# 5 STEPS SOLUTIONS TO QUADRATIC EQUATIONS BY FACTORIZATION

This post is about the 5 STEPS SOLUTIONS TO QUADRATIC EQUATIONS BY FACTORIZATION. The process of solving quadratic equation using factorization method is a simple one. The following examples explain the processes the let to a valid root.

Consider the expression;

ax2  +  bx  +  C  =  0

where a  ≠  0

To solve quadratic equation, you need to find two numbers whose sum is equal to b and  the product is equal to ac. This does not mean that the numbers must be a and c. By so doing, quadratic equation is best express as;

X2   +   (sum of two numbers)X   +     (product of two numbers)     =     0

The only sure way of getting these numbers is by practice. Here I will start with a simple one where

a = 1, b = 7 and c = 10   that is

x2  +  7x  +  10  =  0

Step1

Finding two numbers whose sum is 7 and product is 10

Let see the numbers whose sum is 7

6 and 1

5 and 2

4 and 3

Note there are infinite pair of numbers whose sum is 7 but only a pair has  their product equal to 10 and that is 2 and 5.

Step2

Expand the coefficient of x using the discovered numbers, that is 2 and 5

X2  +  (2  +  5)x  +  10  =  0

X2  +  2x  +  5x  +  10  =  0

step3

Putting bracket

(X2  +  2x)  + ( 5x  +  10 ) =  0

Step4

Factorizing

X(x  +  2)  +  5(x  +2)  =  0

Bring outside elements together

(x   +    5)(x    +   2)=0

Step5

Equate each bracket equal to 0

X  +  5  =  0 and  x  +  2  =  0

X  =  -5  and x  =  -2

Note that the sign on numbers must change each time a number crosses over the equality sign. That is so easy.

Now try your hands on   X2   –   7x   +   6   =   0, answer is -1 and 6

Where a  ≠  1 and a  ≠  0

Consider   3X2   –   13x   +   10   =   0

Remember that the sum of two numbers (the roots) must be equal to b and their product must be equal to ac provided

ax2  +  bx  +  C  =  0,  a  ≠  0.

In   3X2   –   13x   +   10   =   0, ac  =  3×10  =  30

So therefore we need to find two numbers whose product is 30 and sum is -13. This is simple, it is valid that   (-3)x(-10)  =  30 and   (-3)  +  (-10)  =  -13. Then the two numbers are -3 and -10

Expanding

3X2   –   3x  –  10x    +   10   =   0

Bracketing

(3X2   –   3x)  –  (10x    +   10)   =   0

Factorizing

3x(x  –  1)  -10(x  –  1)  =  0

Bringing elements together

(3x  –  10)(x  –  1)  =  0

3x  –  10  =  0  and  x  –  1  =   0

3x  =  10  and  x  =  1

X  = and  x   =   1

Now try your hands on      2X2   –   11x    +   12   =   0

Answer 4 and 