Algebra Examples

$x^{4} - 4 x^{3} - 9 x^{2} + 36 x$

Step 1

Set $x^{4} - 4 x^{3} - 9 x^{2} + 36 x$ equal to $0$ .

$x^{4} - 4 x^{3} - 9 x^{2} + 36 x = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $x$ out of $x^{4} - 4 x^{3} - 9 x^{2} + 36 x$ .

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Step 2.1.1.1

Factor $x$ out of $x^{4}$ .

$x \cdot x^{3} - 4 x^{3} - 9 x^{2} + 36 x = 0$

Step 2.1.1.2

Factor $x$ out of $- 4 x^{3}$ .

$x \cdot x^{3} + x (- 4 x^{2}) - 9 x^{2} + 36 x = 0$

Step 2.1.1.3

Factor $x$ out of $- 9 x^{2}$ .

$x \cdot x^{3} + x (- 4 x^{2}) + x (- 9 x) + 36 x = 0$

Step 2.1.1.4

Factor $x$ out of $36 x$ .

$x \cdot x^{3} + x (- 4 x^{2}) + x (- 9 x) + x \cdot 36 = 0$

Step 2.1.1.5

Factor $x$ out of $x \cdot x^{3} + x (- 4 x^{2})$ .

$x (x^{3} - 4 x^{2}) + x (- 9 x) + x \cdot 36 = 0$

Step 2.1.1.6

Factor $x$ out of $x (x^{3} - 4 x^{2}) + x (- 9 x)$ .

$x (x^{3} - 4 x^{2} - 9 x) + x \cdot 36 = 0$

Step 2.1.1.7

Factor $x$ out of $x (x^{3} - 4 x^{2} - 9 x) + x \cdot 36$ .

$x (x^{3} - 4 x^{2} - 9 x + 36) = 0$

Step 2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1

Group the first two terms and the last two terms.

$x ((x^{3} - 4 x^{2}) - 9 x + 36) = 0$

Step 2.1.2.2

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

$x (x^{2} (x - 4) - 9 (x - 4)) = 0$

Step 2.1.3

Factor the polynomial by factoring out the greatest common factor, $x - 4$ .

$x ((x - 4) (x^{2} - 9)) = 0$

Step 2.1.4

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

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

Step 2.1.5

Factor.

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Step 2.1.5.1

Factor.

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Step 2.1.5.1.1

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 $b = 3$ .

$x ((x - 4) ((x + 3) (x - 3))) = 0$

Step 2.1.5.1.2

Remove unnecessary parentheses.

$x ((x - 4) (x + 3) (x - 3)) = 0$

Step 2.1.5.2

Remove unnecessary parentheses.

$x (x - 4) (x + 3) (x - 3) = 0$

Step 2.2

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

$x = 0$

$x - 4 = 0$

$x + 3 = 0$

$x - 3 = 0$

Step 2.3

Set $x$ equal to $0$ .

$x = 0$

Step 2.4

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 2.4.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.5

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 2.5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2.6

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

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Step 2.6.1

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

$x - 3 = 0$

Step 2.6.2

Add $3$ to both sides of the equation.

$x = 3$

Step 2.7

The final solution is all the values that make $x (x - 4) (x + 3) (x - 3) = 0$ true.

$x = 0, 4, - 3, 3$

Step 3