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Quadratic Expressions And Equations I

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MATHEMATICS REVISION QUESTIONS

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




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