Algebra Examples

$x^{10} - 31 x^{5} - 32 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Rewrite $x^{10}$ as ${(x^{5})}^{2}$ .

${(x^{5})}^{2} - 31 x^{5} - 32 = 0$

Step 1.2

Let $u = x^{5}$ . Substitute $u$ for all occurrences of $x^{5}$ .

$u^{2} - 31 u - 32 = 0$

Step 1.3

Factor $u^{2} - 31 u - 32$ using the AC method.

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Step 1.3.1

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 $- 32$ and whose sum is $- 31$ .

$- 32, 1$

Step 1.3.2

Write the factored form using these integers.

$(u - 32) (u + 1) = 0$

Step 1.4

Replace all occurrences of $u$ with $x^{5}$ .

$(x^{5} - 32) (x^{5} + 1) = 0$

Step 2

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

$x^{5} - 32 = 0$

$x^{5} + 1 = 0$

Step 3

Set $x^{5} - 32$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x^{5} - 32$ equal to $0$ .

$x^{5} - 32 = 0$

Step 3.2

Solve $x^{5} - 32 = 0$ for $x$ .

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Step 3.2.1

Add $32$ to both sides of the equation.

$x^{5} = 32$

Step 3.2.2

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

$x = \sqrt[5]{32}$

Step 3.2.3

Simplify $\sqrt[5]{32}$ .

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Step 3.2.3.1

Rewrite $32$ as $2^{5}$ .

$x = \sqrt[5]{2^{5}}$

Step 3.2.3.2

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

$x = 2$

Step 4

Set $x^{5} + 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{5} + 1$ equal to $0$ .

$x^{5} + 1 = 0$

Step 4.2

Solve $x^{5} + 1 = 0$ for $x$ .

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$x^{5} = - 1$

Step 4.2.2

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

$x = \sqrt[5]{- 1}$

Step 4.2.3

Simplify $\sqrt[5]{- 1}$ .

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Step 4.2.3.1

Rewrite $- 1$ as ${(- 1)}^{5}$ .

$x = \sqrt[5]{{(- 1)}^{5}}$

Step 4.2.3.2

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

$x = - 1$

Step 5

The final solution is all the values that make $(x^{5} - 32) (x^{5} + 1) = 0$ true.

$x = 2, - 1$

Step 6