Algebra Examples

$p (x) = (x^{2} - 9) (x^{2} + x - 2)$

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 $x$ .

$0 = (x^{2} - 9) (x^{2} + x - 2)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $(x^{2} - 9) (x^{2} + x - 2) = 0$ .

$(x^{2} - 9) (x^{2} + x - 2) = 0$

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 $0$ .

$x^{2} - 9 = 0$

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

Step 1.2.3

Set $x^{2} - 9$ equal to $0$ and solve for $x$ .

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Step 1.2.3.1

Set $x^{2} - 9$ equal to $0$ .

$x^{2} - 9 = 0$

Step 1.2.3.2

Solve $x^{2} - 9 = 0$ for $x$ .

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Step 1.2.3.2.1

Add $9$ to both sides of the equation.

$x^{2} = 9$

Step 1.2.3.2.2

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

$x = \pm \sqrt{9}$

Step 1.2.3.2.3

Simplify $\pm \sqrt{9}$ .

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Step 1.2.3.2.3.1

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

Step 1.2.3.2.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 3$

Step 1.2.3.2.4

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

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Step 1.2.3.2.4.1

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

$x = 3$

Step 1.2.3.2.4.2

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

$x = - 3$

Step 1.2.3.2.4.3

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

$x = 3, - 3$

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

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

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Step 1.2.4.2.1

Factor $x^{2} + x - 2$ using the AC method.

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Step 1.2.4.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 1.2.4.2.1.2

Write the factored form using these integers.

$(x - 1) (x + 2) = 0$

Step 1.2.4.2.2

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

$x - 1 = 0$

$x + 2 = 0$

Step 1.2.4.2.3

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 1.2.4.2.3.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 1.2.4.2.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.4.2.4

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

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Step 1.2.4.2.4.1

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

$x + 2 = 0$

Step 1.2.4.2.4.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.2.4.2.5

The final solution is all the values that make $(x - 1) (x + 2) = 0$ true.

$x = 1, - 2$

Step 1.2.5

The final solution is all the values that make $(x^{2} - 9) (x^{2} + x - 2) = 0$ true.

$x = 3, - 3, 1, - 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(3, 0), (- 3, 0), (1, 0), (- 2, 0)$

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 $y$ .

$y = ({(0)}^{2} - 9) ({(0)}^{2} + 0 - 2)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = ({(0)}^{2} - 9) ({(0)}^{2} + 0 - 2)$

Step 2.2.2

Remove parentheses.

$y = (0^{2} - 9) ({(0)}^{2} + 0 - 2)$

Step 2.2.3

Remove parentheses.

$y = (0^{2} - 9) (0^{2} + 0 - 2)$

Step 2.2.4

Remove parentheses.

$y = ({(0)}^{2} - 9) ({(0)}^{2} + 0 - 2)$

Step 2.2.5

Simplify $({(0)}^{2} - 9) ({(0)}^{2} + 0 - 2)$ .

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Step 2.2.5.1

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

$y = (0 - 9) ({(0)}^{2} + 0 - 2)$

Step 2.2.5.2

Subtract $9$ from $0$ .

$y = - 9 ({(0)}^{2} + 0 - 2)$

Step 2.2.5.3

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

$y = - 9 (0 + 0 - 2)$

Step 2.2.5.4

Add $0$ and $0$ .

$y = - 9 (0 - 2)$

Step 2.2.5.5

Subtract $2$ from $0$ .

$y = - 9 \cdot - 2$

Step 2.2.5.6

Multiply $- 9$ by $- 2$ .

$y = 18$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 18)$

Step 3

List the intersections.

x-intercept(s): $(3, 0), (- 3, 0), (1, 0), (- 2, 0)$

y-intercept(s): $(0, 18)$

Step 4