Precalculus Examples

$f (x) = 3 x^{4} + x^{2} - 1$

Step 1

Set $3 x^{4} + x^{2} - 1$ equal to $0$ .

$3 x^{4} + x^{2} - 1 = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$3 u^{2} + u - 1 = 0$

$u = x^{2}$

Step 2.2

Use the quadratic formula to find the solutions.

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

Step 2.3

Substitute the values $a = 3$ , $b = 1$ , and $c = - 1$ into the quadratic formula and solve for $u$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (3 \cdot - 1)}}{2 \cdot 3}$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

One to any power is one.

$u = \frac{- 1 \pm \sqrt{1 - 4 \cdot 3 \cdot - 1}}{2 \cdot 3}$

Step 2.4.1.2

Multiply $- 4 \cdot 3 \cdot - 1$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $3$ .

$u = \frac{- 1 \pm \sqrt{1 - 12 \cdot - 1}}{2 \cdot 3}$

Step 2.4.1.2.2

Multiply $- 12$ by $- 1$ .

$u = \frac{- 1 \pm \sqrt{1 + 12}}{2 \cdot 3}$

Step 2.4.1.3

Add $1$ and $12$ .

$u = \frac{- 1 \pm \sqrt{13}}{2 \cdot 3}$

Step 2.4.2

Multiply $2$ by $3$ .

$u = \frac{- 1 \pm \sqrt{13}}{6}$

Step 2.5

The final answer is the combination of both solutions.

$u = - \frac{1 - \sqrt{13}}{6}, - \frac{1 + \sqrt{13}}{6}$

Step 2.6

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

$x^{2} = 0.43425854$

${(x^{2})}^{1} = - 0.76759187$

Step 2.7

Solve the first equation for $x$ .

$x^{2} = 0.43425854$

Step 2.8

Solve the equation for $x$ .

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Step 2.8.1

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

$x = \pm \sqrt{0.43425854}$

Step 2.8.2

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

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Step 2.8.2.1

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

$x = \sqrt{0.43425854}$

Step 2.8.2.2

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

$x = - \sqrt{0.43425854}$

Step 2.8.2.3

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

$x = \sqrt{0.43425854}, - \sqrt{0.43425854}$

Step 2.9

Solve the second equation for $x$ .

${(x^{2})}^{1} = - 0.76759187$

Step 2.10

Solve the equation for $x$ .

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Step 2.10.1

Remove parentheses.

$x^{2} = - 0.76759187$

Step 2.10.2

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

$x = \pm \sqrt{- 0.76759187}$

Step 2.10.3

Simplify $\pm \sqrt{- 0.76759187}$ .

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Step 2.10.3.1

Rewrite $- 0.76759187$ as $- 1 (0.76759187)$ .

$x = \pm \sqrt{- 1 (0.76759187)}$

Step 2.10.3.2

Rewrite $\sqrt{- 1 (0.76759187)}$ as $\sqrt{- 1} \cdot \sqrt{0.76759187}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{0.76759187}$

Step 2.10.3.3

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

$x = \pm i \sqrt{0.76759187}$

Step 2.10.4

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

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Step 2.10.4.1

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

$x = i \sqrt{0.76759187}$

Step 2.10.4.2

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

$x = - i \sqrt{0.76759187}$

Step 2.10.4.3

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

$x = i \sqrt{0.76759187}, - i \sqrt{0.76759187}$

Step 2.11

The solution to $3 x^{4} + x^{2} - 1 = 0$ is $x = \sqrt{0.43425854}, - \sqrt{0.43425854}, i \sqrt{0.76759187}, - i \sqrt{0.76759187}$ .

$x = \sqrt{0.43425854}, - \sqrt{0.43425854}, i \sqrt{0.76759187}, - i \sqrt{0.76759187}$

Step 3