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Quadratic Inequalities Mastery

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Quadratic Inequalities Mastery

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Quadratic inequalities challenge

Created by

Isagani Nocus
Isagani Nocus
Philippines

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Quadratic Inequalities Mastery
 

Quadratic Inequalities MasteryOnline version

Quadratic inequalities challenge

by Isagani Nocus
1

Solve x^2 - 5x + 6 < 0.

2

Solve x^2 - 4x + 4 ≤ 0.

3

Solve -x^2 + 5x - 6 > 0.

4

Solve x^2 - 4x + 1 ≥ 0.

5

Solve (x-3)(x+1) < 0.

6

Solve x^2 - 4x + 5 > 0.

7

Solve x^2 - 4x + 5 ≤ 0.

8

Solve 2x^2 - 3x - 2 ≥ 0.

9

Solve x^2 - 6x + 8 > 0.

10

Solve x^2 - 3x + 2 ≠ 0.

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The parabola opens up; it is negative between the roots 2 and 3, so (2,3). Outside that interval it’s ≥0.

x^2-4x+4 = (x-2)^2; only value making it nonpositive is x=2.

Multiplying by -1 reverses the inequality; x^2-5x+6<0 holds between 2 and 3.

Roots are x = 2 and x = 0.5; since leading coef is positive, outside the roots the expression is ≥ 0.

Two roots -1 and 3; product negative between the roots.

Discriminant negative; the quadratic is always positive for all x.

Discriminant < 0 and leading coef positive; expression never ≤ 0.

Roots are x = -0.5 and x = 2. Upward parabola is ≥ 0 outside the roots.

Roots at 2 and 4; outside the interval (2,4) the expression is positive.

Quadratic factors to (x-1)(x-2); it is zero at 1 and 2, so non-equal for all other x.

 
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