Precalculus Examples

${(x^{2} - 8)}^{2} - 2 (x^{2} - 8) = - 1$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

${(x^{2} - 8)}^{2} - 2 x^{2} - 2 \cdot - 8 = - 1$

Step 1.2

Multiply $- 2$ by $- 8$ .

${(x^{2} - 8)}^{2} - 2 x^{2} + 16 = - 1$

Step 2

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

$u^{2} - 18 u + 80 = - 1$

$u = x^{2}$

Step 3

Add $1$ to both sides of the equation.

$u^{2} - 18 u + 80 + 1 = 0$

Step 4

Add $80$ and $1$ .

$u^{2} - 18 u + 81 = 0$

Step 5

Factor using the perfect square rule.

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Step 5.1

Rewrite $81$ as $9^{2}$ .

$u^{2} - 18 u + 9^{2} = 0$

Step 5.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$18 u = 2 \cdot u \cdot 9$

Step 5.3

Rewrite the polynomial.

$u^{2} - 2 \cdot u \cdot 9 + 9^{2} = 0$

Step 5.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 9$ .

${(u - 9)}^{2} = 0$

Step 6

Set the $u - 9$ equal to $0$ .

$u - 9 = 0$

Step 7

Add $9$ to both sides of the equation.

$u = 9$

Step 8

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

$x^{2} = 9$

Step 9

Solve the equation for $x$ .

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Step 9.1

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

$x = \pm \sqrt{9}$

Step 9.2

Simplify $\pm \sqrt{9}$ .

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Step 9.2.1

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

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

Step 9.2.2

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

$x = \pm 3$

Step 9.3

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

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Step 9.3.1

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

$x = 3$

Step 9.3.2

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

$x = - 3$

Step 9.3.3

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

$x = 3, - 3$