Algebra Examples

${(x^{2} + 3)}^{2} = 4 x^{2} + 12$

Step 1

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

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Step 1.1

Subtract $4 x^{2}$ from both sides of the equation.

${(x^{2} + 3)}^{2} - 4 x^{2} = 12$

Step 1.2

Simplify each term.

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Step 1.2.1

Rewrite ${(x^{2} + 3)}^{2}$ as $(x^{2} + 3) (x^{2} + 3)$ .

$(x^{2} + 3) (x^{2} + 3) - 4 x^{2} = 12$

Step 1.2.2

Expand $(x^{2} + 3) (x^{2} + 3)$ using the FOIL Method.

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Step 1.2.2.1

Apply the distributive property.

$x^{2} (x^{2} + 3) + 3 (x^{2} + 3) - 4 x^{2} = 12$

Step 1.2.2.2

Apply the distributive property.

$x^{2} x^{2} + x^{2} \cdot 3 + 3 (x^{2} + 3) - 4 x^{2} = 12$

Step 1.2.2.3

Apply the distributive property.

$x^{2} x^{2} + x^{2} \cdot 3 + 3 x^{2} + 3 \cdot 3 - 4 x^{2} = 12$

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.

$x^{2 + 2} + x^{2} \cdot 3 + 3 x^{2} + 3 \cdot 3 - 4 x^{2} = 12$

Step 1.2.3.1.1.2

Add $2$ and $2$ .

$x^{4} + x^{2} \cdot 3 + 3 x^{2} + 3 \cdot 3 - 4 x^{2} = 12$

Step 1.2.3.1.2

Move $3$ to the left of $x^{2}$ .

$x^{4} + 3 \cdot x^{2} + 3 x^{2} + 3 \cdot 3 - 4 x^{2} = 12$

Step 1.2.3.1.3

Multiply $3$ by $3$ .

$x^{4} + 3 x^{2} + 3 x^{2} + 9 - 4 x^{2} = 12$

Step 1.2.3.2

Add $3 x^{2}$ and $3 x^{2}$ .

$x^{4} + 6 x^{2} + 9 - 4 x^{2} = 12$

Step 1.3

Subtract $4 x^{2}$ from $6 x^{2}$ .

$x^{4} + 2 x^{2} + 9 = 12$

Step 2

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

$u^{2} + 2 u + 9 = 12$

$u = x^{2}$

Step 3

Subtract $12$ from both sides of the equation.

$u^{2} + 2 u + 9 - 12 = 0$

Step 4

Subtract $12$ from $9$ .

$u^{2} + 2 u - 3 = 0$

Step 5

Factor $u^{2} + 2 u - 3$ 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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 5.2

Write the factored form using these integers.

$(u - 1) (u + 3) = 0$

Step 6

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

$u - 1 = 0$

$u + 3 = 0$

Step 7

Set $u - 1$ equal to $0$ and solve for $u$ .

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Step 7.1

Set $u - 1$ equal to $0$ .

$u - 1 = 0$

Step 7.2

Add $1$ to both sides of the equation.

$u = 1$

Step 8

Set $u + 3$ equal to $0$ and solve for $u$ .

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Step 8.1

Set $u + 3$ equal to $0$ .

$u + 3 = 0$

Step 8.2

Subtract $3$ from both sides of the equation.

$u = - 3$

Step 9

The final solution is all the values that make $(u - 1) (u + 3) = 0$ true.

$u = 1, - 3$

Step 10

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

$x^{2} = 1$

${(x^{2})}^{1} = - 3$

Step 11

Solve the first equation for $x$ .

$x^{2} = 1$

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.

$x = \pm \sqrt{1}$

Step 12.2

Any root of $1$ is $1$ .

$x = \pm 1$

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.

$x = 1$

Step 12.3.2

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

$x = - 1$

Step 12.3.3

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

$x = 1, - 1$

Step 13

Solve the second equation for $x$ .

${(x^{2})}^{1} = - 3$

Step 14

Solve the equation for $x$ .

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Step 14.1

Remove parentheses.

$x^{2} = - 3$

Step 14.2

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

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

Step 14.3

Simplify $\pm \sqrt{- 3}$ .

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Step 14.3.1

Rewrite $- 3$ as $- 1 (3)$ .

$x = \pm \sqrt{- 1 (3)}$

Step 14.3.2

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{3}$

Step 14.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 14.4

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

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Step 14.4.1

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

$x = i \sqrt{3}$

Step 14.4.2

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

$x = - i \sqrt{3}$

Step 14.4.3

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

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

Step 15

The solution to $x^{4} + 2 x^{2} + 9 = 12$ is $x = 1, - 1, i \sqrt{3}, - i \sqrt{3}$ .

$x = 1, - 1, i \sqrt{3}, - i \sqrt{3}$