Algebra Examples

$x^{4} - x^{3} - 15 x^{2} + 9 x + 54 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Regroup terms.

$- x^{3} + 9 x + x^{4} - 15 x^{2} + 54 = 0$

Step 1.2

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

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Step 1.2.1

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

$- x \cdot x^{2} + 9 x + x^{4} - 15 x^{2} + 54 = 0$

Step 1.2.2

Factor $- x$ out of $9 x$ .

$- x \cdot x^{2} - x \cdot - 9 + x^{4} - 15 x^{2} + 54 = 0$

Step 1.2.3

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

$- x (x^{2} - 9) + x^{4} - 15 x^{2} + 54 = 0$

Step 1.3

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

$- x (x^{2} - 3^{2}) + x^{4} - 15 x^{2} + 54 = 0$

Step 1.4

Factor.

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Step 1.4.1

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 $b = 3$ .

$- x ((x + 3) (x - 3)) + x^{4} - 15 x^{2} + 54 = 0$

Step 1.4.2

Remove unnecessary parentheses.

$- x (x + 3) (x - 3) + x^{4} - 15 x^{2} + 54 = 0$

Step 1.5

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$- x (x + 3) (x - 3) + {(x^{2})}^{2} - 15 x^{2} + 54 = 0$

Step 1.6

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$- x (x + 3) (x - 3) + u^{2} - 15 u + 54 = 0$

Step 1.7

Factor $u^{2} - 15 u + 54$ using the AC method.

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Step 1.7.1

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 $54$ and whose sum is $- 15$ .

$- 9, - 6$

Step 1.7.2

Write the factored form using these integers.

$- x (x + 3) (x - 3) + (u - 9) (u - 6) = 0$

Step 1.8

Replace all occurrences of $u$ with $x^{2}$ .

$- x (x + 3) (x - 3) + (x^{2} - 9) (x^{2} - 6) = 0$

Step 1.9

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

$- x (x + 3) (x - 3) + (x^{2} - 3^{2}) (x^{2} - 6) = 0$

Step 1.10

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 $b = 3$ .

$- x (x + 3) (x - 3) + (x + 3) (x - 3) (x^{2} - 6) = 0$

Step 1.11

Factor $(x + 3) (x - 3)$ out of $- x (x + 3) (x - 3) + (x + 3) (x - 3) (x^{2} - 6)$ .

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Step 1.11.1

Factor $(x + 3) (x - 3)$ out of $- x (x + 3) (x - 3)$ .

$(x + 3) (x - 3) (- x) + (x + 3) (x - 3) (x^{2} - 6) = 0$

Step 1.11.2

Factor $(x + 3) (x - 3)$ out of $(x + 3) (x - 3) (- x) + (x + 3) (x - 3) (x^{2} - 6)$ .

$(x + 3) (x - 3) (- x + x^{2} - 6) = 0$

Step 1.12

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$(x + 3) (x - 3) (- u + u^{2} - 6) = 0$

Step 1.13

Factor $- u + u^{2} - 6$ using the AC method.

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Step 1.13.1

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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 1.13.2

Write the factored form using these integers.

$(x + 3) (x - 3) ((u - 3) (u + 2)) = 0$

Step 1.14

Factor.

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Step 1.14.1

Replace all occurrences of $u$ with $x$ .

$(x + 3) (x - 3) ((x - 3) (x + 2)) = 0$

Step 1.14.2

Remove unnecessary parentheses.

$(x + 3) (x - 3) (x - 3) (x + 2) = 0$

Step 1.15

Combine exponents.

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Step 1.15.1

Raise $x - 3$ to the power of $1$ .

$(x + 3) ((x - 3) (x - 3)) (x + 2) = 0$

Step 1.15.2

Raise $x - 3$ to the power of $1$ .

$(x + 3) ((x - 3) (x - 3)) (x + 2) = 0$

Step 1.15.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(x + 3) {(x - 3)}^{1 + 1} (x + 2) = 0$

Step 1.15.4

Add $1$ and $1$ .

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

Step 2

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

$x + 3 = 0$

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

$x + 2 = 0$

Step 3

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

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Step 3.1

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

$x + 3 = 0$

Step 3.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 4

Set ${(x - 3)}^{2}$ equal to $0$ and solve for $x$ .

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Step 4.1

Set ${(x - 3)}^{2}$ equal to $0$ .

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

Step 4.2

Solve ${(x - 3)}^{2} = 0$ for $x$ .

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Step 4.2.1

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

$x - 3 = 0$

Step 4.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 5

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

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Step 5.1

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

$x + 2 = 0$

Step 5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 6

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

$x = - 3, 3, - 2$

Step 7