Algebra Examples

$x^{7} - 63 x^{5} + 486 x^{3}$

Step 1

Set $x^{7} - 63 x^{5} + 486 x^{3}$ equal to $0$ .

$x^{7} - 63 x^{5} + 486 x^{3} = 0$

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 $x^{3}$ out of $x^{7} - 63 x^{5} + 486 x^{3}$ .

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Step 2.1.1.1

Factor $x^{3}$ out of $x^{7}$ .

$x^{3} x^{4} - 63 x^{5} + 486 x^{3} = 0$

Step 2.1.1.2

Factor $x^{3}$ out of $- 63 x^{5}$ .

$x^{3} x^{4} + x^{3} (- 63 x^{2}) + 486 x^{3} = 0$

Step 2.1.1.3

Factor $x^{3}$ out of $486 x^{3}$ .

$x^{3} x^{4} + x^{3} (- 63 x^{2}) + x^{3} \cdot 486 = 0$

Step 2.1.1.4

Factor $x^{3}$ out of $x^{3} x^{4} + x^{3} (- 63 x^{2})$ .

$x^{3} (x^{4} - 63 x^{2}) + x^{3} \cdot 486 = 0$

Step 2.1.1.5

Factor $x^{3}$ out of $x^{3} (x^{4} - 63 x^{2}) + x^{3} \cdot 486$ .

$x^{3} (x^{4} - 63 x^{2} + 486) = 0$

Step 2.1.2

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

$x^{3} ({(x^{2})}^{2} - 63 x^{2} + 486) = 0$

Step 2.1.3

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

$x^{3} (u^{2} - 63 u + 486) = 0$

Step 2.1.4

Factor $u^{2} - 63 u + 486$ using the AC method.

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Step 2.1.4.1

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 $486$ and whose sum is $- 63$ .

$- 54, - 9$

Step 2.1.4.2

Write the factored form using these integers.

$x^{3} ((u - 54) (u - 9)) = 0$

Step 2.1.5

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

$x^{3} ((x^{2} - 54) (x^{2} - 9)) = 0$

Step 2.1.6

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

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

Step 2.1.7

Factor.

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Step 2.1.7.1

Factor.

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Step 2.1.7.1.1

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 $b = 3$ .

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

Step 2.1.7.1.2

Remove unnecessary parentheses.

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

Step 2.1.7.2

Remove unnecessary parentheses.

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

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 $0$ .

$x^{3} = 0$

$x^{2} - 54 = 0$

$x + 3 = 0$

$x - 3 = 0$

Step 2.3

Set $x^{3}$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x^{3}$ equal to $0$ .

$x^{3} = 0$

Step 2.3.2

Solve $x^{3} = 0$ for $x$ .

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Step 2.3.2.1

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

$x = \sqrt[3]{0}$

Step 2.3.2.2

Simplify $\sqrt[3]{0}$ .

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Step 2.3.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 2.3.2.2.2

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

$x = 0$

Step 2.4

Set $x^{2} - 54$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x^{2} - 54$ equal to $0$ .

$x^{2} - 54 = 0$

Step 2.4.2

Solve $x^{2} - 54 = 0$ for $x$ .

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Step 2.4.2.1

Add $54$ to both sides of the equation.

$x^{2} = 54$

Step 2.4.2.2

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

$x = \pm \sqrt{54}$

Step 2.4.2.3

Simplify $\pm \sqrt{54}$ .

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Step 2.4.2.3.1

Rewrite $54$ as $3^{2} \cdot 6$ .

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Step 2.4.2.3.1.1

Factor $9$ out of $54$ .

$x = \pm \sqrt{9 (6)}$

Step 2.4.2.3.1.2

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

$x = \pm \sqrt{3^{2} \cdot 6}$

Step 2.4.2.3.2

Pull terms out from under the radical.

$x = \pm 3 \sqrt{6}$

Step 2.4.2.4

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

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Step 2.4.2.4.1

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

$x = 3 \sqrt{6}$

Step 2.4.2.4.2

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

$x = - 3 \sqrt{6}$

Step 2.4.2.4.3

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

$x = 3 \sqrt{6}, - 3 \sqrt{6}$

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

$x = 0, 3 \sqrt{6}, - 3 \sqrt{6}, - 3, 3$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 0, 3 \sqrt{6}, - 3 \sqrt{6}, - 3, 3$

Decimal Form:

$x = 0, 7.34846922 \dots, - 7.34846922 \dots, - 3, 3$

Step 4