Algebra Examples

$0 = x^{3} + 9 x^{2} - 7 x - 63$

Step 1

Rewrite the equation as $x^{3} + 9 x^{2} - 7 x - 63 = 0$ .

$x^{3} + 9 x^{2} - 7 x - 63 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Factor out the greatest common factor from each group.

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Step 2.1.1

Group the first two terms and the last two terms.

$(x^{3} + 9 x^{2}) - 7 x - 63 = 0$

Step 2.1.2

Factor out the greatest common factor (GCF) from each group.

$x^{2} (x + 9) - 7 (x + 9) = 0$

Step 2.2

Factor the polynomial by factoring out the greatest common factor, $x + 9$ .

$(x + 9) (x^{2} - 7) = 0$

Step 3

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

$x + 9 = 0$

$x^{2} - 7 = 0$

Step 4

Set $x + 9$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Step 4.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 5

Set $x^{2} - 7$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x^{2} - 7$ equal to $0$ .

$x^{2} - 7 = 0$

Step 5.2

Solve $x^{2} - 7 = 0$ for $x$ .

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Step 5.2.1

Add $7$ to both sides of the equation.

$x^{2} = 7$

Step 5.2.2

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

$x = \pm \sqrt{7}$

Step 5.2.3

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

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Step 5.2.3.1

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

$x = \sqrt{7}$

Step 5.2.3.2

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

$x = - \sqrt{7}$

Step 5.2.3.3

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

$x = \sqrt{7}, - \sqrt{7}$

Step 6

The final solution is all the values that make $(x + 9) (x^{2} - 7) = 0$ true.

$x = - 9, \sqrt{7}, - \sqrt{7}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - 9, \sqrt{7}, - \sqrt{7}$

Decimal Form:

$x = - 9, 2.64575131 \dots, - 2.64575131 \dots$

Step 8