Solving Quadratic Inequalities - High School Math

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Question

Solve the quadratric inequality:

$y\geq -x^2+10x-21$

Answer

Factor and solve.

$y\geq -x^2+10x-21$

$y\leq x^2-10x+21$

Since the equation is less than or equal to, you know the inequality will be OR, not AND.

$y\leq (x-7)(x-3)$

$x\leq3$ or $x\geq7$

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Question

Solve the following quadratic inequality:

$y\leq x^2+4x+3$

Answer

Factor and solve. Since the sign is less than or equal to, we know the inequality will be OR, not AND.

$y\leq x^2+4x+3$

$y\leq (x+3)(x+1)$

$x\leq-3$ or $x\geq -1$

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Question

Solve the following quadratic inequality:

$y\leq -x^2+6x+5$

Answer

Use the quadratic formula to solve.

$y\leq -x^2+6x+5$

$y\geq x^2-6x-5$

Since the inequality is greater than or equal to, we know the inequality will be AND, not OR.

$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$

$x = \frac{-(-6) \pm \sqrt{(-6)^2-4(1)(-5)}}{2(1)}$

$x= \frac{6 \pm \sqrt{56}}{2}$

$x= 3 \pm \sqrt{14}$

$3-\sqrt{14}\leq x \leq 3+\sqrt{14}$

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