#### Topic Content:

- Factorization of Quadratic Expressions I
- Quadratic expression of the form x
^{2}+ bx + c

- Quadratic expression of the form x

### Quadratic expression of the form x^{2} + bx + c

The general form of a quadratic expression is ax^{2} + bx + c, where a, b and c are constants

a \( \scriptsize \neq 0 \)

a is the coefficient of x^{2} and b is the coefficient of x and c is a constant term. When a = 1, ax^{2} + bx + c becomes x^{2} + 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 ax^{2} + bx + c, where a = 1, i.e x^{2} + bx + c

Consider the quadratic equation x^{2} + bx + c = 0

**Step 1:** Multiply the coefficient of **x ^{2}** 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,

x^{2} + (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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