Algebra Examples

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

Step 1

Move all terms containing $x$ to the left side of the equation.

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Step 1.1

Subtract $9 x^{2}$ from both sides of the equation.

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

Step 1.2

Simplify each term.

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Step 1.2.1

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

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

Step 1.2.2

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

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Step 1.2.2.1

Apply the distributive property.

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

Step 1.2.2.2

Apply the distributive property.

$x^{2} x^{2} + x^{2} \cdot - 2 - 2 (x^{2} - 2) + 18 - 9 x^{2} = - 18$

Step 1.2.2.3

Apply the distributive property.

$x^{2} x^{2} + x^{2} \cdot - 2 - 2 x^{2} - 2 \cdot - 2 + 18 - 9 x^{2} = - 18$

Step 1.2.3

Simplify and combine like terms.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

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

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Step 1.2.3.1.1.1

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

$x^{2 + 2} + x^{2} \cdot - 2 - 2 x^{2} - 2 \cdot - 2 + 18 - 9 x^{2} = - 18$

Step 1.2.3.1.1.2

Add $2$ and $2$ .

$x^{4} + x^{2} \cdot - 2 - 2 x^{2} - 2 \cdot - 2 + 18 - 9 x^{2} = - 18$

Step 1.2.3.1.2

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

$x^{4} - 2 \cdot x^{2} - 2 x^{2} - 2 \cdot - 2 + 18 - 9 x^{2} = - 18$

Step 1.2.3.1.3

Multiply $- 2$ by $- 2$ .

$x^{4} - 2 x^{2} - 2 x^{2} + 4 + 18 - 9 x^{2} = - 18$

Step 1.2.3.2

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

$x^{4} - 4 x^{2} + 4 + 18 - 9 x^{2} = - 18$

Step 1.3

Subtract $9 x^{2}$ from $- 4 x^{2}$ .

$x^{4} - 13 x^{2} + 4 + 18 = - 18$

Step 1.4

Add $4$ and $18$ .

$x^{4} - 13 x^{2} + 22 = - 18$

Step 2

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} - 13 u + 22 = - 18$

$u = x^{2}$

Step 3

Add $18$ to both sides of the equation.

$u^{2} - 13 u + 22 + 18 = 0$

Step 4

Add $22$ and $18$ .

$u^{2} - 13 u + 40 = 0$

Step 5

Factor $u^{2} - 13 u + 40$ using the AC method.

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Step 5.1

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 $40$ and whose sum is $- 13$ .

$- 8, - 5$

Step 5.2

Write the factored form using these integers.

$(u - 8) (u - 5) = 0$

Step 6

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

$u - 8 = 0$

$u - 5 = 0$

Step 7

Set $u - 8$ equal to $0$ and solve for $u$ .

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Step 7.1

Set $u - 8$ equal to $0$ .

$u - 8 = 0$

Step 7.2

Add $8$ to both sides of the equation.

$u = 8$

Step 8

Set $u - 5$ equal to $0$ and solve for $u$ .

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Step 8.1

Set $u - 5$ equal to $0$ .

$u - 5 = 0$

Step 8.2

Add $5$ to both sides of the equation.

$u = 5$

Step 9

The final solution is all the values that make $(u - 8) (u - 5) = 0$ true.

$u = 8, 5$

Step 10

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = 8$

${(x^{2})}^{1} = 5$

Step 11

Solve the first equation for $x$ .

$x^{2} = 8$

Step 12

Solve the equation for $x$ .

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Step 12.1

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

$x = \pm \sqrt{8}$

Step 12.2

Simplify $\pm \sqrt{8}$ .

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Step 12.2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 12.2.1.1

Factor $4$ out of $8$ .

$x = \pm \sqrt{4 (2)}$

Step 12.2.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 12.2.2

Pull terms out from under the radical.

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

Step 12.3

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

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Step 12.3.1

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

$x = 2 \sqrt{2}$

Step 12.3.2

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

$x = - 2 \sqrt{2}$

Step 12.3.3

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

$x = 2 \sqrt{2}, - 2 \sqrt{2}$

Step 13

Solve the second equation for $x$ .

${(x^{2})}^{1} = 5$

Step 14

Solve the equation for $x$ .

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Step 14.1

Remove parentheses.

$x^{2} = 5$

Step 14.2

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

$x = \pm \sqrt{5}$

Step 14.3

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

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Step 14.3.1

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

$x = \sqrt{5}$

Step 14.3.2

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

$x = - \sqrt{5}$

Step 14.3.3

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

$x = \sqrt{5}, - \sqrt{5}$

Step 15

The solution to $x^{4} - 13 x^{2} + 22 = - 18$ is $x = 2 \sqrt{2}, - 2 \sqrt{2}, \sqrt{5}, - \sqrt{5}$ .

$x = 2 \sqrt{2}, - 2 \sqrt{2}, \sqrt{5}, - \sqrt{5}$

Step 16

The result can be shown in multiple forms.

Exact Form:

$x = 2 \sqrt{2}, - 2 \sqrt{2}, \sqrt{5}, - \sqrt{5}$

Decimal Form:

$x = 2.82842712 \dots, - 2.82842712 \dots, 2.23606797 \dots, - 2.23606797 \dots$