Finite Math Examples

$x^{2} + 5 x + 9 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 1$ , $b = 5$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (1 \cdot 9)}}{2 \cdot 1}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 3.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

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Step 3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 9}}{2 \cdot 1}$

Step 3.1.2.2

Multiply $- 4$ by $9$ .

$x = \frac{- 5 \pm \sqrt{25 - 36}}{2 \cdot 1}$

Step 3.1.3

Subtract $36$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 11}}{2 \cdot 1}$

Step 3.1.4

Rewrite $- 11$ as $- 1 (11)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 11}}{2 \cdot 1}$

Step 3.1.5

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{11}}{2 \cdot 1}$

Step 3.1.6

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

$x = \frac{- 5 \pm i \sqrt{11}}{2 \cdot 1}$

Step 3.2

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm i \sqrt{11}}{2}$

Step 4

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

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 9}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $9$ .

$x = \frac{- 5 \pm \sqrt{25 - 36}}{2 \cdot 1}$

Step 4.1.3

Subtract $36$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 11}}{2 \cdot 1}$

Step 4.1.4

Rewrite $- 11$ as $- 1 (11)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 11}}{2 \cdot 1}$

Step 4.1.5

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{11}}{2 \cdot 1}$

Step 4.1.6

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

$x = \frac{- 5 \pm i \sqrt{11}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm i \sqrt{11}}{2}$

Step 4.3

Change the $\pm$ to $+$ .

$x = \frac{- 5 + i \sqrt{11}}{2}$

Step 4.4

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 + i \sqrt{11}}{2}$

Step 4.5

Factor $- 1$ out of $i \sqrt{11}$ .

$x = \frac{- 1 \cdot 5 - (- i \sqrt{11})}{2}$

Step 4.6

Factor $- 1$ out of $- 1 (5) - (- i \sqrt{11})$ .

$x = \frac{- 1 (5 - i \sqrt{11})}{2}$

Step 4.7

Move the negative in front of the fraction.

$x = - \frac{5 - i \sqrt{11}}{2}$

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 9}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $9$ .

$x = \frac{- 5 \pm \sqrt{25 - 36}}{2 \cdot 1}$

Step 5.1.3

Subtract $36$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 11}}{2 \cdot 1}$

Step 5.1.4

Rewrite $- 11$ as $- 1 (11)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 11}}{2 \cdot 1}$

Step 5.1.5

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{11}}{2 \cdot 1}$

Step 5.1.6

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

$x = \frac{- 5 \pm i \sqrt{11}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm i \sqrt{11}}{2}$

Step 5.3

Change the $\pm$ to $-$ .

$x = \frac{- 5 - i \sqrt{11}}{2}$

Step 5.4

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 - i \sqrt{11}}{2}$

Step 5.5

Factor $- 1$ out of $- i \sqrt{11}$ .

$x = \frac{- 1 \cdot 5 - (i \sqrt{11})}{2}$

Step 5.6

Factor $- 1$ out of $- 1 (5) - (i \sqrt{11})$ .

$x = \frac{- 1 (5 + i \sqrt{11})}{2}$

Step 5.7

Move the negative in front of the fraction.

$x = - \frac{5 + i \sqrt{11}}{2}$

Step 6

The final answer is the combination of both solutions.

$x = - \frac{5 - i \sqrt{11}}{2}, - \frac{5 + i \sqrt{11}}{2}$