Precalculus Examples

$f (x) = 2 (x^{4} - 8 x^{3} - \frac{13}{2} x^{2} - 4 x - \frac{7}{2})$

Step 1

Set $2 (x^{4} - 8 x^{3} - \frac{13}{2} x^{2} - 4 x - \frac{7}{2})$ equal to $0$ .

$2 (x^{4} - 8 x^{3} - \frac{13}{2} x^{2} - 4 x - \frac{7}{2}) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Simplify $2 (x^{4} - 8 x^{3} - \frac{13}{2} x^{2} - 4 x - \frac{7}{2})$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Combine $x^{2}$ and $\frac{13}{2}$ .

$2 (x^{4} - 8 x^{3} - \frac{x^{2} \cdot 13}{2} - 4 x - \frac{7}{2}) = 0$

Step 2.1.1.2

Move $13$ to the left of $x^{2}$ .

$2 (x^{4} - 8 x^{3} - \frac{13 x^{2}}{2} - 4 x - \frac{7}{2}) = 0$

Step 2.1.2

Apply the distributive property.

$2 x^{4} + 2 (- 8 x^{3}) + 2 (- \frac{13 x^{2}}{2}) + 2 (- 4 x) + 2 (- \frac{7}{2}) = 0$

Step 2.1.3

Simplify.

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Step 2.1.3.1

Multiply $- 8$ by $2$ .

$2 x^{4} - 16 x^{3} + 2 (- \frac{13 x^{2}}{2}) + 2 (- 4 x) + 2 (- \frac{7}{2}) = 0$

Step 2.1.3.2

Cancel the common factor of $2$ .

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Step 2.1.3.2.1

Move the leading negative in $- \frac{13 x^{2}}{2}$ into the numerator.

$2 x^{4} - 16 x^{3} + 2 \frac{- 13 x^{2}}{2} + 2 (- 4 x) + 2 (- \frac{7}{2}) = 0$

Step 2.1.3.2.2

Cancel the common factor.

$2 x^{4} - 16 x^{3} + 2 \frac{- 13 x^{2}}{2} + 2 (- 4 x) + 2 (- \frac{7}{2}) = 0$

Step 2.1.3.2.3

Rewrite the expression.

$2 x^{4} - 16 x^{3} - 13 x^{2} + 2 (- 4 x) + 2 (- \frac{7}{2}) = 0$

Step 2.1.3.3

Multiply $- 4$ by $2$ .

$2 x^{4} - 16 x^{3} - 13 x^{2} - 8 x + 2 (- \frac{7}{2}) = 0$

Step 2.1.3.4

Cancel the common factor of $2$ .

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Step 2.1.3.4.1

Move the leading negative in $- \frac{7}{2}$ into the numerator.

$2 x^{4} - 16 x^{3} - 13 x^{2} - 8 x + 2 (\frac{- 7}{2}) = 0$

Step 2.1.3.4.2

Cancel the common factor.

$2 x^{4} - 16 x^{3} - 13 x^{2} - 8 x + 2 (\frac{- 7}{2}) = 0$

Step 2.1.3.4.3

Rewrite the expression.

$2 x^{4} - 16 x^{3} - 13 x^{2} - 8 x - 7 = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Regroup terms.

$- 16 x^{3} - 8 x + 2 x^{4} - 13 x^{2} - 7 = 0$

Step 2.2.2

Factor $- 8 x$ out of $- 16 x^{3} - 8 x$ .

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Step 2.2.2.1

Factor $- 8 x$ out of $- 16 x^{3}$ .

$- 8 x (2 x^{2}) - 8 x + 2 x^{4} - 13 x^{2} - 7 = 0$

Step 2.2.2.2

Factor $- 8 x$ out of $- 8 x$ .

$- 8 x (2 x^{2}) - 8 x \cdot 1 + 2 x^{4} - 13 x^{2} - 7 = 0$

Step 2.2.2.3

Factor $- 8 x$ out of $- 8 x (2 x^{2}) - 8 x (1)$ .

$- 8 x (2 x^{2} + 1) + 2 x^{4} - 13 x^{2} - 7 = 0$

Step 2.2.3

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$- 8 x (2 x^{2} + 1) + 2 {(x^{2})}^{2} - 13 x^{2} - 7 = 0$

Step 2.2.4

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$- 8 x (2 x^{2} + 1) + 2 u^{2} - 13 u - 7 = 0$

Step 2.2.5

Factor by grouping.

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Step 2.2.5.1

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 $a \cdot c = 2 \cdot - 7 = - 14$ and whose sum is $b = - 13$ .

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Step 2.2.5.1.1

Factor $- 13$ out of $- 13 u$ .

$- 8 x (2 x^{2} + 1) + 2 u^{2} - 13 u - 7 = 0$

Step 2.2.5.1.2

Rewrite $- 13$ as $1$ plus $- 14$

$- 8 x (2 x^{2} + 1) + 2 u^{2} + (1 - 14) u - 7 = 0$

Step 2.2.5.1.3

Apply the distributive property.

$- 8 x (2 x^{2} + 1) + 2 u^{2} + 1 u - 14 u - 7 = 0$

Step 2.2.5.1.4

Multiply $u$ by $1$ .

$- 8 x (2 x^{2} + 1) + 2 u^{2} + u - 14 u - 7 = 0$

Step 2.2.5.2

Factor out the greatest common factor from each group.

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Step 2.2.5.2.1

Group the first two terms and the last two terms.

$- 8 x (2 x^{2} + 1) + 2 u^{2} + u - 14 u - 7 = 0$

Step 2.2.5.2.2

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

$- 8 x (2 x^{2} + 1) + u (2 u + 1) - 7 (2 u + 1) = 0$

Step 2.2.5.3

Factor the polynomial by factoring out the greatest common factor, $2 u + 1$ .

$- 8 x (2 x^{2} + 1) + (2 u + 1) (u - 7) = 0$

Step 2.2.6

Replace all occurrences of $u$ with $x^{2}$ .

$- 8 x (2 x^{2} + 1) + (2 x^{2} + 1) (x^{2} - 7) = 0$

Step 2.2.7

Factor $2 x^{2} + 1$ out of $- 8 x (2 x^{2} + 1) + (2 x^{2} + 1) (x^{2} - 7)$ .

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Step 2.2.7.1

Factor $2 x^{2} + 1$ out of $- 8 x (2 x^{2} + 1)$ .

$(2 x^{2} + 1) (- 8 x) + (2 x^{2} + 1) (x^{2} - 7) = 0$

Step 2.2.7.2

Factor $2 x^{2} + 1$ out of $(2 x^{2} + 1) (- 8 x) + (2 x^{2} + 1) (x^{2} - 7)$ .

$(2 x^{2} + 1) (- 8 x + x^{2} - 7) = 0$

Step 2.2.8

Reorder terms.

$(2 x^{2} + 1) (x^{2} - 8 x - 7) = 0$

Step 2.3

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

$2 x^{2} + 1 = 0$

$x^{2} - 8 x - 7 = 0$

Step 2.4

Set $2 x^{2} + 1$ equal to $0$ and solve for $x$ .

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Step 2.4.1

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

$2 x^{2} + 1 = 0$

Step 2.4.2

Solve $2 x^{2} + 1 = 0$ for $x$ .

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Step 2.4.2.1

Subtract $1$ from both sides of the equation.

$2 x^{2} = - 1$

Step 2.4.2.2

Divide each term in $2 x^{2} = - 1$ by $2$ and simplify.

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Step 2.4.2.2.1

Divide each term in $2 x^{2} = - 1$ by $2$ .

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

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

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

Step 2.4.2.2.2.1.2

Divide $x^{2}$ by $1$ .

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

Step 2.4.2.2.3

Simplify the right side.

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Step 2.4.2.2.3.1

Move the negative in front of the fraction.

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

Step 2.4.2.3

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

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

Step 2.4.2.4

Simplify $\pm \sqrt{- \frac{1}{2}}$ .

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Step 2.4.2.4.1

Rewrite $- \frac{1}{2}$ as $i^{2} \frac{1^{2}}{2}$ .

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Step 2.4.2.4.1.1

Rewrite $- 1$ as $i^{2}$ .

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

Step 2.4.2.4.1.2

Rewrite $1$ as $1^{2}$ .

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

Step 2.4.2.4.2

Pull terms out from under the radical.

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

Step 2.4.2.4.3

One to any power is one.

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

Step 2.4.2.4.4

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Step 2.4.2.4.5

Any root of $1$ is $1$ .

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

Step 2.4.2.4.6

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm i (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 2.4.2.4.7

Combine and simplify the denominator.

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Step 2.4.2.4.7.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm i \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 2.4.2.4.7.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.4.2.4.7.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.4.2.4.7.4

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

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

Step 2.4.2.4.7.5

Add $1$ and $1$ .

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

Step 2.4.2.4.7.6

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

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Step 2.4.2.4.7.6.1

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

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

Step 2.4.2.4.7.6.2

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

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

Step 2.4.2.4.7.6.3

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

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

Step 2.4.2.4.7.6.4

Cancel the common factor of $2$ .

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Step 2.4.2.4.7.6.4.1

Cancel the common factor.

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

Step 2.4.2.4.7.6.4.2

Rewrite the expression.

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

Step 2.4.2.4.7.6.5

Evaluate the exponent.

$x = \pm i \frac{\sqrt{2}}{2}$

Step 2.4.2.4.8

Combine $i$ and $\frac{\sqrt{2}}{2}$ .

$x = \pm \frac{i \sqrt{2}}{2}$

Step 2.4.2.5

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

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Step 2.4.2.5.1

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

$x = \frac{i \sqrt{2}}{2}$

Step 2.4.2.5.2

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

$x = - \frac{i \sqrt{2}}{2}$

Step 2.4.2.5.3

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

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

Step 2.5

Set $x^{2} - 8 x - 7$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x^{2} - 8 x - 7$ equal to $0$ .

$x^{2} - 8 x - 7 = 0$

Step 2.5.2

Solve $x^{2} - 8 x - 7 = 0$ for $x$ .

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Step 2.5.2.1

Use the quadratic formula to find the solutions.

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

Step 2.5.2.2

Substitute the values $a = 1$ , $b = - 8$ , and $c = - 7$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot - 7)}}{2 \cdot 1}$

Step 2.5.2.3

Simplify.

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Step 2.5.2.3.1

Simplify the numerator.

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Step 2.5.2.3.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot - 7}}{2 \cdot 1}$

Step 2.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 7$ .

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Step 2.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 7}}{2 \cdot 1}$

Step 2.5.2.3.1.2.2

Multiply $- 4$ by $- 7$ .

$x = \frac{8 \pm \sqrt{64 + 28}}{2 \cdot 1}$

Step 2.5.2.3.1.3

Add $64$ and $28$ .

$x = \frac{8 \pm \sqrt{92}}{2 \cdot 1}$

Step 2.5.2.3.1.4

Rewrite $92$ as $2^{2} \cdot 23$ .

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Step 2.5.2.3.1.4.1

Factor $4$ out of $92$ .

$x = \frac{8 \pm \sqrt{4 (23)}}{2 \cdot 1}$

Step 2.5.2.3.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 23}}{2 \cdot 1}$

Step 2.5.2.3.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{23}}{2 \cdot 1}$

Step 2.5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{23}}{2}$

Step 2.5.2.3.3

Simplify $\frac{8 \pm 2 \sqrt{23}}{2}$ .

$x = 4 \pm \sqrt{23}$

Step 2.5.2.4

The final answer is the combination of both solutions.

$x = 4 + \sqrt{23}, 4 - \sqrt{23}$

Step 2.6

The final solution is all the values that make $(2 x^{2} + 1) (x^{2} - 8 x - 7) = 0$ true.

$x = \frac{i \sqrt{2}}{2}, - \frac{i \sqrt{2}}{2}, 4 + \sqrt{23}, 4 - \sqrt{23}$

Step 3