Precalculus Examples

$f (x) = x^{4} - 2 x^{3} + 38 x^{2} - 2 x + 37$

Step 1

Set $x^{4} - 2 x^{3} + 38 x^{2} - 2 x + 37$ equal to $0$ .

$x^{4} - 2 x^{3} + 38 x^{2} - 2 x + 37 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Regroup terms.

$- 2 x^{3} - 2 x + x^{4} + 38 x^{2} + 37 = 0$

Step 2.1.2

Factor $- 2 x$ out of $- 2 x^{3} - 2 x$ .

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Step 2.1.2.1

Factor $- 2 x$ out of $- 2 x^{3}$ .

$- 2 x \cdot x^{2} - 2 x + x^{4} + 38 x^{2} + 37 = 0$

Step 2.1.2.2

Factor $- 2 x$ out of $- 2 x$ .

$- 2 x \cdot x^{2} - 2 x \cdot 1 + x^{4} + 38 x^{2} + 37 = 0$

Step 2.1.2.3

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

$- 2 x (x^{2} + 1) + x^{4} + 38 x^{2} + 37 = 0$

Step 2.1.3

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

$- 2 x (x^{2} + 1) + {(x^{2})}^{2} + 38 x^{2} + 37 = 0$

Step 2.1.4

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

$- 2 x (x^{2} + 1) + u^{2} + 38 u + 37 = 0$

Step 2.1.5

Factor $u^{2} + 38 u + 37$ using the AC method.

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Step 2.1.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $37$ and whose sum is $38$ .

$1, 37$

Step 2.1.5.2

Write the factored form using these integers.

$- 2 x (x^{2} + 1) + (u + 1) (u + 37) = 0$

Step 2.1.6

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

$- 2 x (x^{2} + 1) + (x^{2} + 1) (x^{2} + 37) = 0$

Step 2.1.7

Factor $x^{2} + 1$ out of $- 2 x (x^{2} + 1) + (x^{2} + 1) (x^{2} + 37)$ .

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Step 2.1.7.1

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

$(x^{2} + 1) (- 2 x) + (x^{2} + 1) (x^{2} + 37) = 0$

Step 2.1.7.2

Factor $x^{2} + 1$ out of $(x^{2} + 1) (- 2 x) + (x^{2} + 1) (x^{2} + 37)$ .

$(x^{2} + 1) (- 2 x + x^{2} + 37) = 0$

Step 2.1.8

Reorder terms.

$(x^{2} + 1) (x^{2} - 2 x + 37) = 0$

Step 2.2

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

$x^{2} + 1 = 0$

$x^{2} - 2 x + 37 = 0$

Step 2.3

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

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Step 2.3.1

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

$x^{2} + 1 = 0$

Step 2.3.2

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

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Step 2.3.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 2.3.2.2

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

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

Step 2.3.2.3

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

$x = \pm i$

Step 2.3.2.4

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

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Step 2.3.2.4.1

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

$x = i$

Step 2.3.2.4.2

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

$x = - i$

Step 2.3.2.4.3

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

$x = i, - i$

Step 2.4

Set $x^{2} - 2 x + 37$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x^{2} - 2 x + 37$ equal to $0$ .

$x^{2} - 2 x + 37 = 0$

Step 2.4.2

Solve $x^{2} - 2 x + 37 = 0$ for $x$ .

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Step 2.4.2.1

Use the quadratic formula to find the solutions.

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

Step 2.4.2.2

Substitute the values $a = 1$ , $b = - 2$ , and $c = 37$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot 37)}}{2 \cdot 1}$

Step 2.4.2.3

Simplify.

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Step 2.4.2.3.1

Simplify the numerator.

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Step 2.4.2.3.1.1

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 37}}{2 \cdot 1}$

Step 2.4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 37$ .

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Step 2.4.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 37}}{2 \cdot 1}$

Step 2.4.2.3.1.2.2

Multiply $- 4$ by $37$ .

$x = \frac{2 \pm \sqrt{4 - 148}}{2 \cdot 1}$

Step 2.4.2.3.1.3

Subtract $148$ from $4$ .

$x = \frac{2 \pm \sqrt{- 144}}{2 \cdot 1}$

Step 2.4.2.3.1.4

Rewrite $- 144$ as $- 1 (144)$ .

$x = \frac{2 \pm \sqrt{- 1 \cdot 144}}{2 \cdot 1}$

Step 2.4.2.3.1.5

Rewrite $\sqrt{- 1 (144)}$ as $\sqrt{- 1} \cdot \sqrt{144}$ .

$x = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{144}}{2 \cdot 1}$

Step 2.4.2.3.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{144}}{2 \cdot 1}$

Step 2.4.2.3.1.7

Rewrite $144$ as $12^{2}$ .

$x = \frac{2 \pm i \cdot \sqrt{12^{2}}}{2 \cdot 1}$

Step 2.4.2.3.1.8

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

$x = \frac{2 \pm i \cdot 12}{2 \cdot 1}$

Step 2.4.2.3.1.9

Move $12$ to the left of $i$ .

$x = \frac{2 \pm 12 i}{2 \cdot 1}$

Step 2.4.2.3.2

Multiply $2$ by $1$ .

$x = \frac{2 \pm 12 i}{2}$

Step 2.4.2.3.3

Simplify $\frac{2 \pm 12 i}{2}$ .

$x = 1 \pm 6 i$

Step 2.4.2.4

The final answer is the combination of both solutions.

$x = 1 + 6 i, 1 - 6 i$

Step 2.5

The final solution is all the values that make $(x^{2} + 1) (x^{2} - 2 x + 37) = 0$ true.

$x = i, - i, 1 + 6 i, 1 - 6 i$

Step 3