Algebra Examples

${(x^{2} - 4)}^{2} - 2 (x^{2} - 4) = 24$

Step 1

Subtract $24$ from both sides of the equation.

${(x^{2} - 4)}^{2} - 2 (x^{2} - 4) - 24 = 0$

Step 2

Simplify ${(x^{2} - 4)}^{2} - 2 (x^{2} - 4) - 24$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$(x^{2} - 4) (x^{2} - 4) - 2 (x^{2} - 4) - 24 = 0$

Step 2.1.2

Expand $(x^{2} - 4) (x^{2} - 4)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$x^{2} (x^{2} - 4) - 4 (x^{2} - 4) - 2 (x^{2} - 4) - 24 = 0$

Step 2.1.2.2

Apply the distributive property.

$x^{2} x^{2} + x^{2} \cdot - 4 - 4 (x^{2} - 4) - 2 (x^{2} - 4) - 24 = 0$

Step 2.1.2.3

Apply the distributive property.

$x^{2} x^{2} + x^{2} \cdot - 4 - 4 x^{2} - 4 \cdot - 4 - 2 (x^{2} - 4) - 24 = 0$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.1.3.1.1.1

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

$x^{2 + 2} + x^{2} \cdot - 4 - 4 x^{2} - 4 \cdot - 4 - 2 (x^{2} - 4) - 24 = 0$

Step 2.1.3.1.1.2

Add $2$ and $2$ .

$x^{4} + x^{2} \cdot - 4 - 4 x^{2} - 4 \cdot - 4 - 2 (x^{2} - 4) - 24 = 0$

Step 2.1.3.1.2

Move $- 4$ to the left of $x^{2}$ .

$x^{4} - 4 \cdot x^{2} - 4 x^{2} - 4 \cdot - 4 - 2 (x^{2} - 4) - 24 = 0$

Step 2.1.3.1.3

Multiply $- 4$ by $- 4$ .

$x^{4} - 4 x^{2} - 4 x^{2} + 16 - 2 (x^{2} - 4) - 24 = 0$

Step 2.1.3.2

Subtract $4 x^{2}$ from $- 4 x^{2}$ .

$x^{4} - 8 x^{2} + 16 - 2 (x^{2} - 4) - 24 = 0$

Step 2.1.4

Apply the distributive property.

$x^{4} - 8 x^{2} + 16 - 2 x^{2} - 2 \cdot - 4 - 24 = 0$

Step 2.1.5

Multiply $- 2$ by $- 4$ .

$x^{4} - 8 x^{2} + 16 - 2 x^{2} + 8 - 24 = 0$

Step 2.2

Subtract $2 x^{2}$ from $- 8 x^{2}$ .

$x^{4} - 10 x^{2} + 16 + 8 - 24 = 0$

Step 2.3

Add $16$ and $8$ .

$x^{4} - 10 x^{2} + 24 - 24 = 0$

Step 2.4

Combine the opposite terms in $x^{4} - 10 x^{2} + 24 - 24$ .

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Step 2.4.1

Subtract $24$ from $24$ .

$x^{4} - 10 x^{2} + 0 = 0$

Step 2.4.2

Add $x^{4} - 10 x^{2}$ and $0$ .

$x^{4} - 10 x^{2} = 0$

Step 3

Factor $x^{2}$ out of $x^{4} - 10 x^{2}$ .

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Step 3.1

Factor $x^{2}$ out of $x^{4}$ .

$x^{2} x^{2} - 10 x^{2} = 0$

Step 3.2

Factor $x^{2}$ out of $- 10 x^{2}$ .

$x^{2} x^{2} + x^{2} \cdot - 10 = 0$

Step 3.3

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 10$ .

$x^{2} (x^{2} - 10) = 0$

Step 4

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

$x^{2} = 0$

$x^{2} - 10 = 0$

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

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

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Step 5.2.1

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

$x = \pm \sqrt{0}$

Step 5.2.2

Simplify $\pm \sqrt{0}$ .

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Step 5.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 5.2.2.2

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

$x = \pm 0$

Step 5.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6

Set $x^{2} - 10$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x^{2} - 10$ equal to $0$ .

$x^{2} - 10 = 0$

Step 6.2

Solve $x^{2} - 10 = 0$ for $x$ .

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Step 6.2.1

Add $10$ to both sides of the equation.

$x^{2} = 10$

Step 6.2.2

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

$x = \pm \sqrt{10}$

Step 6.2.3

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

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Step 6.2.3.1

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

$x = \sqrt{10}$

Step 6.2.3.2

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

$x = - \sqrt{10}$

Step 6.2.3.3

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

$x = \sqrt{10}, - \sqrt{10}$

Step 7

The final solution is all the values that make $x^{2} (x^{2} - 10) = 0$ true.

$x = 0, \sqrt{10}, - \sqrt{10}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 0, \sqrt{10}, - \sqrt{10}$

Decimal Form:

$x = 0, 3.16227766 \dots, - 3.16227766 \dots$