Precalculus Examples

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

Step 1

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

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

$u = x^{2}$

Step 2

Factor by grouping.

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Step 2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 1 = - 4$ and whose sum is $b = 3$ .

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Step 2.1.1

Factor $3$ out of $3 u$ .

$4 u^{2} + 3 (u) - 1 = 0$

Step 2.1.2

Rewrite $3$ as $- 1$ plus $4$

$4 u^{2} + (- 1 + 4) u - 1 = 0$

Step 2.1.3

Apply the distributive property.

$4 u^{2} - 1 u + 4 u - 1 = 0$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$(4 u^{2} - 1 u) + 4 u - 1 = 0$

Step 2.2.2

Factor out the greatest common factor (GCF) from each group.

$u (4 u - 1) + 1 (4 u - 1) = 0$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $4 u - 1$ .

$(4 u - 1) (u + 1) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$4 u - 1 = 0$

$u + 1 = 0$

Step 4

Set $4 u - 1$ equal to $0$ and solve for $u$ .

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Step 4.1

Set $4 u - 1$ equal to $0$ .

$4 u - 1 = 0$

Step 4.2

Solve $4 u - 1 = 0$ for $u$ .

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Step 4.2.1

Add $1$ to both sides of the equation.

$4 u = 1$

Step 4.2.2

Divide each term in $4 u = 1$ by $4$ and simplify.

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Step 4.2.2.1

Divide each term in $4 u = 1$ by $4$ .

$\frac{4 u}{4} = \frac{1}{4}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{4 u}{4} = \frac{1}{4}$

Step 4.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{1}{4}$

Step 5

Set $u + 1$ equal to $0$ and solve for $u$ .

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Step 5.1

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Step 5.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 6

The final solution is all the values that make $(4 u - 1) (u + 1) = 0$ true.

$u = \frac{1}{4}, - 1$

Step 7

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

$x^{2} = \frac{1}{4}$

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

Step 8

Solve the first equation for $x$ .

$x^{2} = \frac{1}{4}$

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.

$x = \pm \sqrt{\frac{1}{4}}$

Step 9.2

Simplify $\pm \sqrt{\frac{1}{4}}$ .

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Step 9.2.1

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{4}}$

Step 9.2.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{4}}$

Step 9.2.3

Simplify the denominator.

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Step 9.2.3.1

Rewrite $4$ as $2^{2}$ .

$x = \pm \frac{1}{\sqrt{2^{2}}}$

Step 9.2.3.2

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

$x = \pm \frac{1}{2}$

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.

$x = \frac{1}{2}$

Step 9.3.2

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

$x = - \frac{1}{2}$

Step 9.3.3

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

$x = \frac{1}{2}, - \frac{1}{2}$

Step 10

Solve the second equation for $x$ .

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

Step 11

Solve the equation for $x$ .

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Step 11.1

Remove parentheses.

$x^{2} = - 1$

Step 11.2

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

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

Step 11.3

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

$x = \pm i$

Step 11.4

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

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Step 11.4.1

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

$x = i$

Step 11.4.2

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

$x = - i$

Step 11.4.3

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

$x = i, - i$

Step 12

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

$x = \frac{1}{2}, - \frac{1}{2}, i, - i$