Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

Factor the left side.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Factor $u^{2} - 17 u - 18$ using the AC method.

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Step 3.3.1

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 $- 18$ and whose sum is $- 17$ .

$- 18, 1$

Step 3.3.2

Write the factored form using these integers.

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

Step 3.4

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

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