Everything Maths Grade 11

8.2 CHAPTER 8. SOLVINGQUADRATIC INEQUALITIES

f(x) sign of f(x)

Region A x < 2 f(1) = 2 +

Region B x = 2 f(2) = 0 +

Region C 2 < x < 3 f(2,5) =− 2 , 5 -

Region D x = 3 f(3) = 0 +

Region E x > 3 f(4) = 2 +

We see that the functionis positive for x≤ 2 and x≥ 3.

Step 5 : Write the final answerand represent on a number line

We see that x^2 − 5 x + 6≥ 0 is true for x≤ 2 and x≥ 3.

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1 2 3 4

Example 3: Solving Quadratic Inequalities

QUESTION

Solve the quadratic inequality−x^2 − 3 x + 5 > 0.

SOLUTION

Step 1 : Determine how to approach the problem

Let f(x) =−x^2 − 3 x + 5. f(x) cannot be factorised so, use the quadratic

formula to determine the roots of f(x). The x-intercepts are solutionsto the

quadratic equation

−x^2 − 3 x + 5 = 0

x^2 + 3x− 5 = 0

∴ x =

− 3 ±

�

(3)^2 − 4(1)(−5)

2(1)

=

− 3 ±

√

29

2

x 1 =

− 3 −

√

29

2

=− 4 , 2

x 2 =

−3 +

√

29

2

= 1, 2

Step 2 : Determine which ranges correspond to the inequality

We need to figure out which values of x satisfy the inequality. From the answers

we have five regions toconsider.

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A B

− 4 , 2

C D

1 , 2

E