Finite Math Examples

$f (x) = \frac{2 x^{2} + 7 x + 14}{{(x^{2} + 1)}^{2} (x^{2} + 2 x - 3) (2 x^{2} + x - 3)}$

Step 1

Set $\frac{2 x^{2} + 7 x + 14}{{(x^{2} + 1)}^{2} (x^{2} + 2 x - 3) (2 x^{2} + x - 3)}$ equal to $0$ .

$\frac{2 x^{2} + 7 x + 14}{{(x^{2} + 1)}^{2} (x^{2} + 2 x - 3) (2 x^{2} + x - 3)} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Set the numerator equal to zero.

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

Step 2.2

Solve the equation for $x$ .

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Step 2.2.1

Use the quadratic formula to find the solutions.

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

Step 2.2.2

Substitute the values $a = 2$ , $b = 7$ , and $c = 14$ into the quadratic formula and solve for $x$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (2 \cdot 14)}}{2 \cdot 2}$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Simplify the numerator.

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Step 2.2.3.1.1

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

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 2 \cdot 14}}{2 \cdot 2}$

Step 2.2.3.1.2

Multiply $- 4 \cdot 2 \cdot 14$ .

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Step 2.2.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 8 \cdot 14}}{2 \cdot 2}$

Step 2.2.3.1.2.2

Multiply $- 8$ by $14$ .

$x = \frac{- 7 \pm \sqrt{49 - 112}}{2 \cdot 2}$

Step 2.2.3.1.3

Subtract $112$ from $49$ .

$x = \frac{- 7 \pm \sqrt{- 63}}{2 \cdot 2}$

Step 2.2.3.1.4

Rewrite $- 63$ as $- 1 (63)$ .

$x = \frac{- 7 \pm \sqrt{- 1 \cdot 63}}{2 \cdot 2}$

Step 2.2.3.1.5

Rewrite $\sqrt{- 1 (63)}$ as $\sqrt{- 1} \cdot \sqrt{63}$ .

$x = \frac{- 7 \pm \sqrt{- 1} \cdot \sqrt{63}}{2 \cdot 2}$

Step 2.2.3.1.6

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

$x = \frac{- 7 \pm i \cdot \sqrt{63}}{2 \cdot 2}$

Step 2.2.3.1.7

Rewrite $63$ as $3^{2} \cdot 7$ .

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Step 2.2.3.1.7.1

Factor $9$ out of $63$ .

$x = \frac{- 7 \pm i \cdot \sqrt{9 (7)}}{2 \cdot 2}$

Step 2.2.3.1.7.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{- 7 \pm i \cdot \sqrt{3^{2} \cdot 7}}{2 \cdot 2}$

Step 2.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 7 \pm i \cdot (3 \sqrt{7})}{2 \cdot 2}$

Step 2.2.3.1.9

Move $3$ to the left of $i$ .

$x = \frac{- 7 \pm 3 i \sqrt{7}}{2 \cdot 2}$

Step 2.2.3.2

Multiply $2$ by $2$ .

$x = \frac{- 7 \pm 3 i \sqrt{7}}{4}$

Step 2.2.4

The final answer is the combination of both solutions.

$x = - \frac{7 - 3 i \sqrt{7}}{4}, - \frac{7 + 3 i \sqrt{7}}{4}$

$x = \frac{- 7 \pm 3 i \sqrt{7}}{4}$

Step 3