Precalculus Examples

$f (x) = - 2 \sqrt{x - 1} + 3$

Step 1

Set $- 2 \sqrt{x - 1} + 3$ equal to $0$ .

$- 2 \sqrt{x - 1} + 3 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $3$ from both sides of the equation.

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

Step 2.2

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

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

Step 2.3

Simplify each side of the equation.

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Step 2.3.1

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

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

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

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Step 2.3.2.1.1

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

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

Step 2.3.2.1.2

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

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

Step 2.3.2.1.3

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

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Step 2.3.2.1.3.1

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

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

Step 2.3.2.1.3.2

Cancel the common factor of $2$ .

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Step 2.3.2.1.3.2.1

Cancel the common factor.

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

Step 2.3.2.1.3.2.2

Rewrite the expression.

$4 {(x - 1)}^{1} = {(- 3)}^{2}$

Step 2.3.2.1.4

Simplify.

$4 (x - 1) = {(- 3)}^{2}$

Step 2.3.2.1.5

Apply the distributive property.

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

Step 2.3.2.1.6

Multiply $4$ by $- 1$ .

$4 x - 4 = {(- 3)}^{2}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

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

$4 x - 4 = 9$

Step 2.4

Solve for $x$ .

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Step 2.4.1

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

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Step 2.4.1.1

Add $4$ to both sides of the equation.

$4 x = 9 + 4$

Step 2.4.1.2

Add $9$ and $4$ .

$4 x = 13$

Step 2.4.2

Divide each term in $4 x = 13$ by $4$ and simplify.

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Step 2.4.2.1

Divide each term in $4 x = 13$ by $4$ .

$\frac{4 x}{4} = \frac{13}{4}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $4$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{13}{4}$

Step 2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{13}{4}$

Step 3