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
I hope this has help you on the STEPS SOLUTIONS TO QUADRATIC EQUATIONS BY FACTORIZATION.