Precalculus Examples

$x^{4} + 18 x^{2} + 81$

Step 1

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

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

$u = x^{2}$

Step 2

Factor using the perfect square rule.

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Step 2.1

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

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

Step 2.2

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 2.3

Rewrite the polynomial.

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

Step 2.4

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 3

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

$u + 9 = 0$

Step 4

Subtract $9$ from both sides of the equation.

$u = - 9$

Step 5

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

$x^{2} = - 9$

Step 6

Solve the equation for $x$ .

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.2.3

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

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

Step 6.2.4

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

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

Step 6.2.5

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

$x = \pm i \cdot 3$

Step 6.2.6

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

$x = \pm 3 i$

Step 6.3

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

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Step 6.3.1

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

$x = 3 i$

Step 6.3.2

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

$x = - 3 i$

Step 6.3.3

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

$x = 3 i, - 3 i$

Step 7