Precalculus Examples

$f (x) = 7 x^{4} + 49 x^{2} + 14$

Step 1

Set $7 x^{4} + 49 x^{2} + 14$ equal to $0$ .

$7 x^{4} + 49 x^{2} + 14 = 0$

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.

$7 u^{2} + 49 u + 14 = 0$

$u = x^{2}$

Step 2.2

Factor $7$ out of $7 u^{2} + 49 u + 14$ .

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Step 2.2.1

Factor $7$ out of $7 u^{2}$ .

$7 (u^{2}) + 49 u + 14 = 0$

Step 2.2.2

Factor $7$ out of $49 u$ .

$7 (u^{2}) + 7 (7 u) + 14 = 0$

Step 2.2.3

Factor $7$ out of $14$ .

$7 u^{2} + 7 (7 u) + 7 \cdot 2 = 0$

Step 2.2.4

Factor $7$ out of $7 u^{2} + 7 (7 u)$ .

$7 (u^{2} + 7 u) + 7 \cdot 2 = 0$

Step 2.2.5

Factor $7$ out of $7 (u^{2} + 7 u) + 7 \cdot 2$ .

$7 (u^{2} + 7 u + 2) = 0$

Step 2.3

Divide each term in $7 (u^{2} + 7 u + 2) = 0$ by $7$ and simplify.

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Step 2.3.1

Divide each term in $7 (u^{2} + 7 u + 2) = 0$ by $7$ .

$\frac{7 (u^{2} + 7 u + 2)}{7} = \frac{0}{7}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $7$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{7 (u^{2} + 7 u + 2)}{7} = \frac{0}{7}$

Step 2.3.2.1.2

Divide $u^{2} + 7 u + 2$ by $1$ .

$u^{2} + 7 u + 2 = \frac{0}{7}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $0$ by $7$ .

$u^{2} + 7 u + 2 = 0$

Step 2.4

Use the quadratic formula to find the solutions.

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

Step 2.5

Substitute the values $a = 1$ , $b = 7$ , and $c = 2$ into the quadratic formula and solve for $u$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (1 \cdot 2)}}{2 \cdot 1}$

Step 2.6

Simplify.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise $7$ to the power of $2$ .

$u = \frac{- 7 \pm \sqrt{49 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 2.6.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{- 7 \pm \sqrt{49 - 4 \cdot 2}}{2 \cdot 1}$

Step 2.6.1.2.2

Multiply $- 4$ by $2$ .

$u = \frac{- 7 \pm \sqrt{49 - 8}}{2 \cdot 1}$

Step 2.6.1.3

Subtract $8$ from $49$ .

$u = \frac{- 7 \pm \sqrt{41}}{2 \cdot 1}$

Step 2.6.2

Multiply $2$ by $1$ .

$u = \frac{- 7 \pm \sqrt{41}}{2}$

Step 2.7

The final answer is the combination of both solutions.

$u = - \frac{7 - \sqrt{41}}{2}, - \frac{7 + \sqrt{41}}{2}$

Step 2.8

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

$x^{2} = - 0.29843788$

${(x^{2})}^{1} = - 6.70156211$

Step 2.9

Solve the first equation for $x$ .

$x^{2} = - 0.29843788$

Step 2.10

Solve the equation for $x$ .

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Step 2.10.1

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

$x = \pm \sqrt{- 0.29843788}$

Step 2.10.2

Simplify $\pm \sqrt{- 0.29843788}$ .

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Step 2.10.2.1

Rewrite $- 0.29843788$ as $- 1 (0.29843788)$ .

$x = \pm \sqrt{- 1 (0.29843788)}$

Step 2.10.2.2

Rewrite $\sqrt{- 1 (0.29843788)}$ as $\sqrt{- 1} \cdot \sqrt{0.29843788}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{0.29843788}$

Step 2.10.2.3

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

$x = \pm i \sqrt{0.29843788}$

Step 2.10.3

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

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Step 2.10.3.1

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

$x = i \sqrt{0.29843788}$

Step 2.10.3.2

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

$x = - i \sqrt{0.29843788}$

Step 2.10.3.3

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

$x = i \sqrt{0.29843788}, - i \sqrt{0.29843788}$

Step 2.11

Solve the second equation for $x$ .

${(x^{2})}^{1} = - 6.70156211$

Step 2.12

Solve the equation for $x$ .

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Step 2.12.1

Remove parentheses.

$x^{2} = - 6.70156211$

Step 2.12.2

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

$x = \pm \sqrt{- 6.70156211}$

Step 2.12.3

Simplify $\pm \sqrt{- 6.70156211}$ .

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Step 2.12.3.1

Rewrite $- 6.70156211$ as $- 1 (6.70156211)$ .

$x = \pm \sqrt{- 1 (6.70156211)}$

Step 2.12.3.2

Rewrite $\sqrt{- 1 (6.70156211)}$ as $\sqrt{- 1} \cdot \sqrt{6.70156211}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{6.70156211}$

Step 2.12.3.3

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

$x = \pm i \sqrt{6.70156211}$

Step 2.12.4

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

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Step 2.12.4.1

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

$x = i \sqrt{6.70156211}$

Step 2.12.4.2

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

$x = - i \sqrt{6.70156211}$

Step 2.12.4.3

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

$x = i \sqrt{6.70156211}, - i \sqrt{6.70156211}$

Step 2.13

The solution to $7 x^{4} + 49 x^{2} + 14 = 0$ is $x = i \sqrt{0.29843788}, - i \sqrt{0.29843788}, i \sqrt{6.70156211}, - i \sqrt{6.70156211}$ .

$x = i \sqrt{0.29843788}, - i \sqrt{0.29843788}, i \sqrt{6.70156211}, - i \sqrt{6.70156211}$

Step 3