Algebra Examples

$x^{3} + x^{2} = 4 x + 4$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $4 x$ from both sides of the equation.

$x^{3} + x^{2} - 4 x = 4$

Step 1.2

Subtract $4$ from both sides of the equation.

$x^{3} + x^{2} - 4 x - 4 = 0$

Step 2

Factor out the greatest common factor from each group.

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Step 2.1

Group the first two terms and the last two terms.

$(x^{3} + x^{2}) - 4 x - 4 = 0$

Step 2.2

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

$x^{2} (x + 1) - 4 (x + 1) = 0$

Step 3

Factor the polynomial by factoring out the greatest common factor, $x + 1$ .

$(x + 1) (x^{2} - 4) = 0$

Step 4

Rewrite $4$ as $2^{2}$ .

$(x + 1) (x^{2} - 2^{2}) = 0$

Step 5

Factor.

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Step 5.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

Step 5.2

Remove unnecessary parentheses.

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

Step 6

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

$x + 1 = 0$

$x + 2 = 0$

$x - 2 = 0$

Step 7

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

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Step 7.1

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

$x + 1 = 0$

Step 7.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 8

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

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Step 8.1

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

$x + 2 = 0$

Step 8.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 9

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

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Step 9.1

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

$x - 2 = 0$

Step 9.2

Add $2$ to both sides of the equation.

$x = 2$

Step 10

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

$x = - 1, - 2, 2$