Precalculus Examples

$7 + 5 x^{4} - 3 x^{2}$

Step 1

Set $7 + 5 x^{4} - 3 x^{2}$ equal to $0$ .

$7 + 5 x^{4} - 3 x^{2} = 0$

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.

$5 u^{2} - 3 u + 7 = 0$

$u = x^{2}$

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 $a = 5$ , $b = - 3$ , and $c = 7$ into the quadratic formula and solve for $u$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (5 \cdot 7)}}{2 \cdot 5}$

Step 2.4

Simplify.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Raise $- 3$ to the power of $2$ .

$u = \frac{3 \pm \sqrt{9 - 4 \cdot 5 \cdot 7}}{2 \cdot 5}$

Step 2.4.1.2

Multiply $- 4 \cdot 5 \cdot 7$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $5$ .

$u = \frac{3 \pm \sqrt{9 - 20 \cdot 7}}{2 \cdot 5}$

Step 2.4.1.2.2

Multiply $- 20$ by $7$ .

$u = \frac{3 \pm \sqrt{9 - 140}}{2 \cdot 5}$

Step 2.4.1.3

Subtract $140$ from $9$ .

$u = \frac{3 \pm \sqrt{- 131}}{2 \cdot 5}$

Step 2.4.1.4

Rewrite $- 131$ as $- 1 (131)$ .

$u = \frac{3 \pm \sqrt{- 1 \cdot 131}}{2 \cdot 5}$

Step 2.4.1.5

Rewrite $\sqrt{- 1 (131)}$ as $\sqrt{- 1} \cdot \sqrt{131}$ .

$u = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{131}}{2 \cdot 5}$

Step 2.4.1.6

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

$u = \frac{3 \pm i \sqrt{131}}{2 \cdot 5}$

Step 2.4.2

Multiply $2$ by $5$ .

$u = \frac{3 \pm i \sqrt{131}}{10}$

Step 2.5

The final answer is the combination of both solutions.

$u = \frac{3 + i \sqrt{131}}{10}, \frac{3 - i \sqrt{131}}{10}$

Step 2.6

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

$x^{2} = \frac{3 + i \sqrt{131}}{10}$

${(x^{2})}^{1} = \frac{3 - i \sqrt{131}}{10}$

Step 2.7

Solve the first equation for $x$ .

$x^{2} = \frac{3 + i \sqrt{131}}{10}$

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.

$x = \pm \sqrt{\frac{3 + i \sqrt{131}}{10}}$

Step 2.8.2

Simplify $\pm \sqrt{\frac{3 + i \sqrt{131}}{10}}$ .

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Step 2.8.2.1

Rewrite $\sqrt{\frac{3 + i \sqrt{131}}{10}}$ as $\frac{\sqrt{3 + i \sqrt{131}}}{\sqrt{10}}$ .

$x = \pm \frac{\sqrt{3 + i \sqrt{131}}}{\sqrt{10}}$

Step 2.8.2.2

Multiply $\frac{\sqrt{3 + i \sqrt{131}}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$x = \pm \frac{\sqrt{3 + i \sqrt{131}}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 2.8.2.3

Combine and simplify the denominator.

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Step 2.8.2.3.1

Multiply $\frac{\sqrt{3 + i \sqrt{131}}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$x = \pm \frac{\sqrt{3 + i \sqrt{131}} \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 2.8.2.3.2

Raise $\sqrt{10}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3 + i \sqrt{131}} \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

Step 2.8.2.3.3

Raise $\sqrt{10}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3 + i \sqrt{131}} \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

Step 2.8.2.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{3 + i \sqrt{131}} \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 2.8.2.3.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{3 + i \sqrt{131}} \sqrt{10}}{{\sqrt{10}}^{2}}$

Step 2.8.2.3.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 2.8.2.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{3 + i \sqrt{131}} \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 2.8.2.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{3 + i \sqrt{131}} \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 2.8.2.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{3 + i \sqrt{131}} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 2.8.2.3.6.4

Cancel the common factor of $2$ .

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Step 2.8.2.3.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{3 + i \sqrt{131}} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 2.8.2.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{3 + i \sqrt{131}} \sqrt{10}}{10^{1}}$

Step 2.8.2.3.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{3 + i \sqrt{131}} \sqrt{10}}{10}$

Step 2.8.2.4

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{(3 + i \sqrt{131}) \cdot 10}}{10}$

Step 2.8.3

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

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Step 2.8.3.1

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

$x = \frac{\sqrt{(3 + i \sqrt{131}) \cdot 10}}{10}$

Step 2.8.3.2

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

$x = - \frac{\sqrt{(3 + i \sqrt{131}) \cdot 10}}{10}$

Step 2.8.3.3

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

$x = \frac{\sqrt{(3 + i \sqrt{131}) \cdot 10}}{10}, - \frac{\sqrt{(3 + i \sqrt{131}) \cdot 10}}{10}$

Step 2.9

Solve the second equation for $x$ .

${(x^{2})}^{1} = \frac{3 - i \sqrt{131}}{10}$

Step 2.10

Solve the equation for $x$ .

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Step 2.10.1

Remove parentheses.

$x^{2} = \frac{3 - i \sqrt{131}}{10}$

Step 2.10.2

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

$x = \pm \sqrt{\frac{3 - i \sqrt{131}}{10}}$

Step 2.10.3

Simplify $\pm \sqrt{\frac{3 - i \sqrt{131}}{10}}$ .

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Step 2.10.3.1

Rewrite $\sqrt{\frac{3 - i \sqrt{131}}{10}}$ as $\frac{\sqrt{3 - i \sqrt{131}}}{\sqrt{10}}$ .

$x = \pm \frac{\sqrt{3 - i \sqrt{131}}}{\sqrt{10}}$

Step 2.10.3.2

Multiply $\frac{\sqrt{3 - i \sqrt{131}}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$x = \pm \frac{\sqrt{3 - i \sqrt{131}}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 2.10.3.3

Combine and simplify the denominator.

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Step 2.10.3.3.1

Multiply $\frac{\sqrt{3 - i \sqrt{131}}}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$x = \pm \frac{\sqrt{3 - i \sqrt{131}} \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 2.10.3.3.2

Raise $\sqrt{10}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3 - i \sqrt{131}} \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

Step 2.10.3.3.3

Raise $\sqrt{10}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3 - i \sqrt{131}} \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

Step 2.10.3.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{3 - i \sqrt{131}} \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 2.10.3.3.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{3 - i \sqrt{131}} \sqrt{10}}{{\sqrt{10}}^{2}}$

Step 2.10.3.3.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 2.10.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{3 - i \sqrt{131}} \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 2.10.3.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{3 - i \sqrt{131}} \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 2.10.3.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{3 - i \sqrt{131}} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 2.10.3.3.6.4

Cancel the common factor of $2$ .

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Step 2.10.3.3.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{3 - i \sqrt{131}} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 2.10.3.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{3 - i \sqrt{131}} \sqrt{10}}{10^{1}}$

Step 2.10.3.3.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{3 - i \sqrt{131}} \sqrt{10}}{10}$

Step 2.10.3.4

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{(3 - i \sqrt{131}) \cdot 10}}{10}$

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.

$x = \frac{\sqrt{(3 - i \sqrt{131}) \cdot 10}}{10}$

Step 2.10.4.2

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

$x = - \frac{\sqrt{(3 - i \sqrt{131}) \cdot 10}}{10}$

Step 2.10.4.3

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

$x = \frac{\sqrt{(3 - i \sqrt{131}) \cdot 10}}{10}, - \frac{\sqrt{(3 - i \sqrt{131}) \cdot 10}}{10}$

Step 2.11

The solution to $5 x^{4} - 3 x^{2} + 7 = 0$ is $x = \frac{\sqrt{(3 + i \sqrt{131}) \cdot 10}}{10}, - \frac{\sqrt{(3 + i \sqrt{131}) \cdot 10}}{10}, \frac{\sqrt{(3 - i \sqrt{131}) \cdot 10}}{10}, - \frac{\sqrt{(3 - i \sqrt{131}) \cdot 10}}{10}$ .

$x = \frac{\sqrt{(3 + i \sqrt{131}) \cdot 10}}{10}, - \frac{\sqrt{(3 + i \sqrt{131}) \cdot 10}}{10}, \frac{\sqrt{(3 - i \sqrt{131}) \cdot 10}}{10}, - \frac{\sqrt{(3 - i \sqrt{131}) \cdot 10}}{10}$

Step 3