Algebra Examples

$4 \sqrt{- 6 x} - 12$

Step 1

Write $4 \sqrt{- 6 x} - 12$ as an equation.

$y = 4 \sqrt{- 6 x} - 12$

Step 2

Find the x-intercepts.

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Step 2.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 4 \sqrt{- 6 x} - 12$

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as $4 \sqrt{- 6 x} - 12 = 0$ .

$4 \sqrt{- 6 x} - 12 = 0$

Step 2.2.2

Add $12$ to both sides of the equation.

$4 \sqrt{- 6 x} = 12$

Step 2.2.3

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

${(4 \sqrt{- 6 x})}^{2} = 12^{2}$

Step 2.2.4

Simplify each side of the equation.

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Step 2.2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- 6 x}$ as ${(- 6 x)}^{\frac{1}{2}}$ .

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

Step 2.2.4.2

Simplify the left side.

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Step 2.2.4.2.1

Simplify ${(4 {(- 6 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.4.2.1.1

Apply the product rule to $- 6 x$ .

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

Step 2.2.4.2.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.2.4.2.1.2.1

Apply the product rule to $4 {(- 6)}^{\frac{1}{2}} x^{\frac{1}{2}}$ .

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

Step 2.2.4.2.1.2.2

Apply the product rule to $4 {(- 6)}^{\frac{1}{2}}$ .

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

Step 2.2.4.2.1.3

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

$16 {({(- 6)}^{\frac{1}{2}})}^{2} {(x^{\frac{1}{2}})}^{2} = 12^{2}$

Step 2.2.4.2.1.4

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

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Step 2.2.4.2.1.4.1

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

$16 {(- 6)}^{\frac{1}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = 12^{2}$

Step 2.2.4.2.1.4.2

Cancel the common factor of $2$ .

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Step 2.2.4.2.1.4.2.1

Cancel the common factor.

$16 {(- 6)}^{\frac{1}{2} \cdot 2} {(x^{\frac{1}{2}})}^{2} = 12^{2}$

Step 2.2.4.2.1.4.2.2

Rewrite the expression.

$16 {(- 6)}^{1} {(x^{\frac{1}{2}})}^{2} = 12^{2}$

Step 2.2.4.2.1.5

Evaluate the exponent.

$16 \cdot - 6 {(x^{\frac{1}{2}})}^{2} = 12^{2}$

Step 2.2.4.2.1.6

Multiply $16$ by $- 6$ .

$- 96 {(x^{\frac{1}{2}})}^{2} = 12^{2}$

Step 2.2.4.2.1.7

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 2.2.4.2.1.7.1

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

$- 96 x^{\frac{1}{2} \cdot 2} = 12^{2}$

Step 2.2.4.2.1.7.2

Cancel the common factor of $2$ .

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Step 2.2.4.2.1.7.2.1

Cancel the common factor.

$- 96 x^{\frac{1}{2} \cdot 2} = 12^{2}$

Step 2.2.4.2.1.7.2.2

Rewrite the expression.

$- 96 x^{1} = 12^{2}$

Step 2.2.4.2.1.8

Simplify.

$- 96 x = 12^{2}$

Step 2.2.4.3

Simplify the right side.

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Step 2.2.4.3.1

Raise $12$ to the power of $2$ .

$- 96 x = 144$

Step 2.2.5

Divide each term in $- 96 x = 144$ by $- 96$ and simplify.

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Step 2.2.5.1

Divide each term in $- 96 x = 144$ by $- 96$ .

$\frac{- 96 x}{- 96} = \frac{144}{- 96}$

Step 2.2.5.2

Simplify the left side.

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Step 2.2.5.2.1

Cancel the common factor of $- 96$ .

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Step 2.2.5.2.1.1

Cancel the common factor.

$\frac{- 96 x}{- 96} = \frac{144}{- 96}$

Step 2.2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{144}{- 96}$

Step 2.2.5.3

Simplify the right side.

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Step 2.2.5.3.1

Cancel the common factor of $144$ and $- 96$ .

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Step 2.2.5.3.1.1

Factor $48$ out of $144$ .

$x = \frac{48 (3)}{- 96}$

Step 2.2.5.3.1.2

Cancel the common factors.

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Step 2.2.5.3.1.2.1

Factor $48$ out of $- 96$ .

$x = \frac{48 \cdot 3}{48 \cdot - 2}$

Step 2.2.5.3.1.2.2

Cancel the common factor.

$x = \frac{48 \cdot 3}{48 \cdot - 2}$

Step 2.2.5.3.1.2.3

Rewrite the expression.

$x = \frac{3}{- 2}$

Step 2.2.5.3.2

Move the negative in front of the fraction.

$x = - \frac{3}{2}$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(- \frac{3}{2}, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 4 \sqrt{- 6 \cdot 0} - 12$

Step 3.2

Simplify $4 \sqrt{- 6 \cdot 0} - 12$ .

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Multiply $- 6$ by $0$ .

$y = 4 \sqrt{0} - 12$

Step 3.2.1.2

Rewrite $0$ as $0^{2}$ .

$y = 4 \sqrt{0^{2}} - 12$

Step 3.2.1.3

Pull terms out from under the radical, assuming positive real numbers.

$y = 4 \cdot 0 - 12$

Step 3.2.1.4

Multiply $4$ by $0$ .

$y = 0 - 12$

Step 3.2.2

Subtract $12$ from $0$ .

$y = - 12$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 12)$

Step 4

List the intersections.

x-intercept(s): $(- \frac{3}{2}, 0)$

y-intercept(s): $(0, - 12)$

Step 5