Finite Math Examples

$x^{4} + 3 x^{3} + 3 x^{2} - 9 x - 18$

Step 1

Factor the left side of the equation.

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Step 1.1

Regroup terms.

$3 x^{3} - 9 x + x^{4} + 3 x^{2} - 18 = 0$

Step 1.2

Factor $3 x$ out of $3 x^{3} - 9 x$ .

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Step 1.2.1

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

$3 x (x^{2}) - 9 x + x^{4} + 3 x^{2} - 18 = 0$

Step 1.2.2

Factor $3 x$ out of $- 9 x$ .

$3 x (x^{2}) + 3 x (- 3) + x^{4} + 3 x^{2} - 18 = 0$

Step 1.2.3

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

$3 x (x^{2} - 3) + x^{4} + 3 x^{2} - 18 = 0$

Step 1.3

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

$3 x (x^{2} - 3) + {(x^{2})}^{2} + 3 x^{2} - 18 = 0$

Step 1.4

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

$3 x (x^{2} - 3) + u^{2} + 3 u - 18 = 0$

Step 1.5

Factor $u^{2} + 3 u - 18$ using the AC method.

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Step 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 $- 18$ and whose sum is $3$ .

$- 3, 6$

Step 1.5.2

Write the factored form using these integers.

$3 x (x^{2} - 3) + (u - 3) (u + 6) = 0$

Step 1.6

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

$3 x (x^{2} - 3) + (x^{2} - 3) (x^{2} + 6) = 0$

Step 1.7

Factor $x^{2} - 3$ out of $3 x (x^{2} - 3) + (x^{2} - 3) (x^{2} + 6)$ .

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Step 1.7.1

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

$(x^{2} - 3) (3 x) + (x^{2} - 3) (x^{2} + 6) = 0$

Step 1.7.2

Factor $x^{2} - 3$ out of $(x^{2} - 3) (3 x) + (x^{2} - 3) (x^{2} + 6)$ .

$(x^{2} - 3) (3 x + x^{2} + 6) = 0$

Step 1.8

Reorder terms.

$(x^{2} - 3) (x^{2} + 3 x + 6) = 0$

Step 2

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

$x^{2} - 3 = 0$

$x^{2} + 3 x + 6 = 0$

Step 3

Set $x^{2} - 3$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x^{2} - 3$ equal to $0$ .

$x^{2} - 3 = 0$

Step 3.2

Solve $x^{2} - 3 = 0$ for $x$ .

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Step 3.2.1

Add $3$ to both sides of the equation.

$x^{2} = 3$

Step 3.2.2

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

$x = \pm \sqrt{3}$

Step 3.2.3

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

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Step 3.2.3.1

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

$x = \sqrt{3}$

Step 3.2.3.2

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

$x = - \sqrt{3}$

Step 3.2.3.3

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

$x = \sqrt{3}, - \sqrt{3}$

Step 4

Set $x^{2} + 3 x + 6$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{2} + 3 x + 6$ equal to $0$ .

$x^{2} + 3 x + 6 = 0$

Step 4.2

Solve $x^{2} + 3 x + 6 = 0$ for $x$ .

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Step 4.2.1

Use the quadratic formula to find the solutions.

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

Step 4.2.2

Substitute the values $a = 1$ , $b = 3$ , and $c = 6$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 6)}}{2 \cdot 1}$

Step 4.2.3

Simplify.

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Step 4.2.3.1

Simplify the numerator.

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Step 4.2.3.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}$

Step 4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 6$ .

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Step 4.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 6}}{2 \cdot 1}$

Step 4.2.3.1.2.2

Multiply $- 4$ by $6$ .

$x = \frac{- 3 \pm \sqrt{9 - 24}}{2 \cdot 1}$

Step 4.2.3.1.3

Subtract $24$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 15}}{2 \cdot 1}$

Step 4.2.3.1.4

Rewrite $- 15$ as $- 1 (15)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 15}}{2 \cdot 1}$

Step 4.2.3.1.5

Rewrite $\sqrt{- 1 (15)}$ as $\sqrt{- 1} \cdot \sqrt{15}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{15}}{2 \cdot 1}$

Step 4.2.3.1.6

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

$x = \frac{- 3 \pm i \sqrt{15}}{2 \cdot 1}$

Step 4.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm i \sqrt{15}}{2}$

Step 4.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 4.2.4.1

Simplify the numerator.

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Step 4.2.4.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}$

Step 4.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 6$ .

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Step 4.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 6}}{2 \cdot 1}$

Step 4.2.4.1.2.2

Multiply $- 4$ by $6$ .

$x = \frac{- 3 \pm \sqrt{9 - 24}}{2 \cdot 1}$

Step 4.2.4.1.3

Subtract $24$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 15}}{2 \cdot 1}$

Step 4.2.4.1.4

Rewrite $- 15$ as $- 1 (15)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 15}}{2 \cdot 1}$

Step 4.2.4.1.5

Rewrite $\sqrt{- 1 (15)}$ as $\sqrt{- 1} \cdot \sqrt{15}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{15}}{2 \cdot 1}$

Step 4.2.4.1.6

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

$x = \frac{- 3 \pm i \sqrt{15}}{2 \cdot 1}$

Step 4.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm i \sqrt{15}}{2}$

Step 4.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 3 + i \sqrt{15}}{2}$

Step 4.2.4.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + i \sqrt{15}}{2}$

Step 4.2.4.5

Factor $- 1$ out of $i \sqrt{15}$ .

$x = \frac{- 1 \cdot 3 - (- i \sqrt{15})}{2}$

Step 4.2.4.6

Factor $- 1$ out of $- 1 (3) - (- i \sqrt{15})$ .

$x = \frac{- 1 (3 - i \sqrt{15})}{2}$

Step 4.2.4.7

Move the negative in front of the fraction.

$x = - \frac{3 - i \sqrt{15}}{2}$

Step 4.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 4.2.5.1

Simplify the numerator.

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Step 4.2.5.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}$

Step 4.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 6$ .

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Step 4.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 6}}{2 \cdot 1}$

Step 4.2.5.1.2.2

Multiply $- 4$ by $6$ .

$x = \frac{- 3 \pm \sqrt{9 - 24}}{2 \cdot 1}$

Step 4.2.5.1.3

Subtract $24$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 15}}{2 \cdot 1}$

Step 4.2.5.1.4

Rewrite $- 15$ as $- 1 (15)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 15}}{2 \cdot 1}$

Step 4.2.5.1.5

Rewrite $\sqrt{- 1 (15)}$ as $\sqrt{- 1} \cdot \sqrt{15}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{15}}{2 \cdot 1}$

Step 4.2.5.1.6

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

$x = \frac{- 3 \pm i \sqrt{15}}{2 \cdot 1}$

Step 4.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm i \sqrt{15}}{2}$

Step 4.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 3 - i \sqrt{15}}{2}$

Step 4.2.5.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - i \sqrt{15}}{2}$

Step 4.2.5.5

Factor $- 1$ out of $- i \sqrt{15}$ .

$x = \frac{- 1 \cdot 3 - (i \sqrt{15})}{2}$

Step 4.2.5.6

Factor $- 1$ out of $- 1 (3) - (i \sqrt{15})$ .

$x = \frac{- 1 (3 + i \sqrt{15})}{2}$

Step 4.2.5.7

Move the negative in front of the fraction.

$x = - \frac{3 + i \sqrt{15}}{2}$

Step 4.2.6

The final answer is the combination of both solutions.

$x = - \frac{3 - i \sqrt{15}}{2}, - \frac{3 + i \sqrt{15}}{2}$

Step 5

The final solution is all the values that make $(x^{2} - 3) (x^{2} + 3 x + 6) = 0$ true.

$x = \sqrt{3}, - \sqrt{3}, - \frac{3 - i \sqrt{15}}{2}, - \frac{3 + i \sqrt{15}}{2}$

Step 6