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CAT 2025 Lesson : Quadratic Equations - Factorisation Method

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3.4 Solving Equations

There are three commonly used methods to solve quadratic equations. These are explained below.

3.4.1 Factorisation Method

This is used when the roots are small integers or fractions. Most of the questions will be of this type in CAT and other MBA entrance tests.

Method Example
Step 1 In the equation ax2+bx+c=0ax^{2} + bx + c = 0, split b into 22 parts such that sum of the parts = b and product of parts == ac In 6x2+7x+2=06x^{2} + 7x + 2 = 0, b=7b = 7 & ac=12ac = 12
4+3=74 + 3 = 7 and 4×3=124 \times 3 = 12
Step 2 Factorise the equation by taking two terms at a time. 6x2+7x+2=06x^{2} + 7x + 2 = 0
6x2+4x+3x+2=0 6x^{2} + 4x + 3x + 2 = 0
2x(3x+2)+1(3x+2)=0 2x (3x + 2) + 1 (3x + 2) = 0
(2x+1)(3x+2)=0 (2x + 1) (3x + 2) = 0
Step 3 One of the factors has to equal 0 for the equation to be 00. Therefore, equate the factors to 00 to find the 22 roots. 2x+1=0 2x + 1 = 0 or 3x+2=03x + 2 = 0
x=12 x = \dfrac{-1}{2} or 23\dfrac{-2}{3}


Example 6

What are the roots of the equations x2x6=0x^{2} - x - 6 = 0 and 12x2+5x3=012 x^{2} + 5x - 3 = 0?

Solution

x2x6=0x^{2} - x - 6 = 0 (b=1b = -1 and ac=6ac = -6. bb can be split as 3-3 and 22.)
x23x+2x6=0 x^{2} - 3x + 2x - 6 = 0
x(x3)+2(x3)=0x(x - 3) + 2(x - 3) = 0
(x+2)(x3)=0 (x + 2) (x - 3) = 0
x=2,3\bm{x = -2, 3}

12x2+5x3=012x^{2} + 5x - 3 = 0 (b=5b = 5 and ac=36ac = -36. bb can be split as 99 and 4-4.)
12x2+9x4x3=0 12x^{2} + 9x - 4x - 3 = 0
3x(4x+3)1(4x+3)=0 3x (4x + 3) - 1(4x + 3) = 0
(3x1)(4x+3)=0 (3x - 1) (4x + 3) = 0
x=13,34\bm{x = \dfrac{1}{3}, \dfrac{-3}{4}}

Alternatively (Recommended)

The following improvisation helps saves time and is, therefore, recommended.

Step 1: Split
bb into 22 terms as stated earlier (sum of terms =b= b and product of terms =ac= ac)
Step 2: Change their signs.
Step 3: Divide them by
aa and these are your roots.

x2x6=0x^{2} - x - 6 = 0 (bb can be split as 3-3 and 22.)
x=31,21=3,2 x = \dfrac{3}{1}, \dfrac{-2}{1} \bm{= 3, -2}

12x2+5x3=012x^{2} + 5x - 3 = 0 (bb can be split as 99 and 4-4.)

x=912,412=34,13 x = \dfrac{-9}{12}, \dfrac{4}{12} = \bm{\dfrac{-3}{4}, \dfrac{1}{3}}


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