Solve Using the Quadratic Formula x^2+9x+14>0

$x^{2} + 9 x + 14 > 0$

Step 1

Convert the inequality to an equation.

$x^{2} + 9 x + 14 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 1$ , $b = 9$ , and $c = 14$ into the quadratic formula and solve for $x$ .

$\frac{- 9 \pm \sqrt{9^{2} - 4 \cdot (1 \cdot 14)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $9$ to the power of $2$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 1 \cdot 14}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 14$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 9 \pm \sqrt{81 - 4 \cdot 14}}{2 \cdot 1}$

Multiply $- 4$ by $14$ .

$x = \frac{- 9 \pm \sqrt{81 - 56}}{2 \cdot 1}$

Subtract $56$ from $81$ .

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

Rewrite $25$ as $5^{2}$ .

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

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

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

Multiply $2$ by $1$ .

$x = \frac{- 9 \pm 5}{2}$

Step 5

The final answer is the combination of both solutions.

$x = - 2, - 7$

Step 6

Use each root to create test intervals.

$x < - 7$

$- 7 < x < - 2$

$x > - 2$

Step 7

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Test a value on the interval $x < - 7$ to see if it makes the inequality true.

Choose a value on the interval $x < - 7$ and see if this value makes the original inequality true.

$x = - 10$

Replace $x$ with $- 10$ in the original inequality.

${(- 10)}^{2} + 9 (- 10) + 14 > 0$

The left side $24$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $- 7 < x < - 2$ to see if it makes the inequality true.

Choose a value on the interval $- 7 < x < - 2$ and see if this value makes the original inequality true.

$x = - 4$

Replace $x$ with $- 4$ in the original inequality.

${(- 4)}^{2} + 9 (- 4) + 14 > 0$

The left side $- 6$ is not greater than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $x > - 2$ to see if it makes the inequality true.

Choose a value on the interval $x > - 2$ and see if this value makes the original inequality true.

$x = 0$

Replace $x$ with $0$ in the original inequality.

${(0)}^{2} + 9 (0) + 14 > 0$

The left side $14$ is greater than the right side $0$ , which means that the given statement is always true.

True

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 7$ True

$- 7 < x < - 2$ False

$x > - 2$ True

$x < - 7$ True

$- 7 < x < - 2$ False

$x > - 2$ True

Step 8

The solution consists of all of the true intervals.

$x < - 7$ or $x > - 2$

Solve Using the Quadratic Formula x^2+9x+14>0

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