Precalculus Examples

$\sqrt{x - 2} + 5$

Step 1

Write $\sqrt{x - 2} + 5$ as an equation.

$y = \sqrt{x - 2} + 5$

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 $x$ .

$0 = \sqrt{x - 2} + 5$

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as $\sqrt{x - 2} + 5 = 0$ .

$\sqrt{x - 2} + 5 = 0$

Step 2.2.2

Subtract $5$ from both sides of the equation.

$\sqrt{x - 2} = - 5$

Step 2.2.3

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

${\sqrt{x - 2}}^{2} = {(- 5)}^{2}$

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

${({(x - 2)}^{\frac{1}{2}})}^{2} = {(- 5)}^{2}$

Step 2.2.4.2

Simplify the left side.

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Step 2.2.4.2.1

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

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Step 2.2.4.2.1.1

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

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Step 2.2.4.2.1.1.1

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

${(x - 2)}^{\frac{1}{2} \cdot 2} = {(- 5)}^{2}$

Step 2.2.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.4.2.1.1.2.1

Cancel the common factor.

${(x - 2)}^{\frac{1}{2} \cdot 2} = {(- 5)}^{2}$

Step 2.2.4.2.1.1.2.2

Rewrite the expression.

${(x - 2)}^{1} = {(- 5)}^{2}$

Step 2.2.4.2.1.2

Simplify.

$x - 2 = {(- 5)}^{2}$

Step 2.2.4.3

Simplify the right side.

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Step 2.2.4.3.1

Raise $- 5$ to the power of $2$ .

$x - 2 = 25$

Step 2.2.5

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

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Step 2.2.5.1

Add $2$ to both sides of the equation.

$x = 25 + 2$

Step 2.2.5.2

Add $25$ and $2$ .

$x = 27$

Step 2.2.6

Exclude the solutions that do not make $0 = \sqrt{x - 2} + 5$ true.

$No solution$

Step 2.3

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

x-intercept(s): $None$

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 $y$ .

$y = \sqrt{(0) - 2} + 5$

Step 3.2

Simplify $\sqrt{(0) - 2} + 5$ .

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Subtract $2$ from $0$ .

$y = \sqrt{- 2} + 5$

Step 3.2.1.2

Rewrite $- 2$ as $- 1 (2)$ .

$y = \sqrt{- 1 (2)} + 5$

Step 3.2.1.3

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$y = \sqrt{- 1} \cdot \sqrt{2} + 5$

Step 3.2.1.4

Rewrite $\sqrt{- 1}$ as $i$ .

$y = i \sqrt{2} + 5$

Step 3.2.2

Reorder $i \sqrt{2}$ and $5$ .

$y = 5 + i \sqrt{2}$

Step 3.3

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

y-intercept(s): $None$

Step 4

List the intersections.

x-intercept(s): $None$

y-intercept(s): $None$

Step 5