Everything Maths Grade 11

7.5 CHAPTER 7. SOLVINGQUADRATIC EQUATIONS

Step 1 : Product of two brackets

Notice that if x =−^32 then 2 x + 3 = 0

Therefore the two brackets will be:

(2x + 3)(x− 4) = 0

Step 2 : Remove brackets

The equation is:

2 x^2 − 5 x− 12 = 0

Extension: Theory of Quadratic Equations - Advanced

This section is not in the syllabus, but it gives one a good understanding about some of the

solutions of the quadratic equations.

What is the Discriminant of a Quadratic

Equation?

EMBAK

Consider a general quadratic function of the form f(x) = ax^2 + bx + c. The discriminant is

defined as:

Δ = b^2 − 4 ac. (7.18)

This is the expression under the square root inthe formula for the roots of this function. We

have already seen thatwhether the roots existor not depends on whether this factor Δ is

negative or positive.

The Nature of the Roots EMBAL

Real Roots (Δ≥ 0 )

Consider Δ≥ 0 for some quadratic function f(x) = ax^2 + bx + c. In this case there are

solutions to the equation f(x) = 0 given by the formula

x =

−b±

√

b^2 − 4 ac

2 a

=

−b±

√

Δ

2 a

(7.19)

If the expression underthe square root is non-negative then the squareroot exists. These are

the roots of the function f(x).

There various possibilities are summarised in the figure below.