Algebra Examples

$f (x) = - 3 \sqrt{- 2 x - 3}$

Step 1

Set $- 3 \sqrt{- 2 x - 3}$ equal to $0$ .

$- 3 \sqrt{- 2 x - 3} = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

${(- 3 \sqrt{- 2 x - 3})}^{2} = 0^{2}$

Step 2.2

Simplify each side of the equation.

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Step 2.2.1

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

${(- 3 {(- 2 x - 3)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify ${(- 3 {(- 2 x - 3)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1

Apply the product rule to $- 3 {(- 2 x - 3)}^{\frac{1}{2}}$ .

${(- 3)}^{2} {({(- 2 x - 3)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 2.2.2.1.2

Raise $- 3$ to the power of $2$ .

$9 {({(- 2 x - 3)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 2.2.2.1.3

Multiply the exponents in ${({(- 2 x - 3)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.3.1

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

$9 {(- 2 x - 3)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 2.2.2.1.3.2

Cancel the common factor of $2$ .

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Step 2.2.2.1.3.2.1

Cancel the common factor.

$9 {(- 2 x - 3)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 2.2.2.1.3.2.2

Rewrite the expression.

$9 {(- 2 x - 3)}^{1} = 0^{2}$

Step 2.2.2.1.4

Simplify.

$9 (- 2 x - 3) = 0^{2}$

Step 2.2.2.1.5

Apply the distributive property.

$9 (- 2 x) + 9 \cdot - 3 = 0^{2}$

Step 2.2.2.1.6

Multiply.

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Step 2.2.2.1.6.1

Multiply $- 2$ by $9$ .

$- 18 x + 9 \cdot - 3 = 0^{2}$

Step 2.2.2.1.6.2

Multiply $9$ by $- 3$ .

$- 18 x - 27 = 0^{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Raising $0$ to any positive power yields $0$ .

$- 18 x - 27 = 0$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Add $27$ to both sides of the equation.

$- 18 x = 27$

Step 2.3.2

Divide each term in $- 18 x = 27$ by $- 18$ and simplify.

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Step 2.3.2.1

Divide each term in $- 18 x = 27$ by $- 18$ .

$\frac{- 18 x}{- 18} = \frac{27}{- 18}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $- 18$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{- 18 x}{- 18} = \frac{27}{- 18}$

Step 2.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{27}{- 18}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Cancel the common factor of $27$ and $- 18$ .

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Step 2.3.2.3.1.1

Factor $9$ out of $27$ .

$x = \frac{9 (3)}{- 18}$

Step 2.3.2.3.1.2

Cancel the common factors.

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Step 2.3.2.3.1.2.1

Factor $9$ out of $- 18$ .

$x = \frac{9 \cdot 3}{9 \cdot - 2}$

Step 2.3.2.3.1.2.2

Cancel the common factor.

$x = \frac{9 \cdot 3}{9 \cdot - 2}$

Step 2.3.2.3.1.2.3

Rewrite the expression.

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

Step 2.3.2.3.2

Move the negative in front of the fraction.

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

Step 3