Algebra Examples

$f (x) = - 2 x (2 x^{2} + 2) (2 x - 3) (5 x + 2)$

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 $x$ .

$0 = - 2 x (2 x^{2} + 2) (2 x - 3) (5 x + 2)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 2 x (2 x^{2} + 2) (2 x - 3) (5 x + 2) = 0$ .

$- 2 x (2 x^{2} + 2) (2 x - 3) (5 x + 2) = 0$

Step 1.2.2

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

$x = 0$

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

$2 x - 3 = 0$

$5 x + 2 = 0$

Step 1.2.3

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.4

Set $2 x^{2} + 2$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $2 x^{2} + 2$ equal to $0$ .

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

Step 1.2.4.2

Solve $2 x^{2} + 2 = 0$ for $x$ .

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Step 1.2.4.2.1

Subtract $2$ from both sides of the equation.

$2 x^{2} = - 2$

Step 1.2.4.2.2

Divide each term in $2 x^{2} = - 2$ by $2$ and simplify.

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Step 1.2.4.2.2.1

Divide each term in $2 x^{2} = - 2$ by $2$ .

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

Step 1.2.4.2.2.2

Simplify the left side.

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Step 1.2.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.4.2.2.2.1.1

Cancel the common factor.

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

Step 1.2.4.2.2.2.1.2

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

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

Step 1.2.4.2.2.3

Simplify the right side.

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Step 1.2.4.2.2.3.1

Divide $- 2$ by $2$ .

$x^{2} = - 1$

Step 1.2.4.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 1}$

Step 1.2.4.2.4

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

$x = \pm i$

Step 1.2.4.2.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.4.2.5.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i$

Step 1.2.4.2.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i$

Step 1.2.4.2.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i, - i$

Step 1.2.5

Set $2 x - 3$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Step 1.2.5.2

Solve $2 x - 3 = 0$ for $x$ .

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Step 1.2.5.2.1

Add $3$ to both sides of the equation.

$2 x = 3$

Step 1.2.5.2.2

Divide each term in $2 x = 3$ by $2$ and simplify.

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Step 1.2.5.2.2.1

Divide each term in $2 x = 3$ by $2$ .

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

Step 1.2.5.2.2.2

Simplify the left side.

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Step 1.2.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.5.2.2.2.1.1

Cancel the common factor.

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

Step 1.2.5.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 1.2.6

Set $5 x + 2$ equal to $0$ and solve for $x$ .

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Step 1.2.6.1

Set $5 x + 2$ equal to $0$ .

$5 x + 2 = 0$

Step 1.2.6.2

Solve $5 x + 2 = 0$ for $x$ .

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Step 1.2.6.2.1

Subtract $2$ from both sides of the equation.

$5 x = - 2$

Step 1.2.6.2.2

Divide each term in $5 x = - 2$ by $5$ and simplify.

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Step 1.2.6.2.2.1

Divide each term in $5 x = - 2$ by $5$ .

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

Step 1.2.6.2.2.2

Simplify the left side.

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Step 1.2.6.2.2.2.1

Cancel the common factor of $5$ .

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Step 1.2.6.2.2.2.1.1

Cancel the common factor.

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

Step 1.2.6.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 1.2.6.2.2.3

Simplify the right side.

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Step 1.2.6.2.2.3.1

Move the negative in front of the fraction.

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

Step 1.2.7

The final solution is all the values that make $- 2 x (2 x^{2} + 2) (2 x - 3) (5 x + 2) = 0$ true.

$x = 0, i, - i, \frac{3}{2}, - \frac{2}{5}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (\frac{3}{2}, 0), (- \frac{2}{5}, 0)$

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 $y$ .

$y = - 2 \cdot 0 (2 {(0)}^{2} + 2) (2 (0) - 3) (5 (0) + 2)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - 2 \cdot (0 (2 \cdot 0^{2} + 2) (2 (0) - 3) (5 (0) + 2))$

Step 2.2.2

Remove parentheses.

$y = - 2 \cdot (0 (2 {(0)}^{2} + 2) (2 (0) - 3) (5 (0) + 2))$

Step 2.2.3

Simplify $- 2 \cdot (0 (2 {(0)}^{2} + 2) (2 (0) - 3) (5 (0) + 2))$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

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

$y = - 2 \cdot (0 (2 \cdot 0 + 2) (2 (0) - 3) (5 (0) + 2))$

Step 2.2.3.1.2

Multiply $2$ by $0$ .

$y = - 2 \cdot (0 (0 + 2) (2 (0) - 3) (5 (0) + 2))$

Step 2.2.3.2

Simplify the expression.

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Step 2.2.3.2.1

Add $0$ and $2$ .

$y = - 2 \cdot (0 \cdot 2 (2 (0) - 3) (5 (0) + 2))$

Step 2.2.3.2.2

Multiply $0$ by $2$ .

$y = - 2 \cdot (0 (2 (0) - 3) (5 (0) + 2))$

Step 2.2.3.2.3

Multiply $2$ by $0$ .

$y = - 2 \cdot (0 (0 - 3) (5 (0) + 2))$

Step 2.2.3.2.4

Subtract $3$ from $0$ .

$y = - 2 \cdot (0 \cdot - 3 (5 (0) + 2))$

Step 2.2.3.2.5

Multiply $0$ by $- 3$ .

$y = - 2 \cdot (0 (5 (0) + 2))$

Step 2.2.3.2.6

Multiply $5$ by $0$ .

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

Step 2.2.3.2.7

Add $0$ and $2$ .

$y = - 2 \cdot (0 \cdot 2)$

Step 2.2.3.3

Multiply $- 2 (0 \cdot 2)$ .

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Step 2.2.3.3.1

Multiply $0$ by $2$ .

$y = - 2 \cdot 0$

Step 2.2.3.3.2

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): $(0, 0), (\frac{3}{2}, 0), (- \frac{2}{5}, 0)$

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

Step 4