Solve by Factoring (x^2+4)^2+32=12x^2+48

${(x^{2} + 4)}^{2} + 32 = 12 x^{2} + 48$

Step 1

Move all the expressions to the left side of the equation.

Subtract $12 x^{2}$ from both sides of the equation.

${(x^{2} + 4)}^{2} + 32 - 12 x^{2} = 48$

Subtract $48$ from both sides of the equation.

${(x^{2} + 4)}^{2} + 32 - 12 x^{2} - 48 = 0$

Step 2

Simplify ${(x^{2} + 4)}^{2} + 32 - 12 x^{2} - 48$ .

Simplify each term.

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

$(x^{2} + 4) (x^{2} + 4) + 32 - 12 x^{2} - 48 = 0$

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

Apply the distributive property.

$x^{2} (x^{2} + 4) + 4 (x^{2} + 4) + 32 - 12 x^{2} - 48 = 0$

Apply the distributive property.

$x^{2} x^{2} + x^{2} \cdot 4 + 4 (x^{2} + 4) + 32 - 12 x^{2} - 48 = 0$

Apply the distributive property.

$x^{2} x^{2} + x^{2} \cdot 4 + 4 x^{2} + 4 \cdot 4 + 32 - 12 x^{2} - 48 = 0$

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.

$x^{2 + 2} + x^{2} \cdot 4 + 4 x^{2} + 4 \cdot 4 + 32 - 12 x^{2} - 48 = 0$

Add $2$ and $2$ .

$x^{4} + x^{2} \cdot 4 + 4 x^{2} + 4 \cdot 4 + 32 - 12 x^{2} - 48 = 0$

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

$x^{4} + 4 \cdot x^{2} + 4 x^{2} + 4 \cdot 4 + 32 - 12 x^{2} - 48 = 0$

Multiply $4$ by $4$ .

$x^{4} + 4 x^{2} + 4 x^{2} + 16 + 32 - 12 x^{2} - 48 = 0$

Add $4 x^{2}$ and $4 x^{2}$ .

$x^{4} + 8 x^{2} + 16 + 32 - 12 x^{2} - 48 = 0$

Subtract $12 x^{2}$ from $8 x^{2}$ .

$x^{4} - 4 x^{2} + 16 + 32 - 48 = 0$

Add $16$ and $32$ .

$x^{4} - 4 x^{2} + 48 - 48 = 0$

Combine the opposite terms in $x^{4} - 4 x^{2} + 48 - 48$ .

Subtract $48$ from $48$ .

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

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

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

Step 3

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

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

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

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

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

Factor $x^{2}$ out of $x^{2} x^{2} + x^{2} \cdot - 4$ .

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

Step 4

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

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

Step 5

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

Remove unnecessary parentheses.

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

Step 6

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

$x^{2} = 0$

$x + 2 = 0$

$x - 2 = 0$

Step 7

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

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

$x^{2} = 0$

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

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

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

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

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

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Step 8

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 9

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 10

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

$x = 0, - 2, 2$

Solve by Factoring (x^2+4)^2+32=12x^2+48

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