Algebra Examples

$f (x) = \frac{3}{4} \sqrt{x + 2} - 3$

Step 1

Find the x-intercepts.

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Step 1.1

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

$0 = \frac{3}{4} \cdot \sqrt{x + 2} - 3$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\frac{3}{4} \cdot \sqrt{x + 2} - 3 = 0$ .

$\frac{3}{4} \cdot \sqrt{x + 2} - 3 = 0$

Step 1.2.2

Add $3$ to both sides of the equation.

$\frac{3}{4} \cdot \sqrt{x + 2} = 3$

Step 1.2.3

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

${(\frac{3}{4} \cdot \sqrt{x + 2})}^{2} = 3^{2}$

Step 1.2.4

Simplify each side of the equation.

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Step 1.2.4.1

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

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

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Simplify ${(\frac{3}{4} \cdot {(x + 2)}^{\frac{1}{2}})}^{2}$ .

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Step 1.2.4.2.1.1

Combine $\frac{3}{4}$ and ${(x + 2)}^{\frac{1}{2}}$ .

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

Step 1.2.4.2.1.2

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

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Step 1.2.4.2.1.2.1

Apply the product rule to $\frac{3 {(x + 2)}^{\frac{1}{2}}}{4}$ .

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

Step 1.2.4.2.1.2.2

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

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

Step 1.2.4.2.1.3

Simplify the numerator.

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Step 1.2.4.2.1.3.1

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

$\frac{9 {({(x + 2)}^{\frac{1}{2}})}^{2}}{4^{2}} = 3^{2}$

Step 1.2.4.2.1.3.2

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

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Step 1.2.4.2.1.3.2.1

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

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

Step 1.2.4.2.1.3.2.2

Cancel the common factor of $2$ .

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Step 1.2.4.2.1.3.2.2.1

Cancel the common factor.

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

Step 1.2.4.2.1.3.2.2.2

Rewrite the expression.

$\frac{9 {(x + 2)}^{1}}{4^{2}} = 3^{2}$

Step 1.2.4.2.1.3.3

Simplify.

$\frac{9 (x + 2)}{4^{2}} = 3^{2}$

Step 1.2.4.2.1.4

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

$\frac{9 (x + 2)}{16} = 3^{2}$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

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

$\frac{9 (x + 2)}{16} = 9$

Step 1.2.5

Solve for $x$ .

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Step 1.2.5.1

Multiply both sides of the equation by $\frac{16}{9}$ .

$\frac{16}{9} \cdot \frac{9 (x + 2)}{16} = \frac{16}{9} \cdot 9$

Step 1.2.5.2

Simplify both sides of the equation.

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Step 1.2.5.2.1

Simplify the left side.

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Step 1.2.5.2.1.1

Simplify $\frac{16}{9} \cdot \frac{9 (x + 2)}{16}$ .

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Step 1.2.5.2.1.1.1

Cancel the common factor of $16$ .

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Step 1.2.5.2.1.1.1.1

Cancel the common factor.

$\frac{16}{9} \cdot \frac{9 (x + 2)}{16} = \frac{16}{9} \cdot 9$

Step 1.2.5.2.1.1.1.2

Rewrite the expression.

$\frac{1}{9} (9 (x + 2)) = \frac{16}{9} \cdot 9$

Step 1.2.5.2.1.1.2

Cancel the common factor of $9$ .

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Step 1.2.5.2.1.1.2.1

Cancel the common factor.

$\frac{1}{9} (9 (x + 2)) = \frac{16}{9} \cdot 9$

Step 1.2.5.2.1.1.2.2

Rewrite the expression.

$x + 2 = \frac{16}{9} \cdot 9$

Step 1.2.5.2.2

Simplify the right side.

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Step 1.2.5.2.2.1

Cancel the common factor of $9$ .

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Step 1.2.5.2.2.1.1

Cancel the common factor.

$x + 2 = \frac{16}{9} \cdot 9$

Step 1.2.5.2.2.1.2

Rewrite the expression.

$x + 2 = 16$

Step 1.2.5.3

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

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Step 1.2.5.3.1

Subtract $2$ from both sides of the equation.

$x = 16 - 2$

Step 1.2.5.3.2

Subtract $2$ from $16$ .

$x = 14$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(14, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

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

$y = \frac{3}{4} \cdot \sqrt{(0) + 2} - 3$

Step 2.2

Simplify each term.

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Step 2.2.1

Add $0$ and $2$ .

$y = \frac{3}{4} \cdot \sqrt{2} - 3$

Step 2.2.2

Combine $\frac{3}{4}$ and $\sqrt{2}$ .

$y = \frac{3 \sqrt{2}}{4} - 3$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \frac{3 \sqrt{2}}{4} - 3)$

Step 3

List the intersections.

x-intercept(s): $(14, 0)$

y-intercept(s): $(0, \frac{3 \sqrt{2}}{4} - 3)$

Step 4