Precalculus Examples

$x^{2} + 10 x + 25 \leq 0$

Step 1

Convert the inequality to an equation.

$x^{2} + 10 x + 25 = 0$

Step 2

Factor using the perfect square rule.

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Step 2.1

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

$x^{2} + 10 x + 5^{2} = 0$

Step 2.2

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

$10 x = 2 \cdot x \cdot 5$

Step 2.3

Rewrite the polynomial.

$x^{2} + 2 \cdot x \cdot 5 + 5^{2} = 0$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 5$ .

${(x + 5)}^{2} = 0$

Step 3

Set the $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 4

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 5

Convert the inequality to interval notation.

$[- 5, - 5]$

Step 6