Linear Algebra Examples

$\frac{{(x - 3)}^{2}}{81} - \frac{y^{2}}{144} = 1$

Step 1

Simplify $\frac{{(x - 3)}^{2}}{81} - \frac{y^{2}}{144}$ .

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Step 1.1

To write $\frac{{(x - 3)}^{2}}{81}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{{(x - 3)}^{2}}{81} \cdot \frac{16}{16} - \frac{y^{2}}{144} = 1$

Step 1.2

To write $- \frac{y^{2}}{144}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{{(x - 3)}^{2}}{81} \cdot \frac{16}{16} - \frac{y^{2}}{144} \cdot \frac{9}{9} = 1$

Step 1.3

Write each expression with a common denominator of $1296$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{{(x - 3)}^{2}}{81}$ by $\frac{16}{16}$ .

$\frac{{(x - 3)}^{2} \cdot 16}{81 \cdot 16} - \frac{y^{2}}{144} \cdot \frac{9}{9} = 1$

Step 1.3.2

Multiply $81$ by $16$ .

$\frac{{(x - 3)}^{2} \cdot 16}{1296} - \frac{y^{2}}{144} \cdot \frac{9}{9} = 1$

Step 1.3.3

Multiply $\frac{y^{2}}{144}$ by $\frac{9}{9}$ .

$\frac{{(x - 3)}^{2} \cdot 16}{1296} - \frac{y^{2} \cdot 9}{144 \cdot 9} = 1$

Step 1.3.4

Multiply $144$ by $9$ .

$\frac{{(x - 3)}^{2} \cdot 16}{1296} - \frac{y^{2} \cdot 9}{1296} = 1$

Step 1.4

Combine the numerators over the common denominator.

$\frac{{(x - 3)}^{2} \cdot 16 - y^{2} \cdot 9}{1296} = 1$

Step 1.5

Simplify the numerator.

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Step 1.5.1

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

$\frac{{((x - 3) \cdot 4)}^{2} - y^{2} \cdot 9}{1296} = 1$

Step 1.5.2

Rewrite $y^{2} \cdot 9$ as ${(y \cdot 3)}^{2}$ .

$\frac{{((x - 3) \cdot 4)}^{2} - {(y \cdot 3)}^{2}}{1296} = 1$

Step 1.5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = (x - 3) \cdot 4$ and $b = y \cdot 3$ .

$\frac{((x - 3) \cdot 4 + y \cdot 3) ((x - 3) \cdot 4 - (y \cdot 3))}{1296} = 1$

Step 1.5.4

Simplify.

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Step 1.5.4.1

Apply the distributive property.

$\frac{(x \cdot 4 - 3 \cdot 4 + y \cdot 3) ((x - 3) \cdot 4 - (y \cdot 3))}{1296} = 1$

Step 1.5.4.2

Move $4$ to the left of $x$ .

$\frac{(4 \cdot x - 3 \cdot 4 + y \cdot 3) ((x - 3) \cdot 4 - (y \cdot 3))}{1296} = 1$

Step 1.5.4.3

Multiply $- 3$ by $4$ .

$\frac{(4 \cdot x - 12 + y \cdot 3) ((x - 3) \cdot 4 - (y \cdot 3))}{1296} = 1$

Step 1.5.4.4

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

$\frac{(4 x - 12 + 3 \cdot y) ((x - 3) \cdot 4 - (y \cdot 3))}{1296} = 1$

Step 1.5.4.5

Apply the distributive property.

$\frac{(4 x - 12 + 3 y) (x \cdot 4 - 3 \cdot 4 - (y \cdot 3))}{1296} = 1$

Step 1.5.4.6

Move $4$ to the left of $x$ .

$\frac{(4 x - 12 + 3 y) (4 \cdot x - 3 \cdot 4 - (y \cdot 3))}{1296} = 1$

Step 1.5.4.7

Multiply $- 3$ by $4$ .

$\frac{(4 x - 12 + 3 y) (4 \cdot x - 12 - (y \cdot 3))}{1296} = 1$

Step 1.5.4.8

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

$\frac{(4 x - 12 + 3 y) (4 x - 12 - (3 \cdot y))}{1296} = 1$

Step 1.5.4.9

Multiply $3$ by $- 1$ .

$\frac{(4 x - 12 + 3 y) (4 x - 12 - 3 y)}{1296} = 1$

Step 2

Multiply both sides by $1296$ .

$\frac{(4 x - 12 + 3 y) (4 x - 12 - 3 y)}{1296} \cdot 1296 = 1 \cdot 1296$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{(4 x - 12 + 3 y) (4 x - 12 - 3 y)}{1296} \cdot 1296$ .

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Step 3.1.1.1

Cancel the common factor of $1296$ .

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Step 3.1.1.1.1

Cancel the common factor.

$\frac{(4 x - 12 + 3 y) (4 x - 12 - 3 y)}{1296} \cdot 1296 = 1 \cdot 1296$

Step 3.1.1.1.2

Rewrite the expression.

$(4 x - 12 + 3 y) (4 x - 12 - 3 y) = 1 \cdot 1296$

Step 3.1.1.2

Expand $(4 x - 12 + 3 y) (4 x - 12 - 3 y)$ by multiplying each term in the first expression by each term in the second expression.

$4 x (4 x) + 4 x \cdot - 12 + 4 x (- 3 y) - 12 (4 x) - 12 \cdot - 12 - 12 (- 3 y) + 3 y (4 x) + 3 y \cdot - 12 + 3 y (- 3 y) = 1 \cdot 1296$

Step 3.1.1.3

Simplify terms.

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Step 3.1.1.3.1

Combine the opposite terms in $4 x (4 x) + 4 x \cdot - 12 + 4 x (- 3 y) - 12 (4 x) - 12 \cdot - 12 - 12 (- 3 y) + 3 y (4 x) + 3 y \cdot - 12 + 3 y (- 3 y)$ .

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Step 3.1.1.3.1.1

Reorder the factors in the terms $4 x (- 3 y)$ and $3 y (4 x)$ .

$4 x (4 x) + 4 x \cdot - 12 - 3 \cdot 4 x y - 12 (4 x) - 12 \cdot - 12 - 12 (- 3 y) + 3 \cdot 4 x y + 3 y \cdot - 12 + 3 y (- 3 y) = 1 \cdot 1296$

Step 3.1.1.3.1.2

Add $- 3 \cdot 4 x y$ and $3 \cdot 4 x y$ .

$4 x (4 x) + 4 x \cdot - 12 - 12 (4 x) - 12 \cdot - 12 - 12 (- 3 y) + 0 + 3 y \cdot - 12 + 3 y (- 3 y) = 1 \cdot 1296$

Step 3.1.1.3.1.3

Add $4 x (4 x) + 4 x \cdot - 12 - 12 (4 x) - 12 \cdot - 12 - 12 (- 3 y)$ and $0$ .

$4 x (4 x) + 4 x \cdot - 12 - 12 (4 x) - 12 \cdot - 12 - 12 (- 3 y) + 3 y \cdot - 12 + 3 y (- 3 y) = 1 \cdot 1296$

Step 3.1.1.3.2

Simplify each term.

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Step 3.1.1.3.2.1

Rewrite using the commutative property of multiplication.

$4 \cdot 4 x \cdot x + 4 x \cdot - 12 - 12 (4 x) - 12 \cdot - 12 - 12 (- 3 y) + 3 y \cdot - 12 + 3 y (- 3 y) = 1 \cdot 1296$

Step 3.1.1.3.2.2

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

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Step 3.1.1.3.2.2.1

Move $x$ .

$4 \cdot 4 (x \cdot x) + 4 x \cdot - 12 - 12 (4 x) - 12 \cdot - 12 - 12 (- 3 y) + 3 y \cdot - 12 + 3 y (- 3 y) = 1 \cdot 1296$

Step 3.1.1.3.2.2.2

Multiply $x$ by $x$ .

$4 \cdot 4 x^{2} + 4 x \cdot - 12 - 12 (4 x) - 12 \cdot - 12 - 12 (- 3 y) + 3 y \cdot - 12 + 3 y (- 3 y) = 1 \cdot 1296$

Step 3.1.1.3.2.3

Multiply $4$ by $4$ .

$16 x^{2} + 4 x \cdot - 12 - 12 (4 x) - 12 \cdot - 12 - 12 (- 3 y) + 3 y \cdot - 12 + 3 y (- 3 y) = 1 \cdot 1296$

Step 3.1.1.3.2.4

Multiply $- 12$ by $4$ .

$16 x^{2} - 48 x - 12 (4 x) - 12 \cdot - 12 - 12 (- 3 y) + 3 y \cdot - 12 + 3 y (- 3 y) = 1 \cdot 1296$

Step 3.1.1.3.2.5

Multiply $4$ by $- 12$ .

$16 x^{2} - 48 x - 48 x - 12 \cdot - 12 - 12 (- 3 y) + 3 y \cdot - 12 + 3 y (- 3 y) = 1 \cdot 1296$

Step 3.1.1.3.2.6

Multiply $- 12$ by $- 12$ .

$16 x^{2} - 48 x - 48 x + 144 - 12 (- 3 y) + 3 y \cdot - 12 + 3 y (- 3 y) = 1 \cdot 1296$

Step 3.1.1.3.2.7

Multiply $- 3$ by $- 12$ .

$16 x^{2} - 48 x - 48 x + 144 + 36 y + 3 y \cdot - 12 + 3 y (- 3 y) = 1 \cdot 1296$

Step 3.1.1.3.2.8

Multiply $- 12$ by $3$ .

$16 x^{2} - 48 x - 48 x + 144 + 36 y - 36 y + 3 y (- 3 y) = 1 \cdot 1296$

Step 3.1.1.3.2.9

Rewrite using the commutative property of multiplication.

$16 x^{2} - 48 x - 48 x + 144 + 36 y - 36 y + 3 \cdot - 3 y \cdot y = 1 \cdot 1296$

Step 3.1.1.3.2.10

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

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Step 3.1.1.3.2.10.1

Move $y$ .

$16 x^{2} - 48 x - 48 x + 144 + 36 y - 36 y + 3 \cdot - 3 (y \cdot y) = 1 \cdot 1296$

Step 3.1.1.3.2.10.2

Multiply $y$ by $y$ .

$16 x^{2} - 48 x - 48 x + 144 + 36 y - 36 y + 3 \cdot - 3 y^{2} = 1 \cdot 1296$

Step 3.1.1.3.2.11

Multiply $3$ by $- 3$ .

$16 x^{2} - 48 x - 48 x + 144 + 36 y - 36 y - 9 y^{2} = 1 \cdot 1296$

Step 3.1.1.3.3

Simplify by adding terms.

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Step 3.1.1.3.3.1

Combine the opposite terms in $16 x^{2} - 48 x - 48 x + 144 + 36 y - 36 y - 9 y^{2}$ .

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Step 3.1.1.3.3.1.1

Subtract $36 y$ from $36 y$ .

$16 x^{2} - 48 x - 48 x + 144 + 0 - 9 y^{2} = 1 \cdot 1296$

Step 3.1.1.3.3.1.2

Add $16 x^{2} - 48 x - 48 x + 144$ and $0$ .

$16 x^{2} - 48 x - 48 x + 144 - 9 y^{2} = 1 \cdot 1296$

Step 3.1.1.3.3.2

Subtract $48 x$ from $- 48 x$ .

$16 x^{2} - 96 x + 144 - 9 y^{2} = 1 \cdot 1296$

Step 3.1.1.3.3.3

Reorder.

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Step 3.1.1.3.3.3.1

Move $144$ .

$16 x^{2} - 96 x - 9 y^{2} + 144 = 1 \cdot 1296$

Step 3.1.1.3.3.3.2

Move $- 96 x$ .

$16 x^{2} - 9 y^{2} - 96 x + 144 = 1 \cdot 1296$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $1296$ by $1$ .

$16 x^{2} - 9 y^{2} - 96 x + 144 = 1296$

Step 4

Solve for $y$ .

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Step 4.1

Move all terms not containing $y$ to the right side of the equation.

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Step 4.1.1

Subtract $16 x^{2}$ from both sides of the equation.

$- 9 y^{2} - 96 x + 144 = 1296 - 16 x^{2}$

Step 4.1.2

Add $96 x$ to both sides of the equation.

$- 9 y^{2} + 144 = 1296 - 16 x^{2} + 96 x$

Step 4.1.3

Subtract $144$ from both sides of the equation.

$- 9 y^{2} = 1296 - 16 x^{2} + 96 x - 144$

Step 4.1.4

Subtract $144$ from $1296$ .

$- 9 y^{2} = - 16 x^{2} + 96 x + 1152$

Step 4.2

Divide each term in $- 9 y^{2} = - 16 x^{2} + 96 x + 1152$ by $- 9$ and simplify.

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Step 4.2.1

Divide each term in $- 9 y^{2} = - 16 x^{2} + 96 x + 1152$ by $- 9$ .

$\frac{- 9 y^{2}}{- 9} = \frac{- 16 x^{2}}{- 9} + \frac{96 x}{- 9} + \frac{1152}{- 9}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 9$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 9 y^{2}}{- 9} = \frac{- 16 x^{2}}{- 9} + \frac{96 x}{- 9} + \frac{1152}{- 9}$

Step 4.2.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{- 16 x^{2}}{- 9} + \frac{96 x}{- 9} + \frac{1152}{- 9}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Dividing two negative values results in a positive value.

$y^{2} = \frac{16 x^{2}}{9} + \frac{96 x}{- 9} + \frac{1152}{- 9}$

Step 4.2.3.1.2

Cancel the common factor of $96$ and $- 9$ .

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Step 4.2.3.1.2.1

Factor $3$ out of $96 x$ .

$y^{2} = \frac{16 x^{2}}{9} + \frac{3 (32 x)}{- 9} + \frac{1152}{- 9}$

Step 4.2.3.1.2.2

Cancel the common factors.

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Step 4.2.3.1.2.2.1

Factor $3$ out of $- 9$ .

$y^{2} = \frac{16 x^{2}}{9} + \frac{3 (32 x)}{3 (- 3)} + \frac{1152}{- 9}$

Step 4.2.3.1.2.2.2

Cancel the common factor.

$y^{2} = \frac{16 x^{2}}{9} + \frac{3 (32 x)}{3 \cdot - 3} + \frac{1152}{- 9}$

Step 4.2.3.1.2.2.3

Rewrite the expression.

$y^{2} = \frac{16 x^{2}}{9} + \frac{32 x}{- 3} + \frac{1152}{- 9}$

Step 4.2.3.1.3

Move the negative in front of the fraction.

$y^{2} = \frac{16 x^{2}}{9} - \frac{32 x}{3} + \frac{1152}{- 9}$

Step 4.2.3.1.4

Divide $1152$ by $- 9$ .

$y^{2} = \frac{16 x^{2}}{9} - \frac{32 x}{3} - 128$

Step 4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{\frac{16 x^{2}}{9} - \frac{32 x}{3} - 128}$

Step 4.4

Simplify $\pm \sqrt{\frac{16 x^{2}}{9} - \frac{32 x}{3} - 128}$ .

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Step 4.4.1

To write $- \frac{32 x}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \pm \sqrt{\frac{16 x^{2}}{9} - \frac{32 x}{3} \cdot \frac{3}{3} - 128}$

Step 4.4.2

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.4.2.1

Multiply $\frac{32 x}{3}$ by $\frac{3}{3}$ .

$y = \pm \sqrt{\frac{16 x^{2}}{9} - \frac{32 x \cdot 3}{3 \cdot 3} - 128}$

Step 4.4.2.2

Multiply $3$ by $3$ .

$y = \pm \sqrt{\frac{16 x^{2}}{9} - \frac{32 x \cdot 3}{9} - 128}$

Step 4.4.3

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{16 x^{2} - 32 x \cdot 3}{9} - 128}$

Step 4.4.4

Simplify the numerator.

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Step 4.4.4.1

Factor $16 x$ out of $16 x^{2} - 32 x \cdot 3$ .

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Step 4.4.4.1.1

Factor $16 x$ out of $16 x^{2}$ .

$y = \pm \sqrt{\frac{16 x (x) - 32 x \cdot 3}{9} - 128}$

Step 4.4.4.1.2

Factor $16 x$ out of $- 32 x \cdot 3$ .

$y = \pm \sqrt{\frac{16 x (x) + 16 x (- 2 \cdot 3)}{9} - 128}$

Step 4.4.4.1.3

Factor $16 x$ out of $16 x (x) + 16 x (- 2 \cdot 3)$ .

$y = \pm \sqrt{\frac{16 x (x - 2 \cdot 3)}{9} - 128}$

Step 4.4.4.2

Multiply $- 2$ by $3$ .

$y = \pm \sqrt{\frac{16 x (x - 6)}{9} - 128}$

Step 4.4.5

To write $- 128$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$y = \pm \sqrt{\frac{16 x (x - 6)}{9} - 128 \cdot \frac{9}{9}}$

Step 4.4.6

Simplify terms.

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Step 4.4.6.1

Combine $- 128$ and $\frac{9}{9}$ .

$y = \pm \sqrt{\frac{16 x (x - 6)}{9} + \frac{- 128 \cdot 9}{9}}$

Step 4.4.6.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{16 x (x - 6) - 128 \cdot 9}{9}}$

Step 4.4.7

Simplify the numerator.

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Step 4.4.7.1

Factor $16$ out of $16 x (x - 6) - 128 \cdot 9$ .

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Step 4.4.7.1.1

Factor $16$ out of $16 x (x - 6)$ .

$y = \pm \sqrt{\frac{16 (x (x - 6)) - 128 \cdot 9}{9}}$

Step 4.4.7.1.2

Factor $16$ out of $- 128 \cdot 9$ .

$y = \pm \sqrt{\frac{16 (x (x - 6)) + 16 (- 8 \cdot 9)}{9}}$

Step 4.4.7.1.3

Factor $16$ out of $16 (x (x - 6)) + 16 (- 8 \cdot 9)$ .

$y = \pm \sqrt{\frac{16 (x (x - 6) - 8 \cdot 9)}{9}}$

Step 4.4.7.2

Apply the distributive property.

$y = \pm \sqrt{\frac{16 (x \cdot x + x \cdot - 6 - 8 \cdot 9)}{9}}$

Step 4.4.7.3

Multiply $x$ by $x$ .

$y = \pm \sqrt{\frac{16 (x^{2} + x \cdot - 6 - 8 \cdot 9)}{9}}$

Step 4.4.7.4

Move $- 6$ to the left of $x$ .

$y = \pm \sqrt{\frac{16 (x^{2} - 6 \cdot x - 8 \cdot 9)}{9}}$

Step 4.4.7.5

Multiply $- 8$ by $9$ .

$y = \pm \sqrt{\frac{16 (x^{2} - 6 x - 72)}{9}}$

Step 4.4.7.6

Factor $x^{2} - 6 x - 72$ using the AC method.

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Step 4.4.7.6.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 72$ and whose sum is $- 6$ .

$- 12, 6$

Step 4.4.7.6.2

Write the factored form using these integers.

$y = \pm \sqrt{\frac{16 ((x - 12) (x + 6))}{9}}$

$y = \pm \sqrt{\frac{16 (x - 12) (x + 6)}{9}}$

Step 4.4.8

Rewrite $\frac{16 (x - 12) (x + 6)}{9}$ as ${(\frac{2^{2}}{3})}^{2} ((x - 12) (x + 6))$ .

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Step 4.4.8.1

Factor the perfect power ${(2^{2})}^{2}$ out of $16 (x - 12) (x + 6)$ .

$y = \pm \sqrt{\frac{{(2^{2})}^{2} ((x - 12) (x + 6))}{9}}$

Step 4.4.8.2

Factor the perfect power $3^{2}$ out of $9$ .

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

Step 4.4.8.3

Rearrange the fraction $\frac{{(2^{2})}^{2} ((x - 12) (x + 6))}{3^{2} \cdot 1}$ .

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

Step 4.4.9

Pull terms out from under the radical.

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

Step 4.4.10

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

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

Step 4.4.11

Combine $\frac{4}{3}$ and $\sqrt{(x - 12) (x + 6)}$ .

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

Step 4.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.5.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 4.5.2

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 4.5.3

The complete solution is the result of both the positive and negative portions of the solution.

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

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

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

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

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

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

Step 5

Set the radicand in $\sqrt{(x - 12) (x + 6)}$ greater than or equal to $0$ to find where the expression is defined.

$(x - 12) (x + 6) \geq 0$

Step 6

Solve for $x$ .

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Step 6.1

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

$x - 12 = 0$

$x + 6 = 0$

Step 6.2

Set $x - 12$ equal to $0$ and solve for $x$ .

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Step 6.2.1

Set $x - 12$ equal to $0$ .

$x - 12 = 0$

Step 6.2.2

Add $12$ to both sides of the equation.

$x = 12$

Step 6.3

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

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Step 6.3.1

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

$x + 6 = 0$

Step 6.3.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 6.4

The final solution is all the values that make $(x - 12) (x + 6) \geq 0$ true.

$x = 12, - 6$

Step 6.5

Use each root to create test intervals.

$x < - 6$

$- 6 < x < 12$

$x > 12$

Step 6.6

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 6.6.1

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

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Step 6.6.1.1

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

$x = - 8$

Step 6.6.1.2

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

$((- 8) - 12) ((- 8) + 6) \geq 0$

Step 6.6.1.3

The left side $40$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 6.6.2

Test a value on the interval $- 6 < x < 12$ to see if it makes the inequality true.

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Step 6.6.2.1

Choose a value on the interval $- 6 < x < 12$ and see if this value makes the original inequality true.

$x = 0$

Step 6.6.2.2

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

$((0) - 12) ((0) + 6) \geq 0$

Step 6.6.2.3

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

False

Step 6.6.3

Test a value on the interval $x > 12$ to see if it makes the inequality true.

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Step 6.6.3.1

Choose a value on the interval $x > 12$ and see if this value makes the original inequality true.

$x = 14$

Step 6.6.3.2

Replace $x$ with $14$ in the original inequality.

$((14) - 12) ((14) + 6) \geq 0$

Step 6.6.3.3

The left side $40$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 6.6.4

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

$x < - 6$ True

$- 6 < x < 12$ False

$x > 12$ True

$x < - 6$ True

$- 6 < x < 12$ False

$x > 12$ True

Step 6.7

The solution consists of all of the true intervals.

$x \leq - 6$ or $x \geq 12$

Step 7

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

Interval Notation:

$(- \infty, - 6] \cup [12, \infty)$

Set-Builder Notation:

${x | x \leq - 6, x \geq 12}$

Step 8