Precalculus Examples

$9 x^{2} + 16 y^{2} - 36 x + 96 y + 36 = 0$

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 $a = 16$ , $b = 96$ , and $c = 9 x^{2} - 36 x + 36$ into the quadratic formula and solve for $y$ .

$\frac{- 96 \pm \sqrt{96^{2} - 4 \cdot (16 \cdot (9 x^{2} - 36 x + 36))}}{2 \cdot 16}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Rewrite $4 \cdot 16 \cdot (9 x^{2} - 36 x + 36)$ as ${(2 \cdot 4 \cdot (3 x - 6))}^{2}$ .

$y = \frac{- 96 \pm \sqrt{96^{2} - {(2 \cdot 4 \cdot (3 x - 6))}^{2}}}{2 \cdot 16}$

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 $a = 96$ and $b = 2 \cdot 4 \cdot (3 x - 6)$ .

$y = \frac{- 96 \pm \sqrt{(96 + 2 \cdot 4 \cdot (3 x - 6)) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 3.1.3

Simplify.

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Step 3.1.3.1

Multiply $2$ by $4$ .

$y = \frac{- 96 \pm \sqrt{(96 + 8 \cdot (3 x - 6)) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 3.1.3.2

Apply the distributive property.

$y = \frac{- 96 \pm \sqrt{(96 + 8 (3 x) + 8 \cdot - 6) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 3.1.3.3

Multiply $3$ by $8$ .

$y = \frac{- 96 \pm \sqrt{(96 + 24 x + 8 \cdot - 6) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 3.1.3.4

Multiply $8$ by $- 6$ .

$y = \frac{- 96 \pm \sqrt{(96 + 24 x - 48) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 3.1.3.5

Subtract $48$ from $96$ .

$y = \frac{- 96 \pm \sqrt{(24 x + 48) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 3.1.3.6

Factor $24$ out of $24 x + 48$ .

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Step 3.1.3.6.1

Factor $24$ out of $24 x$ .

$y = \frac{- 96 \pm \sqrt{(24 (x) + 48) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 3.1.3.6.2

Factor $24$ out of $48$ .

$y = \frac{- 96 \pm \sqrt{(24 x + 24 \cdot 2) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 3.1.3.6.3

Factor $24$ out of $24 x + 24 \cdot 2$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 3.1.3.7

Combine exponents.

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Step 3.1.3.7.1

Multiply $2$ by $4$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (96 - (8 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 3.1.3.7.2

Multiply $8$ by $- 1$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (96 - 8 (3 x - 6))}}{2 \cdot 16}$

Step 3.1.4

Simplify each term.

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Step 3.1.4.1

Apply the distributive property.

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (96 - 8 (3 x) - 8 \cdot - 6)}}{2 \cdot 16}$

Step 3.1.4.2

Multiply $3$ by $- 8$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (96 - 24 x - 8 \cdot - 6)}}{2 \cdot 16}$

Step 3.1.4.3

Multiply $- 8$ by $- 6$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (96 - 24 x + 48)}}{2 \cdot 16}$

Step 3.1.5

Add $96$ and $48$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (- 24 x + 144)}}{2 \cdot 16}$

Step 3.1.6

Factor $24$ out of $- 24 x + 144$ .

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Step 3.1.6.1

Factor $24$ out of $- 24 x$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (24 (- x) + 144)}}{2 \cdot 16}$

Step 3.1.6.2

Factor $24$ out of $144$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (24 (- x) + 24 (6))}}{2 \cdot 16}$

Step 3.1.6.3

Factor $24$ out of $24 (- x) + 24 (6)$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (24 (- x + 6))}}{2 \cdot 16}$

$y = \frac{- 96 \pm \sqrt{24 (x + 2) \cdot (24 (- x + 6))}}{2 \cdot 16}$

Step 3.1.7

Multiply $24$ by $24$ .

$y = \frac{- 96 \pm \sqrt{576 (x + 2) (- x + 6)}}{2 \cdot 16}$

Step 3.1.8

Rewrite $576 (x + 2) (- x + 6)$ as $24^{2} ((x + 2) (- x + 6))$ .

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Step 3.1.8.1

Rewrite $576$ as $24^{2}$ .

$y = \frac{- 96 \pm \sqrt{24^{2} (x + 2) (- x + 6)}}{2 \cdot 16}$

Step 3.1.8.2

Add parentheses.

$y = \frac{- 96 \pm \sqrt{24^{2} ((x + 2) (- x + 6))}}{2 \cdot 16}$

Step 3.1.9

Pull terms out from under the radical.

$y = \frac{- 96 \pm 24 \sqrt{(x + 2) (- x + 6)}}{2 \cdot 16}$

Step 3.2

Multiply $2$ by $16$ .

$y = \frac{- 96 \pm 24 \sqrt{(x + 2) (- x + 6)}}{32}$

Step 3.3

Simplify $\frac{- 96 \pm 24 \sqrt{(x + 2) (- x + 6)}}{32}$ .

$y = \frac{- 12 \pm 3 \sqrt{(x + 2) (- x + 6)}}{4}$

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 $4 \cdot 16 \cdot (9 x^{2} - 36 x + 36)$ as ${(2 \cdot 4 \cdot (3 x - 6))}^{2}$ .

$y = \frac{- 96 \pm \sqrt{96^{2} - {(2 \cdot 4 \cdot (3 x - 6))}^{2}}}{2 \cdot 16}$

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 $a = 96$ and $b = 2 \cdot 4 \cdot (3 x - 6)$ .

$y = \frac{- 96 \pm \sqrt{(96 + 2 \cdot 4 \cdot (3 x - 6)) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 4.1.3

Simplify.

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Step 4.1.3.1

Multiply $2$ by $4$ .

$y = \frac{- 96 \pm \sqrt{(96 + 8 \cdot (3 x - 6)) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 4.1.3.2

Apply the distributive property.

$y = \frac{- 96 \pm \sqrt{(96 + 8 (3 x) + 8 \cdot - 6) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 4.1.3.3

Multiply $3$ by $8$ .

$y = \frac{- 96 \pm \sqrt{(96 + 24 x + 8 \cdot - 6) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 4.1.3.4

Multiply $8$ by $- 6$ .

$y = \frac{- 96 \pm \sqrt{(96 + 24 x - 48) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 4.1.3.5

Subtract $48$ from $96$ .

$y = \frac{- 96 \pm \sqrt{(24 x + 48) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 4.1.3.6

Factor $24$ out of $24 x + 48$ .

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Step 4.1.3.6.1

Factor $24$ out of $24 x$ .

$y = \frac{- 96 \pm \sqrt{(24 (x) + 48) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 4.1.3.6.2

Factor $24$ out of $48$ .

$y = \frac{- 96 \pm \sqrt{(24 x + 24 \cdot 2) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 4.1.3.6.3

Factor $24$ out of $24 x + 24 \cdot 2$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 4.1.3.7

Combine exponents.

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Step 4.1.3.7.1

Multiply $2$ by $4$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (96 - (8 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 4.1.3.7.2

Multiply $8$ by $- 1$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (96 - 8 (3 x - 6))}}{2 \cdot 16}$

Step 4.1.4

Simplify each term.

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Step 4.1.4.1

Apply the distributive property.

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (96 - 8 (3 x) - 8 \cdot - 6)}}{2 \cdot 16}$

Step 4.1.4.2

Multiply $3$ by $- 8$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (96 - 24 x - 8 \cdot - 6)}}{2 \cdot 16}$

Step 4.1.4.3

Multiply $- 8$ by $- 6$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (96 - 24 x + 48)}}{2 \cdot 16}$

Step 4.1.5

Add $96$ and $48$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (- 24 x + 144)}}{2 \cdot 16}$

Step 4.1.6

Factor $24$ out of $- 24 x + 144$ .

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Step 4.1.6.1

Factor $24$ out of $- 24 x$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (24 (- x) + 144)}}{2 \cdot 16}$

Step 4.1.6.2

Factor $24$ out of $144$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (24 (- x) + 24 (6))}}{2 \cdot 16}$

Step 4.1.6.3

Factor $24$ out of $24 (- x) + 24 (6)$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (24 (- x + 6))}}{2 \cdot 16}$

$y = \frac{- 96 \pm \sqrt{24 (x + 2) \cdot (24 (- x + 6))}}{2 \cdot 16}$

Step 4.1.7

Multiply $24$ by $24$ .

$y = \frac{- 96 \pm \sqrt{576 (x + 2) (- x + 6)}}{2 \cdot 16}$

Step 4.1.8

Rewrite $576 (x + 2) (- x + 6)$ as $24^{2} ((x + 2) (- x + 6))$ .

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Step 4.1.8.1

Rewrite $576$ as $24^{2}$ .

$y = \frac{- 96 \pm \sqrt{24^{2} (x + 2) (- x + 6)}}{2 \cdot 16}$

Step 4.1.8.2

Add parentheses.

$y = \frac{- 96 \pm \sqrt{24^{2} ((x + 2) (- x + 6))}}{2 \cdot 16}$

Step 4.1.9

Pull terms out from under the radical.

$y = \frac{- 96 \pm 24 \sqrt{(x + 2) (- x + 6)}}{2 \cdot 16}$

Step 4.2

Multiply $2$ by $16$ .

$y = \frac{- 96 \pm 24 \sqrt{(x + 2) (- x + 6)}}{32}$

Step 4.3

Simplify $\frac{- 96 \pm 24 \sqrt{(x + 2) (- x + 6)}}{32}$ .

$y = \frac{- 12 \pm 3 \sqrt{(x + 2) (- x + 6)}}{4}$

Step 4.4

Change the $\pm$ to $+$ .

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

Step 4.5

Factor $3$ out of $- 12 + 3 \sqrt{(x + 2) (- x + 6)}$ .

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Step 4.5.1

Factor $3$ out of $- 12$ .

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

Step 4.5.2

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

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

Step 4.6

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

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

Step 4.7

Factor $- 1$ out of $\sqrt{(x + 2) (- x + 6)}$ .

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

Step 4.8

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

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

Step 4.9

Move the negative in front of the fraction.

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

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 $4 \cdot 16 \cdot (9 x^{2} - 36 x + 36)$ as ${(2 \cdot 4 \cdot (3 x - 6))}^{2}$ .

$y = \frac{- 96 \pm \sqrt{96^{2} - {(2 \cdot 4 \cdot (3 x - 6))}^{2}}}{2 \cdot 16}$

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 $a = 96$ and $b = 2 \cdot 4 \cdot (3 x - 6)$ .

$y = \frac{- 96 \pm \sqrt{(96 + 2 \cdot 4 \cdot (3 x - 6)) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 5.1.3

Simplify.

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Step 5.1.3.1

Multiply $2$ by $4$ .

$y = \frac{- 96 \pm \sqrt{(96 + 8 \cdot (3 x - 6)) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 5.1.3.2

Apply the distributive property.

$y = \frac{- 96 \pm \sqrt{(96 + 8 (3 x) + 8 \cdot - 6) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 5.1.3.3

Multiply $3$ by $8$ .

$y = \frac{- 96 \pm \sqrt{(96 + 24 x + 8 \cdot - 6) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 5.1.3.4

Multiply $8$ by $- 6$ .

$y = \frac{- 96 \pm \sqrt{(96 + 24 x - 48) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 5.1.3.5

Subtract $48$ from $96$ .

$y = \frac{- 96 \pm \sqrt{(24 x + 48) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 5.1.3.6

Factor $24$ out of $24 x + 48$ .

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Step 5.1.3.6.1

Factor $24$ out of $24 x$ .

$y = \frac{- 96 \pm \sqrt{(24 (x) + 48) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 5.1.3.6.2

Factor $24$ out of $48$ .

$y = \frac{- 96 \pm \sqrt{(24 x + 24 \cdot 2) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 5.1.3.6.3

Factor $24$ out of $24 x + 24 \cdot 2$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (96 - (2 \cdot 4 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 5.1.3.7

Combine exponents.

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Step 5.1.3.7.1

Multiply $2$ by $4$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (96 - (8 \cdot (3 x - 6)))}}{2 \cdot 16}$

Step 5.1.3.7.2

Multiply $8$ by $- 1$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (96 - 8 (3 x - 6))}}{2 \cdot 16}$

Step 5.1.4

Simplify each term.

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Step 5.1.4.1

Apply the distributive property.

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (96 - 8 (3 x) - 8 \cdot - 6)}}{2 \cdot 16}$

Step 5.1.4.2

Multiply $3$ by $- 8$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (96 - 24 x - 8 \cdot - 6)}}{2 \cdot 16}$

Step 5.1.4.3

Multiply $- 8$ by $- 6$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (96 - 24 x + 48)}}{2 \cdot 16}$

Step 5.1.5

Add $96$ and $48$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (- 24 x + 144)}}{2 \cdot 16}$

Step 5.1.6

Factor $24$ out of $- 24 x + 144$ .

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Step 5.1.6.1

Factor $24$ out of $- 24 x$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (24 (- x) + 144)}}{2 \cdot 16}$

Step 5.1.6.2

Factor $24$ out of $144$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (24 (- x) + 24 (6))}}{2 \cdot 16}$

Step 5.1.6.3

Factor $24$ out of $24 (- x) + 24 (6)$ .

$y = \frac{- 96 \pm \sqrt{24 (x + 2) (24 (- x + 6))}}{2 \cdot 16}$

$y = \frac{- 96 \pm \sqrt{24 (x + 2) \cdot (24 (- x + 6))}}{2 \cdot 16}$

Step 5.1.7

Multiply $24$ by $24$ .

$y = \frac{- 96 \pm \sqrt{576 (x + 2) (- x + 6)}}{2 \cdot 16}$

Step 5.1.8

Rewrite $576 (x + 2) (- x + 6)$ as $24^{2} ((x + 2) (- x + 6))$ .

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Step 5.1.8.1

Rewrite $576$ as $24^{2}$ .

$y = \frac{- 96 \pm \sqrt{24^{2} (x + 2) (- x + 6)}}{2 \cdot 16}$

Step 5.1.8.2

Add parentheses.

$y = \frac{- 96 \pm \sqrt{24^{2} ((x + 2) (- x + 6))}}{2 \cdot 16}$

Step 5.1.9

Pull terms out from under the radical.

$y = \frac{- 96 \pm 24 \sqrt{(x + 2) (- x + 6)}}{2 \cdot 16}$

Step 5.2

Multiply $2$ by $16$ .

$y = \frac{- 96 \pm 24 \sqrt{(x + 2) (- x + 6)}}{32}$

Step 5.3

Simplify $\frac{- 96 \pm 24 \sqrt{(x + 2) (- x + 6)}}{32}$ .

$y = \frac{- 12 \pm 3 \sqrt{(x + 2) (- x + 6)}}{4}$

Step 5.4

Change the $\pm$ to $-$ .

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

Step 5.5

Factor $- 3$ out of $- 12 - 3 \sqrt{(x + 2) (- x + 6)}$ .

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Step 5.5.1

Reorder $- 12$ and $- 3 \sqrt{(x + 2) (- x + 6)}$ .

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

Step 5.5.2

Factor $- 3$ out of $- 3 \sqrt{(x + 2) (- x + 6)}$ .

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

Step 5.5.3

Factor $- 3$ out of $- 12$ .

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

Step 5.5.4

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

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

Step 5.6

Move the negative in front of the fraction.

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

Step 6

The final answer is the combination of both solutions.

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

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

Step 7

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 8

Solve for $x$ .

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Step 8.1

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 8.2

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

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Step 8.2.1

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

$x + 2 = 0$

Step 8.2.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 8.3

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

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Step 8.3.1

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

$- x + 6 = 0$

Step 8.3.2

Solve $- x + 6 = 0$ for $x$ .

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Step 8.3.2.1

Subtract $6$ from both sides of the equation.

$- x = - 6$

Step 8.3.2.2

Divide each term in $- x = - 6$ by $- 1$ and simplify.

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Step 8.3.2.2.1

Divide each term in $- x = - 6$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 6}{- 1}$

Step 8.3.2.2.2

Simplify the left side.

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Step 8.3.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 6}{- 1}$

Step 8.3.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 6}{- 1}$

Step 8.3.2.2.3

Simplify the right side.

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Step 8.3.2.2.3.1

Divide $- 6$ by $- 1$ .

$x = 6$

Step 8.4

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

$x = - 2, 6$

Step 8.5

Use each root to create test intervals.

$x < - 2$

$- 2 < x < 6$

$x > 6$

Step 8.6

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 8.6.1

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

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Step 8.6.1.1

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

$x = - 4$

Step 8.6.1.2

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

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

Step 8.6.1.3

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

False

Step 8.6.2

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

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Step 8.6.2.1

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

$x = 0$

Step 8.6.2.2

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

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

Step 8.6.2.3

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

True

Step 8.6.3

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

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Step 8.6.3.1

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

$x = 8$

Step 8.6.3.2

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

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

Step 8.6.3.3

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

False

Step 8.6.4

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

$x < - 2$ False

$- 2 < x < 6$ True

$x > 6$ False

$x < - 2$ False

$- 2 < x < 6$ True

$x > 6$ False

Step 8.7

The solution consists of all of the true intervals.

$- 2 \leq x \leq 6$

Step 9

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

Interval Notation:

$[- 2, 6]$

Set-Builder Notation:

${x | - 2 \leq x \leq 6}$

Step 10

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

Interval Notation:

$[- 6, 0]$

Set-Builder Notation:

${y | - 6 \leq y \leq 0}$

Step 11

Determine the domain and range.

Domain: $[- 2, 6], {x | - 2 \leq x \leq 6}$

Range: $[- 6, 0], {y | - 6 \leq y \leq 0}$

Step 12