Question

question #10
solve the quadratic inequality. state the solution as an inequality. $-x^{2}+4xleq - 12$
$xleq - 6$ or $xgeq 2$
$-6leq xleq 2$
$xleq - 2$ or $xgeq 6$
$-2leq xleq 6$

Explanation:

Step1: Rewrite the inequality

First, rewrite $-x^{2}+4x\leq - 12$ as $x^{2}-4x - 12\geq0$.

Step2: Factor the quadratic

Factor $x^{2}-4x - 12$ to get $(x - 6)(x+2)\geq0$.

Step3: Find the roots

Set $(x - 6)(x + 2)=0$. The roots are $x=6$ and $x=-2$.

Step4: Test intervals

Test the intervals $x\lt - 2$, $-2\lt x\lt6$, and $x\gt6$. For $x\lt - 2$, let $x=-3$, then $(-3 - 6)(-3 + 2)=(-9)\times(-1)=9\geq0$. For $-2\lt x\lt6$, let $x = 0$, then $(0 - 6)(0 + 2)=-12\lt0$. For $x\gt6$, let $x = 7$, then $(7 - 6)(7 + 2)=9\geq0$.

Answer:

$x\leq - 2$ or $x\geq6$