Solve the Inequality for z z^2-12z+36>=0

$z^{2} - 12 z + 36 \geq 0$

Convert the inequality to an equation.

$z^{2} - 12 z + 36 = 0$

Factor using the perfect square rule.

Rewrite $36$ as $6^{2}$ .

$z^{2} - 12 z + 6^{2} = 0$

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

$12 z = 2 \cdot z \cdot 6$

Rewrite the polynomial.

$z^{2} - 2 \cdot z \cdot 6 + 6^{2} = 0$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = z$ and $b = 6$ .

${(z - 6)}^{2} = 0$

Set the $z - 6$ equal to $0$ .

$z - 6 = 0$

Add $6$ to both sides of the equation.

$z = 6$

Use each root to create test intervals.

$z < 6$

$z > 6$

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 $z < 6$ to see if it makes the inequality true.

Choose a value on the interval $z < 6$ and see if this value makes the original inequality true.

$z = 0$

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

${(0)}^{2} - 12 \cdot 0 + 36 \geq 0$

The left side $36$ is greater than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $z > 6$ to see if it makes the inequality true.

Choose a value on the interval $z > 6$ and see if this value makes the original inequality true.

$z = 8$

Replace $z$ with $8$ in the original inequality.

${(8)}^{2} - 12 \cdot 8 + 36 \geq 0$

The left side $4$ 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.

$z < 6$ True

$z > 6$ True

$z < 6$ True

$z > 6$ True

The solution consists of all of the true intervals.

$z \leq 6$ or $z \geq 6$

Combine the intervals.

All real numbers

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$

Solve the Inequality for z z^2-12z+36>=0

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