Topic Content:
- Factorization of Quadratic Expressions I
- Quadratic expression of the form x2 + bx + c
Quadratic expression of the form x2 + bx + c
The general form of a quadratic expression is ax2 + bx + c, where a, b and c are constants
a \( \scriptsize \neq 0 \)
a is the coefficient of x2 and b is the coefficient of x and c is a constant term. When a = 1, ax2 + bx + c becomes x2 + bx + c, and is called a simple trinomial.
In general, quadratic equations are raised to the power of 2. It is important to note that there are three algebraic methods of solving quadratic equations.
These are:
- Factorization.
- Method of completing the square.
- Using the factor formula.
- The fourth method is by geometry i.e. using graph (graphical method).
The factorization method is considered first.
Factorization of Quadratic Equation by Splitting the Middle Term:
In order to factorise a quadratic expression of the form ax2 + bx + c, where a = 1, i.e x2 + bx + c
Consider the quadratic equation x2 + bx + c = 0
Step 1: Multiply the coefficient of x2 and the constant term c i.e. 1 × c = c
Step 2: Now, find two numbers such that their product is equal to c and sum equals to b.
Product: (number 1)(number 2) = c
Sum: (number 1) + (number 2) = b
Step 3 (Factorise): Now, split the middle term using these two numbers,
x2 + (number 1)x + (number 2)x + c = 0
Step 4 (Simplify): Take the common factors out and simplify.
For the following examples a = 1 (simple trinomial)
Example 6.4.1:
Factorise these expressions:
(a) \( \scriptsize x^2 \: + \: 7x \: + \: 12 \)
(b) \( \scriptsize x^2 \: – \: 12x \: + \:11 \)
(c) \( \scriptsize 20 \: – \: 9x \: + \: x^2 \)
(d) \( \scriptsize x^2 \: – \: x \: – \: 132 \)
Solution:
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