Algebra Examples

$g (x) = \frac{2}{3} x (x + 8)$

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 $x$ .

$0 = \frac{2}{3} \cdot (x (x + 8))$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\frac{2}{3} \cdot (x (x + 8)) = 0$ .

$\frac{2}{3} \cdot (x (x + 8)) = 0$

Step 1.2.2

Multiply both sides of the equation by $\frac{3}{2}$ .

$\frac{3}{2} (\frac{2}{3} \cdot (x (x + 8))) = \frac{3}{2} \cdot 0$

Step 1.2.3

Simplify both sides of the equation.

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Step 1.2.3.1

Simplify the left side.

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Step 1.2.3.1.1

Simplify $\frac{3}{2} (\frac{2}{3} \cdot (x (x + 8)))$ .

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Step 1.2.3.1.1.1

Apply the distributive property.

$\frac{3}{2} (\frac{2}{3} \cdot (x \cdot x + x \cdot 8)) = \frac{3}{2} \cdot 0$

Step 1.2.3.1.1.2

Simplify the expression.

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Step 1.2.3.1.1.2.1

Multiply $x$ by $x$ .

$\frac{3}{2} (\frac{2}{3} \cdot (x^{2} + x \cdot 8)) = \frac{3}{2} \cdot 0$

Step 1.2.3.1.1.2.2

Move $8$ to the left of $x$ .

$\frac{3}{2} (\frac{2}{3} \cdot (x^{2} + 8 \cdot x)) = \frac{3}{2} \cdot 0$

Step 1.2.3.1.1.3

Apply the distributive property.

$\frac{3}{2} (\frac{2}{3} x^{2} + \frac{2}{3} (8 x)) = \frac{3}{2} \cdot 0$

Step 1.2.3.1.1.4

Combine $\frac{2}{3}$ and $x^{2}$ .

$\frac{3}{2} (\frac{2 x^{2}}{3} + \frac{2}{3} (8 x)) = \frac{3}{2} \cdot 0$

Step 1.2.3.1.1.5

Multiply $\frac{2}{3} (8 x)$ .

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Step 1.2.3.1.1.5.1

Combine $8$ and $\frac{2}{3}$ .

$\frac{3}{2} (\frac{2 x^{2}}{3} + \frac{8 \cdot 2}{3} x) = \frac{3}{2} \cdot 0$

Step 1.2.3.1.1.5.2

Multiply $8$ by $2$ .

$\frac{3}{2} (\frac{2 x^{2}}{3} + \frac{16}{3} x) = \frac{3}{2} \cdot 0$

Step 1.2.3.1.1.5.3

Combine $\frac{16}{3}$ and $x$ .

$\frac{3}{2} (\frac{2 x^{2}}{3} + \frac{16 x}{3}) = \frac{3}{2} \cdot 0$

Step 1.2.3.1.1.6

Simplify terms.

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Step 1.2.3.1.1.6.1

Apply the distributive property.

$\frac{3}{2} \cdot \frac{2 x^{2}}{3} + \frac{3}{2} \cdot \frac{16 x}{3} = \frac{3}{2} \cdot 0$

Step 1.2.3.1.1.6.2

Combine.

$\frac{3 (2 x^{2})}{2 \cdot 3} + \frac{3}{2} \cdot \frac{16 x}{3} = \frac{3}{2} \cdot 0$

Step 1.2.3.1.1.6.3

Cancel the common factor of $3$ .

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Step 1.2.3.1.1.6.3.1

Cancel the common factor.

$\frac{3 (2 x^{2})}{2 \cdot 3} + \frac{3}{2} \cdot \frac{16 x}{3} = \frac{3}{2} \cdot 0$

Step 1.2.3.1.1.6.3.2

Rewrite the expression.

$\frac{3 (2 x^{2})}{2 \cdot 3} + \frac{1}{2} (16 x) = \frac{3}{2} \cdot 0$

Step 1.2.3.1.1.6.4

Cancel the common factor of $2$ .

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Step 1.2.3.1.1.6.4.1

Factor $2$ out of $16 x$ .

$\frac{3 (2 x^{2})}{2 \cdot 3} + \frac{1}{2} (2 (8 x)) = \frac{3}{2} \cdot 0$

Step 1.2.3.1.1.6.4.2

Cancel the common factor.

$\frac{3 (2 x^{2})}{2 \cdot 3} + \frac{1}{2} (2 (8 x)) = \frac{3}{2} \cdot 0$

Step 1.2.3.1.1.6.4.3

Rewrite the expression.

$\frac{3 (2 x^{2})}{2 \cdot 3} + 8 x = \frac{3}{2} \cdot 0$

Step 1.2.3.1.1.7

Simplify each term.

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Step 1.2.3.1.1.7.1

Cancel the common factor of $3$ .

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Step 1.2.3.1.1.7.1.1

Cancel the common factor.

$\frac{3 (2 x^{2})}{2 \cdot 3} + 8 x = \frac{3}{2} \cdot 0$

Step 1.2.3.1.1.7.1.2

Rewrite the expression.

$\frac{2 x^{2}}{2} + 8 x = \frac{3}{2} \cdot 0$

Step 1.2.3.1.1.7.2

Cancel the common factor of $2$ .

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Step 1.2.3.1.1.7.2.1

Cancel the common factor.

$\frac{2 x^{2}}{2} + 8 x = \frac{3}{2} \cdot 0$

Step 1.2.3.1.1.7.2.2

Divide $x^{2}$ by $1$ .

$x^{2} + 8 x = \frac{3}{2} \cdot 0$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Multiply $\frac{3}{2}$ by $0$ .

$x^{2} + 8 x = 0$

Step 1.2.4

Factor $x$ out of $x^{2} + 8 x$ .

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Step 1.2.4.1

Factor $x$ out of $x^{2}$ .

$x \cdot x + 8 x = 0$

Step 1.2.4.2

Factor $x$ out of $8 x$ .

$x \cdot x + x \cdot 8 = 0$

Step 1.2.4.3

Factor $x$ out of $x \cdot x + x \cdot 8$ .

$x (x + 8) = 0$

Step 1.2.5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x + 8 = 0$

Step 1.2.6

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.7

Set $x + 8$ equal to $0$ and solve for $x$ .

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Step 1.2.7.1

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Step 1.2.7.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 1.2.8

The final solution is all the values that make $x (x + 8) = 0$ true.

$x = 0, - 8$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 8, 0)$

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 $y$ .

$y = \frac{2}{3} \cdot ((0) ((0) + 8))$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{2}{3} \cdot (0 (0 + 8))$

Step 2.2.2

Remove parentheses.

$y = \frac{2}{3} \cdot (0 (0 + 8))$

Step 2.2.3

Remove parentheses.

$y = \frac{2}{3} \cdot ((0) ((0) + 8))$

Step 2.2.4

Simplify $\frac{2}{3} \cdot ((0) ((0) + 8))$ .

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Step 2.2.4.1

Cancel the common factor of $3$ .

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Step 2.2.4.1.1

Factor $3$ out of $(0) ((0) + 8)$ .

$y = \frac{2}{3} \cdot (3 ((0) ((0) + 8)))$

Step 2.2.4.1.2

Cancel the common factor.

$y = \frac{2}{3} \cdot (3 ((0) ((0) + 8)))$

Step 2.2.4.1.3

Rewrite the expression.

$y = 2 \cdot ((0) ((0) + 8))$

Step 2.2.4.2

Multiply by zero.

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Step 2.2.4.2.1

Multiply $0$ by $(0) + 8$ .

$y = 2 \cdot 0$

Step 2.2.4.2.2

Multiply $2$ by $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (- 8, 0)$

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

Step 4