Precalculus Examples

$x^{4} - 5 x^{2} - 4 = 0$

Step 1

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

$u^{2} - 5 u - 4 = 0$

$u = x^{2}$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 1$ , $b = - 5$ , and $c = - 4$ into the quadratic formula and solve for $u$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (1 \cdot - 4)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 5$ to the power of $2$ .

$u = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 4}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot - 4$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{5 \pm \sqrt{25 - 4 \cdot - 4}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $- 4$ .

$u = \frac{5 \pm \sqrt{25 + 16}}{2 \cdot 1}$

Step 4.1.3

Add $25$ and $16$ .

$u = \frac{5 \pm \sqrt{41}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$u = \frac{5 \pm \sqrt{41}}{2}$

Step 5

The final answer is the combination of both solutions.

$u = \frac{5 + \sqrt{41}}{2}, \frac{5 - \sqrt{41}}{2}$

Step 6

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

$x^{2} = 5.70156211$

${(x^{2})}^{1} = - 0.70156211$

Step 7

Solve the first equation for $x$ .

$x^{2} = 5.70156211$

Step 8

Solve the equation for $x$ .

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Step 8.1

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

$x = \pm \sqrt{5.70156211}$

Step 8.2

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

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Step 8.2.1

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

$x = \sqrt{5.70156211}$

Step 8.2.2

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

$x = - \sqrt{5.70156211}$

Step 8.2.3

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

$x = \sqrt{5.70156211}, - \sqrt{5.70156211}$

Step 9

Solve the second equation for $x$ .

${(x^{2})}^{1} = - 0.70156211$

Step 10

Solve the equation for $x$ .

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Step 10.1

Remove parentheses.

$x^{2} = - 0.70156211$

Step 10.2

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

$x = \pm \sqrt{- 0.70156211}$

Step 10.3

Simplify $\pm \sqrt{- 0.70156211}$ .

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Step 10.3.1

Rewrite $- 0.70156211$ as $- 1 (0.70156211)$ .

$x = \pm \sqrt{- 1 (0.70156211)}$

Step 10.3.2

Rewrite $\sqrt{- 1 (0.70156211)}$ as $\sqrt{- 1} \cdot \sqrt{0.70156211}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{0.70156211}$

Step 10.3.3

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

$x = \pm i \sqrt{0.70156211}$

Step 10.4

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

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Step 10.4.1

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

$x = i \sqrt{0.70156211}$

Step 10.4.2

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

$x = - i \sqrt{0.70156211}$

Step 10.4.3

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

$x = i \sqrt{0.70156211}, - i \sqrt{0.70156211}$

Step 11

The solution to $x^{4} - 5 x^{2} - 4 = 0$ is $x = \sqrt{5.70156211}, - \sqrt{5.70156211}, i \sqrt{0.70156211}, - i \sqrt{0.70156211}$ .

$x = \sqrt{5.70156211}, - \sqrt{5.70156211}, i \sqrt{0.70156211}, - i \sqrt{0.70156211}$