Quadratic Equation: Part-2

Methods of solving Quadratic Equation:
There are three methods of solving Quadratic Equation namely:
1) Factorization method
2) Completing square method
3) Formula method

Explanation with examples
1) Factorization Method:
Q1) Consider a Rectangular Plot of Area 300 m2. Where Breadth is one more than the twice the length. Find the Length and breadth?
Solution: Let length= x breadth= 2x+1
By given condition : x(2x+1)= 300
2 x2 +x =300
2 x2 +x- 300=0
Factorization Method:
Let us spilt the middle term x as (-24x, 25x) (such that -24x+25x=x and (-24x) X (25x)= 300x2
so, 2 x2 +x- 300=0
2x2 -24x+25x-300=0
2x(x-12)+25(x-12)=0
(2x+25)(x-12)=0
hence, (2x+25)=0 or (x-12)=0
x=12.5 or x= 12
since length can’t be negative hence x=12
length= 12 m breadth= 25m

Overview:

middle term (split as)
/\
/ \
sum of numbers = middle term

product of numbers= product of first and last term

2) Completing Square method
Question: Consider the following equation: x2 +4x -5=0. Solve the equation
Solution: consider x2 +4x =x2 +(4/2)x +(4/2)x
= x2 + 2x+ 2x
= x(x+2)+2x
= x(x+2) +2x +(2 X 2)-(2 X 2)
=x(x+2)+ 2(x+2) -4
=(x+2)(x+2)-4
= (x+2)2 -4
hence x2 +4x -5= (x+2)2 -4 -5

= (x+2)2 -9 =0
(x+2)2 =9

(x+2)= (+3 or -3)
hence x= 1 or x= -5

3) Formula Method:
Solution of the Equation ax2 + bx + c =0 by formula method is given by
x =-b±squareroot( bsquare-4ac)2a
where b2 -4ac is called as Discriminant

Consider a equation: x2 -6x +8=0
a= 1 b= -6 c= 8
b2 -4ac= (-6)2 -4(1)(8)= 36–32= 4
x= -(-6) ± squareroot(4) /2
x=(6+4)/2 or x=(6–4)/2
x=5 or x= 1

4)Nature of Roots: Depending on the value of b2 -4ac we have three types of nature of roots
If b2 -4ac= 0 Roots are real and equal
If b2 -4ac > 0 Roots are real and unequal.
If b2 -4ac <0 Roots are complex.
Question:
Determine the nature of roots of roots of the following equation:
1) 2x2 -5x +7 =0 a=2 b= -5 c= 7 b2 -4ac= -31 Hence roots are not real
2) x2 +2x -9= 0 a= 1 b= 2 c= -9 b2 -4ac= 40 hence roots are real and unequal.

Assignment Questions:
Q1) Determine the nature of the roots: 3x2 -5x +7=0
Q2) Find the roots by Factorization method : Find two numbers whose sum is 27 and product is 182
Q3) Find the roots of following equation by completing square method and Formula method: 2x2 +x-4=0