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5. Type V- Factorization of expression in Quadratic form

Learn to factorize the expression of the form

$a x^{2} + bx + c$ .

Procedure to factorize the expression.

Step 1: Determine the coefficient

$a, b$ and

$c$ .

Step 2: Calculate the product of

$a$ and

$c$ . Product

$= ac$ and sum

$= b$ . Thus the middle coefficient is the sum and extreme product is the product value.

Step 3: Express the middle term as sum of two terms such that the sum satisfies the middle term and the product satisfies the extreme product.

Step 4: Now group the expression into two factors by taking the common expression outside.

Example:

$x^2+5x+6$

We have

$a =1$ ,

$b = 5$ and

$c = 6$ .

Here the product

$=$

$a \times c$

$=$

$(1 \times 6)$

$= 6$ and sum

$= b = 5$ .

We need to choose two number such that the sum of two numbers is

$5$ and the product of two numbers is

$6$ .

$(2+3) = 5$ and

$(2 \times 3$

$= 6$

We can write as follows.

$x^2+5x+6$

$=$

$x^2+3x+2x+6$

$=$

$x(x+3)+2(x+3)$

$=$

$(x+3)(x+2)$

$2x^2-5x-3$

We have

$a = 2$ ,

$b = -5$ and

$c = -3$ .

Here the product

$=$

$a \times c$

$=$

$(2 \times -3)$

$=$

$-6$ and sum

$= b = -5$ .

We need to choose two number such that the sum of two numbers is

$-5$ and the product of two numbers is

$-6$ .

$(-6+1) = -5$ and

$(-6\times 1)$ )

$=-6$

We can rewrite as follows.

$2x^2-5x-3$

$=$

$2x^2-6x+x-3$

$= 2x(x-3)+1(x-3)$

$= (x-3)(2x+1)$

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5. Type V- Factorization of expression in Quadratic form

Theory: