Algebra Examples

$x^{5} = 18 x^{3} - 81 x$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$18 x^{3} - 81 x = x^{5}$

Step 2

Subtract $x^{5}$ from both sides of the equation.

$18 x^{3} - 81 x - x^{5} = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $- x$ out of $18 x^{3} - 81 x - x^{5}$ .

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Step 3.1.1

Reorder the expression.

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Step 3.1.1.1

Move $- 81 x$ .

$18 x^{3} - x^{5} - 81 x = 0$

Step 3.1.1.2

Reorder $18 x^{3}$ and $- x^{5}$ .

$- x^{5} + 18 x^{3} - 81 x = 0$

Step 3.1.2

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

$- x \cdot x^{4} + 18 x^{3} - 81 x = 0$

Step 3.1.3

Factor $- x$ out of $18 x^{3}$ .

$- x \cdot x^{4} - x (- 18 x^{2}) - 81 x = 0$

Step 3.1.4

Factor $- x$ out of $- 81 x$ .

$- x \cdot x^{4} - x (- 18 x^{2}) - x \cdot 81 = 0$

Step 3.1.5

Factor $- x$ out of $- x (x^{4}) - x (- 18 x^{2})$ .

$- x (x^{4} - 18 x^{2}) - x \cdot 81 = 0$

Step 3.1.6

Factor $- x$ out of $- x (x^{4} - 18 x^{2}) - x (81)$ .

$- x (x^{4} - 18 x^{2} + 81) = 0$

Step 3.2

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

$- x ({(x^{2})}^{2} - 18 x^{2} + 81) = 0$

Step 3.3

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

$- x (u^{2} - 18 u + 81) = 0$

Step 3.4

Factor using the perfect square rule.

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Step 3.4.1

Rewrite $81$ as $9^{2}$ .

$- x (u^{2} - 18 u + 9^{2}) = 0$

Step 3.4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$18 u = 2 \cdot u \cdot 9$

Step 3.4.3

Rewrite the polynomial.

$- x (u^{2} - 2 \cdot u \cdot 9 + 9^{2}) = 0$

Step 3.4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 9$ .

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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 $b = 3$ .

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

Step 3.8

Factor.

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Step 3.8.1

Apply the product rule to $(x + 3) (x - 3)$ .

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

Step 3.8.2

Remove unnecessary parentheses.

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

Step 4

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

$x = 0$

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

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

Step 5

Set $x$ equal to $0$ .

$x = 0$

Step 6

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

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

$x + 3 = 0$

Step 6.2.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 7

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

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Step 7.1

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

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

Step 7.2

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

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Step 7.2.1

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

$x - 3 = 0$

Step 7.2.2

Add $3$ to both sides of the equation.

$x = 3$

Step 8

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

$x = 0, - 3, 3$