Solve for x (x^2+6)^2-21=4x^2+24

${(x^{2} + 6)}^{2} - 21 = 4 x^{2} + 24$

Step 1

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

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

${(x^{2} + 6)}^{2} - 21 - 4 x^{2} = 24$

Simplify each term.

Rewrite ${(x^{2} + 6)}^{2}$ as $(x^{2} + 6) (x^{2} + 6)$ .

$(x^{2} + 6) (x^{2} + 6) - 21 - 4 x^{2} = 24$

Expand $(x^{2} + 6) (x^{2} + 6)$ using the FOIL Method.

Apply the distributive property.

$x^{2} (x^{2} + 6) + 6 (x^{2} + 6) - 21 - 4 x^{2} = 24$

Apply the distributive property.

$x^{2} x^{2} + x^{2} \cdot 6 + 6 (x^{2} + 6) - 21 - 4 x^{2} = 24$

Apply the distributive property.

$x^{2} x^{2} + x^{2} \cdot 6 + 6 x^{2} + 6 \cdot 6 - 21 - 4 x^{2} = 24$

Simplify and combine like terms.

Simplify each term.

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

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

$x^{2 + 2} + x^{2} \cdot 6 + 6 x^{2} + 6 \cdot 6 - 21 - 4 x^{2} = 24$

Add $2$ and $2$ .

$x^{4} + x^{2} \cdot 6 + 6 x^{2} + 6 \cdot 6 - 21 - 4 x^{2} = 24$

Move $6$ to the left of $x^{2}$ .

$x^{4} + 6 \cdot x^{2} + 6 x^{2} + 6 \cdot 6 - 21 - 4 x^{2} = 24$

Multiply $6$ by $6$ .

$x^{4} + 6 x^{2} + 6 x^{2} + 36 - 21 - 4 x^{2} = 24$

Add $6 x^{2}$ and $6 x^{2}$ .

$x^{4} + 12 x^{2} + 36 - 21 - 4 x^{2} = 24$

Subtract $4 x^{2}$ from $12 x^{2}$ .

$x^{4} + 8 x^{2} + 36 - 21 = 24$

Subtract $21$ from $36$ .

$x^{4} + 8 x^{2} + 15 = 24$

Step 2

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

$u^{2} + 8 u + 15 = 24$

$u = x^{2}$

Step 3

Subtract $24$ from both sides of the equation.

$u^{2} + 8 u + 15 - 24 = 0$

Step 4

Subtract $24$ from $15$ .

$u^{2} + 8 u - 9 = 0$

Step 5

Factor $u^{2} + 8 u - 9$ using the AC method.

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 $- 9$ and whose sum is $8$ .

$- 1, 9$

Write the factored form using these integers.

$(u - 1) (u + 9) = 0$

Step 6

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

$u - 1 = 0$

$u + 9 = 0$

Step 7

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

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

$u - 1 = 0$

Add $1$ to both sides of the equation.

$u = 1$

Step 8

Set $u + 9$ equal to $0$ and solve for $u$ .

Set $u + 9$ equal to $0$ .

$u + 9 = 0$

Subtract $9$ from both sides of the equation.

$u = - 9$

Step 9

The final solution is all the values that make $(u - 1) (u + 9) = 0$ true.

$u = 1, - 9$

Step 10

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

$x^{2} = 1$

${(x^{2})}^{1} = - 9$

Step 11

Solve the first equation for $x$ .

$x^{2} = 1$

Step 12

Solve the equation for $x$ .

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

$x = \pm \sqrt{1}$

Any root of $1$ is $1$ .

$x = \pm 1$

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

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

$x = 1$

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

$x = - 1$

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 $x$ .

${(x^{2})}^{1} = - 9$

Step 14

Solve the equation for $x$ .

Remove parentheses.

$x^{2} = - 9$

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

$x = \pm \sqrt{- 9}$

Simplify $\pm \sqrt{- 9}$ .

Rewrite $- 9$ as $- 1 (9)$ .

$x = \pm \sqrt{- 1 (9)}$

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{9}$

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

$x = \pm i \cdot \sqrt{9}$

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

$x = \pm i \cdot \sqrt{3^{2}}$

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

$x = \pm i \cdot 3$

Move $3$ to the left of $i$ .

$x = \pm 3 i$

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

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

$x = 3 i$

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

$x = - 3 i$

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

$x = 3 i, - 3 i$

Step 15

The solution to $x^{4} + 8 x^{2} + 15 = 24$ is $x = 1, - 1, 3 i, - 3 i$ .

$x = 1, - 1, 3 i, - 3 i$

Solve for x (x^2+6)^2-21=4x^2+24

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