Solve by Factoring x^4-16x^2=x^2+18

$x^{4} - 16 x^{2} = x^{2} + 18$

Step 1

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

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

$x^{4} - 16 x^{2} - x^{2} = 18$

Subtract $18$ from both sides of the equation.

$x^{4} - 16 x^{2} - x^{2} - 18 = 0$

Step 2

Subtract $x^{2}$ from $- 16 x^{2}$ .

$x^{4} - 17 x^{2} - 18 = 0$

Step 3

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

${(x^{2})}^{2} - 17 x^{2} - 18 = 0$

Step 4

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$u^{2} - 17 u - 18 = 0$

Step 5

Factor $u^{2} - 17 u - 18$ 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 $- 18$ and whose sum is $- 17$ .

$- 18, 1$

Write the factored form using these integers.

$(u - 18) (u + 1) = 0$

Step 6

Replace all occurrences of $u$ with $x^{2}$ .

$(x^{2} - 18) (x^{2} + 1) = 0$

Step 7

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

$x^{2} - 18 = 0$

$x^{2} + 1 = 0$

Step 8

Set $x^{2} - 18$ equal to $0$ and solve for $x$ .

Set $x^{2} - 18$ equal to $0$ .

$x^{2} - 18 = 0$

Solve $x^{2} - 18 = 0$ for $x$ .

Add $18$ to both sides of the equation.

$x^{2} = 18$

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

$x = \pm \sqrt{18}$

Simplify $\pm \sqrt{18}$ .

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$x = \pm \sqrt{9 (2)}$

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

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

Pull terms out from under the radical.

$x = \pm 3 \sqrt{2}$

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.

$x = 3 \sqrt{2}$

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

$x = - 3 \sqrt{2}$

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

$x = 3 \sqrt{2}, - 3 \sqrt{2}$

Step 9

Set $x^{2} + 1$ equal to $0$ and solve for $x$ .

Set $x^{2} + 1$ equal to $0$ .

$x^{2} + 1 = 0$

Solve $x^{2} + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

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

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

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

$x = \pm i$

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.

$x = i$

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

$x = - i$

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

$x = i, - i$

Step 10

The final solution is all the values that make $(x^{2} - 18) (x^{2} + 1) = 0$ true.

$x = 3 \sqrt{2}, - 3 \sqrt{2}, i, - i$

Solve by Factoring x^4-16x^2=x^2+18

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