Quadratic Expressions

Expansion of Algebraic Expressions

Earlier, you read that:

a(b + c) = ab + ac

Example 1

Expand (2x – 3)(3x + 4)

= 2x(3x + 4) – 3(3x + 4)
= 6x² + 8x – 9x -12
= 6x² – x – 12

Quadratic Identities

They are represented as:

1. (a + b)(a + b)
2. (a – b)(a – b)
3. (a + b)(a – b)

Example 2

Consider the third identity Expand (a + b)(a – b)

= a(a – b) + b(a – b)
= a² – ab + ba – b²
= a² – b²

This is referred to as the difference of two squares.

Example 3

Using Quadratic identity Expand (3x + 4)(3x – 4)

From difference of two squares:
(a + b)(a – b) = a² – b²
= (3x)² – 4²
= 9x² – 16

Factorization of quadratic expressions

– This is the opposite of expansion (explained above).

While a(b + c) = ab + ac is expansion
ab + ac = ab + ac is factorization

Example 4

Factorize 3x² – 9x

3x is common hence:
= 3x(x – 3)

Example 5

Factorize x² + 7x + 12

Use the rule: ax² + bx + c
p x q = c and p + q = b
p x q = 12 and p + q = 7
The numbers are 4 and 3 since 4 x 3 = 12 and 4 + 3 = 7
x² + 3x + 4x + 12
= x(x + 3) + 4(x + 3)
= (x + 4)(x + 3)

Solving Quadratic Equations By Factorization

– If a x b = 0, either a = 0 or b = 0.
Example 6

Solve the equation 2x² + 7x + 6 = 0

2x² + 7x + 6 = 0
2x² + 3x + 4x + 6 = 0
x(2x+3) + 2(2x + 3) = 0
(2x + 3)(x + 2) = 0First Part:

2x + 3 = 0
2x = -3
x = -3/2Second Part:

x + 2 = 0
x = -2

Hence x = -2 and x = -3/2

Sum and product of roots of an equation

The solution obtained in solving quadratic equations may also be termed as roots or a quadratic equation.
Note:

Hence:

Example 7

Given that the roots of the equation 3x² – 4x – 4 = 0 are a and b find 1/a + 1/b

Formation of a quadratic equation

Given the roots an equation, a quadratic equation can be formed.
Note:
Example 8

Form the quadratic equation whose roots are:
1/2 or 2/3

x = ½ or x = 2/3
2x = 1 or 3x = 2
2x – 1 = 0 or 3x – 2 = 0
(2x – 1)(3x – 2) = 0
2x(3x – 2) – 1(3x – 2) = 0
6x² – 4x – 3x + 2 = 0
6x² – 7x + 2 = 0