out of $- 48 x^{2} - 96 x + 384$ .

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Step 5.1.6.1.1

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

$y = \frac{6 \pm \sqrt{48 (- x^{2}) - 96 x + 384}}{2 \cdot 3}$

Step 5.1.6.1.2

Factor $48$ out of $- 96 x$ .

$y = \frac{6 \pm \sqrt{48 (- x^{2}) + 48 (- 2 x) + 384}}{2 \cdot 3}$

Step 5.1.6.1.3

Factor $48$ out of $384$ .

$y = \frac{6 \pm \sqrt{48 (- x^{2}) + 48 (- 2 x) + 48 (8)}}{2 \cdot 3}$

Step 5.1.6.1.4

Factor $48$ out of $48 (- x^{2}) + 48 (- 2 x)$ .

$y = \frac{6 \pm \sqrt{48 (- x^{2} - 2 x) + 48 (8)}}{2 \cdot 3}$

Step 5.1.6.1.5

Factor $48$ out of $48 (- x^{2} - 2 x) + 48 (8)$ .

$y = \frac{6 \pm \sqrt{48 (- x^{2} - 2 x + 8)}}{2 \cdot 3}$

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 $a \cdot c = - 1 \cdot 8 = - 8$ and whose sum is $b = - 2$ .

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Step 5.1.6.2.1.1

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

$y = \frac{6 \pm \sqrt{48 (- x^{2} - 2 x + 8)}}{2 \cdot 3}$

Step 5.1.6.2.1.2

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

$y = \frac{6 \pm \sqrt{48 (- x^{2} + (2 - 4) x + 8)}}{2 \cdot 3}$

Step 5.1.6.2.1.3

Apply the distributive property.

$y = \frac{6 \pm \sqrt{48 (- x^{2} + 2 x - 4 x + 8)}}{2 \cdot 3}$

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.

$y = \frac{6 \pm \sqrt{48 ((- x^{2} + 2 x) - 4 x + 8)}}{2 \cdot 3}$

Step 5.1.6.2.2.2

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

$y = \frac{6 \pm \sqrt{48 (x (- x + 2) + 4 (- x + 2))}}{2 \cdot 3}$

Step 5.1.6.2.3

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

$y = \frac{6 \pm \sqrt{48 ((- x + 2) (x + 4))}}{2 \cdot 3}$

$y = \frac{6 \pm \sqrt{48 (- x + 2) (x + 4)}}{2 \cdot 3}$

Step 5.1.7

Rewrite $48 (- x + 2) (x + 4)$ as ${(2^{2})}^{2} \cdot (3 (- x + 2) (x + 4))$ .

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Step 5.1.7.1

Factor $16$ out of $48$ .

$y = \frac{6 \pm \sqrt{16 (3) (- x + 2) (x + 4)}}{2 \cdot 3}$

Step 5.1.7.2

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

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

Step 5.1.7.3

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

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

Step 5.1.7.4

Add parentheses.

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

Step 5.1.7.5

Add parentheses.

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

Step 5.1.8

Pull terms out from under the radical.

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

Step 5.1.9

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

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

Step 5.2

Multiply $2$ by $3$ .

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

Step 5.3

Simplify $\frac{6 \pm 4 \sqrt{3 (- x + 2) (x + 4)}}{6}$ .

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

Step 5.4

Change the $\pm$ to $-$ .

$y = \frac{3 - 2 \sqrt{3 (- x + 2) (x + 4)}}{3}$

Step 6

The final answer is the combination of both solutions.

$y = \frac{3 + 2 \sqrt{3 (- x + 2) (x + 4)}}{3}$

$y = \frac{3 - 2 \sqrt{3 (- x + 2) (x + 4)}}{3}$

Step 7

Interchange the variables. Create an equation for each expression.

$x = \frac{3 + 2 \sqrt{3 (- y + 2) (y + 4)}}{3}, x = \frac{3 - 2 \sqrt{3 (- y + 2) (y + 4)}}{3}$

Step 8

Solve for $y$ .

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Step 8.1

Rewrite the equation as $\frac{3 + 2 \sqrt{3 (- y + 2) (y + 4)}}{3} = x$ .

$\frac{3 + 2 \sqrt{3 (- y + 2) (y + 4)}}{3} = x$

Step 8.2

Multiply both sides by $3$ .

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

Step 8.3

Simplify.

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Step 8.3.1

Simplify the left side.

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Step 8.3.1.1

Simplify $\frac{3 + 2 \sqrt{3 (- y + 2) (y + 4)}}{3} \cdot 3$ .

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Step 8.3.1.1.1

Cancel the common factor of $3$ .

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Step 8.3.1.1.1.1

Cancel the common factor.

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

Step 8.3.1.1.1.2

Rewrite the expression.

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

Step 8.3.1.1.2

Reorder $3$ and $2 \sqrt{3 (- y + 2) (y + 4)}$ .

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

Step 8.3.2

Simplify the right side.

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Step 8.3.2.1

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

$2 \sqrt{3 (- y + 2) (y + 4)} + 3 = 3 x$

Step 8.4

Solve for $y$ .

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Step 8.4.1

Subtract $3$ from both sides of the equation.

$2 \sqrt{3 (- y + 2) (y + 4)} = 3 x - 3$

Step 8.4.2

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

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

Step 8.4.3

Simplify each side of the equation.

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Step 8.4.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 (- y + 2) (y + 4)}$ as ${(3 (- y + 2) (y + 4))}^{\frac{1}{2}}$ .

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

Step 8.4.3.2

Simplify the left side.

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Step 8.4.3.2.1

Simplify ${(2 {(3 (- y + 2) (y + 4))}^{\frac{1}{2}})}^{2}$ .

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Step 8.4.3.2.1.1

Simplify by multiplying through.

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Step 8.4.3.2.1.1.1

Apply the distributive property.

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

Step 8.4.3.2.1.1.2

Multiply.

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Step 8.4.3.2.1.1.2.1

Multiply $- 1$ by $3$ .

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

Step 8.4.3.2.1.1.2.2

Multiply $3$ by $2$ .

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

Step 8.4.3.2.1.2

Expand $(- 3 y + 6) (y + 4)$ using the FOIL Method.

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Step 8.4.3.2.1.2.1

Apply the distributive property.

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

Step 8.4.3.2.1.2.2

Apply the distributive property.

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

Step 8.4.3.2.1.2.3

Apply the distributive property.

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

Step 8.4.3.2.1.3

Simplify and combine like terms.

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Step 8.4.3.2.1.3.1

Simplify each term.

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Step 8.4.3.2.1.3.1.1

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

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Step 8.4.3.2.1.3.1.1.1

Move $y$ .

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

Step 8.4.3.2.1.3.1.1.2

Multiply $y$ by $y$ .

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

Step 8.4.3.2.1.3.1.2

Multiply $4$ by $- 3$ .

${(2 {(- 3 y^{2} - 12 y + 6 y + 6 \cdot 4)}^{\frac{1}{2}})}^{2} = {(3 x - 3)}^{2}$

Step 8.4.3.2.1.3.1.3

Multiply $6$ by $4$ .

${(2 {(- 3 y^{2} - 12 y + 6 y + 24)}^{\frac{1}{2}})}^{2} = {(3 x - 3)}^{2}$

Step 8.4.3.2.1.3.2

Add $- 12 y$ and $6 y$ .

${(2 {(- 3 y^{2} - 6 y + 24)}^{\frac{1}{2}})}^{2} = {(3 x - 3)}^{2}$

Step 8.4.3.2.1.4

Apply the product rule to $2 {(- 3 y^{2} - 6 y + 24)}^{\frac{1}{2}}$ .

$2^{2} {({(- 3 y^{2} - 6 y + 24)}^{\frac{1}{2}})}^{2} = {(3 x - 3)}^{2}$

Step 8.4.3.2.1.5

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

$4 {({(- 3 y^{2} - 6 y + 24)}^{\frac{1}{2}})}^{2} = {(3 x - 3)}^{2}$

Step 8.4.3.2.1.6

Multiply the exponents in ${({(- 3 y^{2} - 6 y + 24)}^{\frac{1}{2}})}^{2}$ .

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Step 8.4.3.2.1.6.1

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

$4 {(- 3 y^{2} - 6 y + 24)}^{\frac{1}{2} \cdot 2} = {(3 x - 3)}^{2}$

Step 8.4.3.2.1.6.2

Cancel the common factor of $2$ .

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Step 8.4.3.2.1.6.2.1

Cancel the common factor.

$4 {(- 3 y^{2} - 6 y + 24)}^{\frac{1}{2} \cdot 2} = {(3 x - 3)}^{2}$

Step 8.4.3.2.1.6.2.2

Rewrite the expression.

$4 {(- 3 y^{2} - 6 y + 24)}^{1} = {(3 x - 3)}^{2}$

Step 8.4.3.2.1.7

Simplify.

$4 (- 3 y^{2} - 6 y + 24) = {(3 x - 3)}^{2}$

Step 8.4.3.2.1.8

Apply the distributive property.

$4 (- 3 y^{2}) + 4 (- 6 y) + 4 \cdot 24 = {(3 x - 3)}^{2}$

Step 8.4.3.2.1.9

Simplify.

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Step 8.4.3.2.1.9.1

Multiply $- 3$ by $4$ .

$- 12 y^{2} + 4 (- 6 y) + 4 \cdot 24 = {(3 x - 3)}^{2}$

Step 8.4.3.2.1.9.2

Multiply $- 6$ by $4$ .

$- 12 y^{2} - 24 y + 4 \cdot 24 = {(3 x - 3)}^{2}$

Step 8.4.3.2.1.9.3

Multiply $4$ by $24$ .

$- 12 y^{2} - 24 y + 96 = {(3 x - 3)}^{2}$

Step 8.4.3.3

Simplify the right side.

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Step 8.4.3.3.1

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

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Step 8.4.3.3.1.1

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

$- 12 y^{2} - 24 y + 96 = (3 x - 3) (3 x - 3)$

Step 8.4.3.3.1.2

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

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Step 8.4.3.3.1.2.1

Apply the distributive property.

$- 12 y^{2} - 24 y + 96 = 3 x (3 x - 3) - 3 (3 x - 3)$

Step 8.4.3.3.1.2.2

Apply the distributive property.

$- 12 y^{2} - 24 y + 96 = 3 x (3 x) + 3 x \cdot - 3 - 3 (3 x - 3)$

Step 8.4.3.3.1.2.3

Apply the distributive property.

$- 12 y^{2} - 24 y + 96 = 3 x (3 x) + 3 x \cdot - 3 - 3 (3 x) - 3 \cdot - 3$

Step 8.4.3.3.1.3

Simplify and combine like terms.

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Step 8.4.3.3.1.3.1

Simplify each term.

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Step 8.4.3.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$- 12 y^{2} - 24 y + 96 = 3 \cdot 3 x \cdot x + 3 x \cdot - 3 - 3 (3 x) - 3 \cdot - 3$

Step 8.4.3.3.1.3.1.2

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

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Step 8.4.3.3.1.3.1.2.1

Move $x$ .

$- 12 y^{2} - 24 y + 96 = 3 \cdot 3 (x \cdot x) + 3 x \cdot - 3 - 3 (3 x) - 3 \cdot - 3$

Step 8.4.3.3.1.3.1.2.2

Multiply $x$ by $x$ .

$- 12 y^{2} - 24 y + 96 = 3 \cdot 3 x^{2} + 3 x \cdot - 3 - 3 (3 x) - 3 \cdot - 3$

Step 8.4.3.3.1.3.1.3

Multiply $3$ by $3$ .

$- 12 y^{2} - 24 y + 96 = 9 x^{2} + 3 x \cdot - 3 - 3 (3 x) - 3 \cdot - 3$

Step 8.4.3.3.1.3.1.4

Multiply $- 3$ by $3$ .

$- 12 y^{2} - 24 y + 96 = 9 x^{2} - 9 x - 3 (3 x) - 3 \cdot - 3$

Step 8.4.3.3.1.3.1.5

Multiply $3$ by $- 3$ .

$- 12 y^{2} - 24 y + 96 = 9 x^{2} - 9 x - 9 x - 3 \cdot - 3$

Step 8.4.3.3.1.3.1.6

Multiply $- 3$ by $- 3$ .

$- 12 y^{2} - 24 y + 96 = 9 x^{2} - 9 x - 9 x + 9$

Step 8.4.3.3.1.3.2

Subtract $9 x$ from $- 9 x$ .

$- 12 y^{2} - 24 y + 96 = 9 x^{2} - 18 x + 9$

Step 8.4.4

Solve for $y$ .

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Step 8.4.4.1

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

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Step 8.4.4.1.1

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

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Step 8.4.4.1.1.1

Subtract $9 x^{2}$ from both sides of the equation.

$- 12 y^{2} - 24 y + 96 - 9 x^{2} = - 18 x + 9$

Step 8.4.4.1.1.2

Add $18 x$ to both sides of the equation.

$- 12 y^{2} - 24 y + 96 - 9 x^{2} + 18 x = 9$

Step 8.4.4.1.1.3

Subtract $9$ from both sides of the equation.

$- 12 y^{2} - 24 y + 96 - 9 x^{2} + 18 x - 9 = 0$

Step 8.4.4.1.2

Subtract $9$ from $96$ .

$- 12 y^{2} - 24 y - 9 x^{2} + 18 x + 87 = 0$

Step 8.4.4.2

Use the quadratic formula to find the solutions.

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

Step 8.4.4.3

Substitute the values $a = - 12$ , $b = - 24$ , and $c = - 9 x^{2} + 18 x + 87$ into the quadratic formula and solve for $y$ .

$\frac{24 \pm \sqrt{{(- 24)}^{2} - 4 \cdot (- 12 \cdot (- 9 x^{2} + 18 x + 87))}}{2 \cdot - 12}$

Step 8.4.4.4

Simplify.

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Step 8.4.4.4.1

Simplify the numerator.

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Step 8.4.4.4.1.1

Raise $- 24$ to the power of $2$ .

$y = \frac{24 \pm \sqrt{576 - 4 \cdot - 12 \cdot (- 9 x^{2} + 18 x + 87)}}{2 \cdot - 12}$

Step 8.4.4.4.1.2

Multiply $- 4$ by $- 12$ .

$y = \frac{24 \pm \sqrt{576 + 48 \cdot (- 9 x^{2} + 18 x + 87)}}{2 \cdot - 12}$

Step 8.4.4.4.1.3

Apply the distributive property.

$y = \frac{24 \pm \sqrt{576 + 48 (- 9 x^{2}) + 48 (18 x) + 48 \cdot 87}}{2 \cdot - 12}$

Step 8.4.4.4.1.4

Simplify.

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Step 8.4.4.4.1.4.1

Multiply $- 9$ by $48$ .

$y = \frac{24 \pm \sqrt{576 - 432 x^{2} + 48 (18 x) + 48 \cdot 87}}{2 \cdot - 12}$

Step 8.4.4.4.1.4.2

Multiply $18$ by $48$ .

$y = \frac{24 \pm \sqrt{576 - 432 x^{2} + 864 x + 48 \cdot 87}}{2 \cdot - 12}$

Step 8.4.4.4.1.4.3

Multiply $48$ by $87$ .

$y = \frac{24 \pm \sqrt{576 - 432 x^{2} + 864 x + 4176}}{2 \cdot - 12}$

Step 8.4.4.4.1.5

Add $576$ and $4176$ .

$y = \frac{24 \pm \sqrt{- 432 x^{2} + 864 x + 4752}}{2 \cdot - 12}$

Step 8.4.4.4.1.6

Factor $432$ out of $- 432 x^{2} + 864 x + 4752$ .

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Step 8.4.4.4.1.6.1

Factor $432$ out of $- 432 x^{2}$ .

$y = \frac{24 \pm \sqrt{432 (- x^{2}) + 864 x + 4752}}{2 \cdot - 12}$

Step 8.4.4.4.1.6.2

Factor $432$ out of $864 x$ .

$y = \frac{24 \pm \sqrt{432 (- x^{2}) + 432 (2 x) + 4752}}{2 \cdot - 12}$

Step 8.4.4.4.1.6.3

Factor $432$ out of $4752$ .

$y = \frac{24 \pm \sqrt{432 (- x^{2}) + 432 (2 x) + 432 (11)}}{2 \cdot - 12}$

Step 8.4.4.4.1.6.4

Factor $432$ out of $432 (- x^{2}) + 432 (2 x)$ .

$y = \frac{24 \pm \sqrt{432 (- x^{2} + 2 x) + 432 (11)}}{2 \cdot - 12}$

Step 8.4.4.4.1.6.5

Factor $432$ out of $432 (- x^{2} + 2 x) + 432 (11)$ .

$y = \frac{24 \pm \sqrt{432 (- x^{2} + 2 x + 11)}}{2 \cdot - 12}$

Step 8.4.4.4.1.7

Rewrite $432 (- x^{2} + 2 x + 11)$ as $12^{2} \cdot (3 (- x^{2} + 2 x + 11))$ .

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Step 8.4.4.4.1.7.1

Factor $144$ out of $432$ .

$y = \frac{24 \pm \sqrt{144 (3) (- x^{2} + 2 x + 11)}}{2 \cdot - 12}$

Step 8.4.4.4.1.7.2

Rewrite $144$ as $12^{2}$ .

$y = \frac{24 \pm \sqrt{12^{2} \cdot (3 (- x^{2} + 2 x + 11))}}{2 \cdot - 12}$

Step 8.4.4.4.1.7.3

Add parentheses.

$y = \frac{24 \pm \sqrt{12^{2} \cdot (3 (- x^{2} + 2 x + 11))}}{2 \cdot - 12}$

Step 8.4.4.4.1.8

Pull terms out from under the radical.

$y = \frac{24 \pm 12 \sqrt{3 (- x^{2} + 2 x + 11)}}{2 \cdot - 12}$

Step 8.4.4.4.2

Multiply $2$ by $- 12$ .

$y = \frac{24 \pm 12 \sqrt{3 (- x^{2} + 2 x + 11)}}{- 24}$

Step 8.4.4.4.3

Simplify $\frac{24 \pm 12 \sqrt{3 (- x^{2} + 2 x + 11)}}{- 24}$ .

$y = \frac{2 \pm \sqrt{3 (- x^{2} + 2 x + 11)}}{- 2}$

Step 8.4.4.4.4

Move the negative in front of the fraction.

$y = - \frac{2 \pm \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$

Step 8.4.4.5

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

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Step 8.4.4.5.1

Simplify the numerator.

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Step 8.4.4.5.1.1

Raise $- 24$ to the power of $2$ .

$y = \frac{24 \pm \sqrt{576 - 4 \cdot - 12 \cdot (- 9 x^{2} + 18 x + 87)}}{2 \cdot - 12}$

Step 8.4.4.5.1.2

Multiply $- 4$ by $- 12$ .

$y = \frac{24 \pm \sqrt{576 + 48 \cdot (- 9 x^{2} + 18 x + 87)}}{2 \cdot - 12}$

Step 8.4.4.5.1.3

Apply the distributive property.

$y = \frac{24 \pm \sqrt{576 + 48 (- 9 x^{2}) + 48 (18 x) + 48 \cdot 87}}{2 \cdot - 12}$

Step 8.4.4.5.1.4

Simplify.

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Step 8.4.4.5.1.4.1

Multiply $- 9$ by $48$ .

$y = \frac{24 \pm \sqrt{576 - 432 x^{2} + 48 (18 x) + 48 \cdot 87}}{2 \cdot - 12}$

Step 8.4.4.5.1.4.2

Multiply $18$ by $48$ .

$y = \frac{24 \pm \sqrt{576 - 432 x^{2} + 864 x + 48 \cdot 87}}{2 \cdot - 12}$

Step 8.4.4.5.1.4.3

Multiply $48$ by $87$ .

$y = \frac{24 \pm \sqrt{576 - 432 x^{2} + 864 x + 4176}}{2 \cdot - 12}$

Step 8.4.4.5.1.5

Add $576$ and $4176$ .

$y = \frac{24 \pm \sqrt{- 432 x^{2} + 864 x + 4752}}{2 \cdot - 12}$

Step 8.4.4.5.1.6

Factor $432$ out of $- 432 x^{2} + 864 x + 4752$ .

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Step 8.4.4.5.1.6.1

Factor $432$ out of $- 432 x^{2}$ .

$y = \frac{24 \pm \sqrt{432 (- x^{2}) + 864 x + 4752}}{2 \cdot - 12}$

Step 8.4.4.5.1.6.2

Factor $432$ out of $864 x$ .

$y = \frac{24 \pm \sqrt{432 (- x^{2}) + 432 (2 x) + 4752}}{2 \cdot - 12}$

Step 8.4.4.5.1.6.3

Factor $432$ out of $4752$ .

$y = \frac{24 \pm \sqrt{432 (- x^{2}) + 432 (2 x) + 432 (11)}}{2 \cdot - 12}$

Step 8.4.4.5.1.6.4

Factor $432$ out of $432 (- x^{2}) + 432 (2 x)$ .

$y = \frac{24 \pm \sqrt{432 (- x^{2} + 2 x) + 432 (11)}}{2 \cdot - 12}$

Step 8.4.4.5.1.6.5

Factor $432$ out of $432 (- x^{2} + 2 x) + 432 (11)$ .

$y = \frac{24 \pm \sqrt{432 (- x^{2} + 2 x + 11)}}{2 \cdot - 12}$

Step 8.4.4.5.1.7

Rewrite $432 (- x^{2} + 2 x + 11)$ as $12^{2} \cdot (3 (- x^{2} + 2 x + 11))$ .

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Step 8.4.4.5.1.7.1

Factor $144$ out of $432$ .

$y = \frac{24 \pm \sqrt{144 (3) (- x^{2} + 2 x + 11)}}{2 \cdot - 12}$

Step 8.4.4.5.1.7.2

Rewrite $144$ as $12^{2}$ .

$y = \frac{24 \pm \sqrt{12^{2} \cdot (3 (- x^{2} + 2 x + 11))}}{2 \cdot - 12}$

Step 8.4.4.5.1.7.3

Add parentheses.

$y = \frac{24 \pm \sqrt{12^{2} \cdot (3 (- x^{2} + 2 x + 11))}}{2 \cdot - 12}$

Step 8.4.4.5.1.8

Pull terms out from under the radical.

$y = \frac{24 \pm 12 \sqrt{3 (- x^{2} + 2 x + 11)}}{2 \cdot - 12}$

Step 8.4.4.5.2

Multiply $2$ by $- 12$ .

$y = \frac{24 \pm 12 \sqrt{3 (- x^{2} + 2 x + 11)}}{- 24}$

Step 8.4.4.5.3

Simplify $\frac{24 \pm 12 \sqrt{3 (- x^{2} + 2 x + 11)}}{- 24}$ .

$y = \frac{2 \pm \sqrt{3 (- x^{2} + 2 x + 11)}}{- 2}$

Step 8.4.4.5.4

Move the negative in front of the fraction.

$y = - \frac{2 \pm \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$

Step 8.4.4.5.5

Change the $\pm$ to $+$ .

$y = - \frac{2 + \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$

Step 8.4.4.6

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

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Step 8.4.4.6.1

Simplify the numerator.

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Step 8.4.4.6.1.1

Raise $- 24$ to the power of $2$ .

$y = \frac{24 \pm \sqrt{576 - 4 \cdot - 12 \cdot (- 9 x^{2} + 18 x + 87)}}{2 \cdot - 12}$

Step 8.4.4.6.1.2

Multiply $- 4$ by $- 12$ .

$y = \frac{24 \pm \sqrt{576 + 48 \cdot (- 9 x^{2} + 18 x + 87)}}{2 \cdot - 12}$

Step 8.4.4.6.1.3

Apply the distributive property.

$y = \frac{24 \pm \sqrt{576 + 48 (- 9 x^{2}) + 48 (18 x) + 48 \cdot 87}}{2 \cdot - 12}$

Step 8.4.4.6.1.4

Simplify.

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Step 8.4.4.6.1.4.1

Multiply $- 9$ by $48$ .

$y = \frac{24 \pm \sqrt{576 - 432 x^{2} + 48 (18 x) + 48 \cdot 87}}{2 \cdot - 12}$

Step 8.4.4.6.1.4.2

Multiply $18$ by $48$ .

$y = \frac{24 \pm \sqrt{576 - 432 x^{2} + 864 x + 48 \cdot 87}}{2 \cdot - 12}$

Step 8.4.4.6.1.4.3

Multiply $48$ by $87$ .

$y = \frac{24 \pm \sqrt{576 - 432 x^{2} + 864 x + 4176}}{2 \cdot - 12}$

Step 8.4.4.6.1.5

Add $576$ and $4176$ .

$y = \frac{24 \pm \sqrt{- 432 x^{2} + 864 x + 4752}}{2 \cdot - 12}$

Step 8.4.4.6.1.6

Factor $432$ out of $- 432 x^{2} + 864 x + 4752$ .

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Step 8.4.4.6.1.6.1

Factor $432$ out of $- 432 x^{2}$ .

$y = \frac{24 \pm \sqrt{432 (- x^{2}) + 864 x + 4752}}{2 \cdot - 12}$

Step 8.4.4.6.1.6.2

Factor $432$ out of $864 x$ .

$y = \frac{24 \pm \sqrt{432 (- x^{2}) + 432 (2 x) + 4752}}{2 \cdot - 12}$

Step 8.4.4.6.1.6.3

Factor $432$ out of $4752$ .

$y = \frac{24 \pm \sqrt{432 (- x^{2}) + 432 (2 x) + 432 (11)}}{2 \cdot - 12}$

Step 8.4.4.6.1.6.4

Factor $432$ out of $432 (- x^{2}) + 432 (2 x)$ .

$y = \frac{24 \pm \sqrt{432 (- x^{2} + 2 x) + 432 (11)}}{2 \cdot - 12}$

Step 8.4.4.6.1.6.5

Factor $432$ out of $432 (- x^{2} + 2 x) + 432 (11)$ .

$y = \frac{24 \pm \sqrt{432 (- x^{2} + 2 x + 11)}}{2 \cdot - 12}$

Step 8.4.4.6.1.7

Rewrite $432 (- x^{2} + 2 x + 11)$ as $12^{2} \cdot (3 (- x^{2} + 2 x + 11))$ .

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Step 8.4.4.6.1.7.1

Factor $144$ out of $432$ .

$y = \frac{24 \pm \sqrt{144 (3) (- x^{2} + 2 x + 11)}}{2 \cdot - 12}$

Step 8.4.4.6.1.7.2

Rewrite $144$ as $12^{2}$ .

$y = \frac{24 \pm \sqrt{12^{2} \cdot (3 (- x^{2} + 2 x + 11))}}{2 \cdot - 12}$

Step 8.4.4.6.1.7.3

Add parentheses.

$y = \frac{24 \pm \sqrt{12^{2} \cdot (3 (- x^{2} + 2 x + 11))}}{2 \cdot - 12}$

Step 8.4.4.6.1.8

Pull terms out from under the radical.

$y = \frac{24 \pm 12 \sqrt{3 (- x^{2} + 2 x + 11)}}{2 \cdot - 12}$

Step 8.4.4.6.2

Multiply $2$ by $- 12$ .

$y = \frac{24 \pm 12 \sqrt{3 (- x^{2} + 2 x + 11)}}{- 24}$

Step 8.4.4.6.3

Simplify $\frac{24 \pm 12 \sqrt{3 (- x^{2} + 2 x + 11)}}{- 24}$ .

$y = \frac{2 \pm \sqrt{3 (- x^{2} + 2 x + 11)}}{- 2}$

Step 8.4.4.6.4

Move the negative in front of the fraction.

$y = - \frac{2 \pm \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$

Step 8.4.4.6.5

Change the $\pm$ to $-$ .

$y = - \frac{2 - \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$

Step 8.4.4.7

The final answer is the combination of both solutions.

$y = - \frac{2 + \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$

$y = - \frac{2 - \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$

$y = - \frac{2 + \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$

$y = - \frac{2 - \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$

$y = - \frac{2 + \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$

$y = - \frac{2 - \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$

$y = - \frac{2 + \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$

$y = - \frac{2 - \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$

Step 9

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

$f^{- 1} (x) = - \frac{2 + \sqrt{3 (- x^{2} + 2 x + 11)}}{2}, - \frac{2 - \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$

Step 10

Verify if $f^{- 1} (x) = - \frac{2 + \sqrt{3 (- x^{2} + 2 x + 11)}}{2}, - \frac{2 - \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$ is the inverse of $f (x) = \frac{3 + 2 \sqrt{3 (- x + 2) (x + 4)}}{3}, \frac{3 - 2 \sqrt{3 (- x + 2) (x + 4)}}{3}$ .

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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 $f (x) = \frac{3 + 2 \sqrt{3 (- x + 2) (x + 4)}}{3}, \frac{3 - 2 \sqrt{3 (- x + 2) (x + 4)}}{3}$ and $f^{- 1} (x) = - \frac{2 + \sqrt{3 (- x^{2} + 2 x + 11)}}{2}, - \frac{2 - \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$ and compare them.

Step 10.2

Find the range of $f (x) = \frac{3 + 2 \sqrt{3 (- x + 2) (x + 4)}}{3}, \frac{3 - 2 \sqrt{3 (- x + 2) (x + 4)}}{3}$ .

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Step 10.2.1

Find the range of $\frac{3 + 2 \sqrt{3 (- x + 2) (x + 4)}}{3}$ .

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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:

$[1, 1 + 2 \sqrt{3}]$

Step 10.2.2

Find the range of $\frac{3 - 2 \sqrt{3 (- x + 2) (x + 4)}}{3}$ .

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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:

$[1 - 2 \sqrt{3}, 1]$

Step 10.2.3

Find the union of $[1, 1 + 2 \sqrt{3}], [1 - 2 \sqrt{3}, 1]$ .

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Step 10.2.3.1

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

$[1 - 2 \sqrt{3}, 1 + 2 \sqrt{3}]$

Step 10.3

Find the domain of $- \frac{2 + \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$ .

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Step 10.3.1

Set the radicand in $\sqrt{3 (- x^{2} + 2 x + 11)}$ greater than or equal to $0$ to find where the expression is defined.

$3 (- x^{2} + 2 x + 11) \geq 0$

Step 10.3.2

Solve for $x$ .

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Step 10.3.2.1

Divide each term in $3 (- x^{2} + 2 x + 11) \geq 0$ by $3$ and simplify.

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Step 10.3.2.1.1

Divide each term in $3 (- x^{2} + 2 x + 11) \geq 0$ by $3$ .

$\frac{3 (- x^{2} + 2 x + 11)}{3} \geq \frac{0}{3}$

Step 10.3.2.1.2

Simplify the left side.

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Step 10.3.2.1.2.1

Cancel the common factor of $3$ .

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Step 10.3.2.1.2.1.1

Cancel the common factor.

$\frac{3 (- x^{2} + 2 x + 11)}{3} \geq \frac{0}{3}$

Step 10.3.2.1.2.1.2

Divide $- x^{2} + 2 x + 11$ by $1$ .

$- x^{2} + 2 x + 11 \geq \frac{0}{3}$

Step 10.3.2.1.3

Simplify the right side.

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Step 10.3.2.1.3.1

Divide $0$ by $3$ .

$- x^{2} + 2 x + 11 \geq 0$

Step 10.3.2.2

Convert the inequality to an equation.

$- x^{2} + 2 x + 11 = 0$

Step 10.3.2.3

Use the quadratic formula to find the solutions.

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

Step 10.3.2.4

Substitute the values $a = - 1$ , $b = 2$ , and $c = 11$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (- 1 \cdot 11)}}{2 \cdot - 1}$

Step 10.3.2.5

Simplify.

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Step 10.3.2.5.1

Simplify the numerator.

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Step 10.3.2.5.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 1 \cdot 11}}{2 \cdot - 1}$

Step 10.3.2.5.1.2

Multiply $- 4 \cdot - 1 \cdot 11$ .

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Step 10.3.2.5.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 2 \pm \sqrt{4 + 4 \cdot 11}}{2 \cdot - 1}$

Step 10.3.2.5.1.2.2

Multiply $4$ by $11$ .

$x = \frac{- 2 \pm \sqrt{4 + 44}}{2 \cdot - 1}$

Step 10.3.2.5.1.3

Add $4$ and $44$ .

$x = \frac{- 2 \pm \sqrt{48}}{2 \cdot - 1}$

Step 10.3.2.5.1.4

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 10.3.2.5.1.4.1

Factor $16$ out of $48$ .

$x = \frac{- 2 \pm \sqrt{16 (3)}}{2 \cdot - 1}$

Step 10.3.2.5.1.4.2

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

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

Step 10.3.2.5.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 4 \sqrt{3}}{2 \cdot - 1}$

Step 10.3.2.5.2

Multiply $2$ by $- 1$ .

$x = \frac{- 2 \pm 4 \sqrt{3}}{- 2}$

Step 10.3.2.5.3

Simplify $\frac{- 2 \pm 4 \sqrt{3}}{- 2}$ .

$x = 1 \pm 2 \sqrt{3}$

Step 10.3.2.6

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

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Step 10.3.2.6.1

Simplify the numerator.

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Step 10.3.2.6.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 1 \cdot 11}}{2 \cdot - 1}$

Step 10.3.2.6.1.2

Multiply $- 4 \cdot - 1 \cdot 11$ .

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Step 10.3.2.6.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 2 \pm \sqrt{4 + 4 \cdot 11}}{2 \cdot - 1}$

Step 10.3.2.6.1.2.2

Multiply $4$ by $11$ .

$x = \frac{- 2 \pm \sqrt{4 + 44}}{2 \cdot - 1}$

Step 10.3.2.6.1.3

Add $4$ and $44$ .

$x = \frac{- 2 \pm \sqrt{48}}{2 \cdot - 1}$

Step 10.3.2.6.1.4

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 10.3.2.6.1.4.1

Factor $16$ out of $48$ .

$x = \frac{- 2 \pm \sqrt{16 (3)}}{2 \cdot - 1}$

Step 10.3.2.6.1.4.2

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

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

Step 10.3.2.6.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 4 \sqrt{3}}{2 \cdot - 1}$

Step 10.3.2.6.2

Multiply $2$ by $- 1$ .

$x = \frac{- 2 \pm 4 \sqrt{3}}{- 2}$

Step 10.3.2.6.3

Simplify $\frac{- 2 \pm 4 \sqrt{3}}{- 2}$ .

$x = 1 \pm 2 \sqrt{3}$

Step 10.3.2.6.4

Change the $\pm$ to $+$ .

$x = 1 + 2 \sqrt{3}$

Step 10.3.2.7

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

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Step 10.3.2.7.1

Simplify the numerator.

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Step 10.3.2.7.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 1 \cdot 11}}{2 \cdot - 1}$

Step 10.3.2.7.1.2

Multiply $- 4 \cdot - 1 \cdot 11$ .

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Step 10.3.2.7.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 2 \pm \sqrt{4 + 4 \cdot 11}}{2 \cdot - 1}$

Step 10.3.2.7.1.2.2

Multiply $4$ by $11$ .

$x = \frac{- 2 \pm \sqrt{4 + 44}}{2 \cdot - 1}$

Step 10.3.2.7.1.3

Add $4$ and $44$ .

$x = \frac{- 2 \pm \sqrt{48}}{2 \cdot - 1}$

Step 10.3.2.7.1.4

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 10.3.2.7.1.4.1

Factor $16$ out of $48$ .

$x = \frac{- 2 \pm \sqrt{16 (3)}}{2 \cdot - 1}$

Step 10.3.2.7.1.4.2

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

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

Step 10.3.2.7.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 4 \sqrt{3}}{2 \cdot - 1}$

Step 10.3.2.7.2

Multiply $2$ by $- 1$ .

$x = \frac{- 2 \pm 4 \sqrt{3}}{- 2}$

Step 10.3.2.7.3

Simplify $\frac{- 2 \pm 4 \sqrt{3}}{- 2}$ .

$x = 1 \pm 2 \sqrt{3}$

Step 10.3.2.7.4

Change the $\pm$ to $-$ .

$x = 1 - 2 \sqrt{3}$

Step 10.3.2.8

Consolidate the solutions.

$x = 1 + 2 \sqrt{3}, 1 - 2 \sqrt{3}$

Step 10.3.2.9

Use each root to create test intervals.

$x < 1 - 2 \sqrt{3}$

$1 - 2 \sqrt{3} < x < 1 + 2 \sqrt{3}$

$x > 1 + 2 \sqrt{3}$

Step 10.3.2.10

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 10.3.2.10.1

Test a value on the interval $x < 1 - 2 \sqrt{3}$ to see if it makes the inequality true.

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Step 10.3.2.10.1.1

Choose a value on the interval $x < 1 - 2 \sqrt{3}$ and see if this value makes the original inequality true.

$x = - 5$

Step 10.3.2.10.1.2

Replace $x$ with $- 5$ in the original inequality.

$3 (- {(- 5)}^{2} + 2 (- 5) + 11) \geq 0$

Step 10.3.2.10.1.3

The left side $- 72$ is less than the right side $0$ , which means that the given statement is false.

False

Step 10.3.2.10.2

Test a value on the interval $1 - 2 \sqrt{3} < x < 1 + 2 \sqrt{3}$ to see if it makes the inequality true.

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Step 10.3.2.10.2.1

Choose a value on the interval $1 - 2 \sqrt{3} < x < 1 + 2 \sqrt{3}$ and see if this value makes the original inequality true.

$x = 0$

Step 10.3.2.10.2.2

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

$3 (- {(0)}^{2} + 2 (0) + 11) \geq 0$

Step 10.3.2.10.2.3

The left side $33$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 10.3.2.10.3

Test a value on the interval $x > 1 + 2 \sqrt{3}$ to see if it makes the inequality true.

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Step 10.3.2.10.3.1

Choose a value on the interval $x > 1 + 2 \sqrt{3}$ and see if this value makes the original inequality true.

$x = 7$

Step 10.3.2.10.3.2

Replace $x$ with $7$ in the original inequality.

$3 (- {(7)}^{2} + 2 (7) + 11) \geq 0$

Step 10.3.2.10.3.3

The left side $- 72$ is less than the right side $0$ , which means that the given statement is false.

False

Step 10.3.2.10.4

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

$x < 1 - 2 \sqrt{3}$ False

$1 - 2 \sqrt{3} < x < 1 + 2 \sqrt{3}$ True

$x > 1 + 2 \sqrt{3}$ False

$x < 1 - 2 \sqrt{3}$ False

$1 - 2 \sqrt{3} < x < 1 + 2 \sqrt{3}$ True

$x > 1 + 2 \sqrt{3}$ False

Step 10.3.2.11

The solution consists of all of the true intervals.

$1 - 2 \sqrt{3} \leq x \leq 1 + 2 \sqrt{3}$

Step 10.3.3

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

$[1 - 2 \sqrt{3}, 1 + 2 \sqrt{3}]$

Step 10.4

Find the domain of $\frac{3 + 2 \sqrt{3 (- x + 2) (x + 4)}}{3}$ .

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Step 10.4.1

Set the radicand in $\sqrt{3 (- x + 2) (x + 4)}$ greater than or equal to $0$ to find where the expression is defined.

$3 (- x + 2) (x + 4) \geq 0$

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 $0$ .

$- x + 2 = 0$

$x + 4 = 0$

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 $x + 4$ equal to $0$ and solve for $x$ .

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Step 10.4.2.3.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 10.4.2.3.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 10.4.2.4

The final solution is all the values that make $3 (- x + 2) (x + 4) \geq 0$ true.

$x = 2, - 4$

Step 10.4.2.5

Use each root to create test intervals.

$x < - 4$

$- 4 < x < 2$

$x > 2$

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 $x < - 4$ 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 $x < - 4$ and see if this value makes the original inequality true.

$x = - 6$

Step 10.4.2.6.1.2

Replace $x$ with $- 6$ in the original inequality.

$3 (- (- 6) + 2) ((- 6) + 4) \geq 0$

Step 10.4.2.6.1.3

The left side $- 48$ 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 $- 4 < x < 2$ 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 $- 4 < x < 2$ 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.

$3 (- (0) + 2) ((0) + 4) \geq 0$

Step 10.4.2.6.2.3

The left side $24$ 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.

$3 (- (4) + 2) ((4) + 4) \geq 0$

Step 10.4.2.6.3.3

The left side $- 48$ 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.

$x < - 4$ False

$- 4 < x < 2$ True

$x > 2$ False

$x < - 4$ False

$- 4 < x < 2$ True

$x > 2$ False

Step 10.4.2.7

The solution consists of all of the true intervals.

$- 4 \leq x \leq 2$

Step 10.4.3

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

$[- 4, 2]$

Step 10.5

Since the domain of $f^{- 1} (x) = - \frac{2 + \sqrt{3 (- x^{2} + 2 x + 11)}}{2}, - \frac{2 - \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$ is the range of $f (x) = \frac{3 + 2 \sqrt{3 (- x + 2) (x + 4)}}{3}, \frac{3 - 2 \sqrt{3 (- x + 2) (x + 4)}}{3}$ and the range of $f^{- 1} (x) = - \frac{2 + \sqrt{3 (- x^{2} + 2 x + 11)}}{2}, - \frac{2 - \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$ is the domain of $f (x) = \frac{3 + 2 \sqrt{3 (- x + 2) (x + 4)}}{3}, \frac{3 - 2 \sqrt{3 (- x + 2) (x + 4)}}{3}$ , then $f^{- 1} (x) = - \frac{2 + \sqrt{3 (- x^{2} + 2 x + 11)}}{2}, - \frac{2 - \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$ is the inverse of $f (x) = \frac{3 + 2 \sqrt{3 (- x + 2) (x + 4)}}{3}, \frac{3 - 2 \sqrt{3 (- x + 2) (x + 4)}}{3}$ .

$f^{- 1} (x) = - \frac{2 + \sqrt{3 (- x^{2} + 2 x + 11)}}{2}, - \frac{2 - \sqrt{3 (- x^{2} + 2 x + 11)}}{2}$

Step 11