Solve the Inequality for x x^2+14x+49>=0

$x^{2} + 14 x + 49 \geq 0$

Step 1

Convert the inequality to an equation.

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

Step 2

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 $b = 7$ .

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

Step 3

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

$x + 7 = 0$

Step 4

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 5

Use each root to create test intervals.

$x < - 7$

$x > - 7$

Step 6

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.

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

The left side $9$ is greater than the right side $0$ , which means that the given statement is always true.

True

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.

${(0)}^{2} + 14 (0) + 49 \geq 0$

The left side $49$ 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

$x > - 7$ True

$x < - 7$ True

$x > - 7$ True

Step 7

The solution consists of all of the true intervals.

$x \leq - 7$ or $x \geq - 7$

Step 8

Combine the intervals.

All real numbers

Step 9

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$

Step 10

Solve the Inequality for x x^2+14x+49>=0

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