Solve the Inequality for x x^2+14x+49<0

$x^{2} + 14 x + 49 < 0$

Convert the inequality to an equation.

$x^{2} + 14 x + 49 = 0$

Factor using the perfect square rule.

Rewrite $49$ as $7^{2}$ .

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

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$14 x = 2 \cdot x \cdot 7$

Rewrite the polynomial.

$x^{2} + 2 \cdot x \cdot 7 + 7^{2} = 0$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 7$ .

${(x + 7)}^{2} = 0$

Set the $x + 7$ equal to $0$ .

$x + 7 = 0$

Subtract $7$ from both sides of the equation.

$x = - 7$

Use each root to create test intervals.

$x < - 7$

$x > - 7$

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.

${(- 10)}^{2} + 14 (- 10) + 49 < 0$

The left side $9$ is not less than the right side $0$ , which means that the given statement is false.

False

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.

$x = 0$

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

${(0)}^{2} + 14 (0) + 49 < 0$

The left side $49$ is not less than the right side $0$ , which means that the given statement is false.

False

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

$x < - 7$ False

$x > - 7$ False

$x < - 7$ False

$x > - 7$ False

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution