Question

1) Finding zeros (roots) by factorization method
2) Finding zeros by the method of completing the square.
3) Finding zeros by quadratic formula
4) Determining the nature of roots

Could someone give me an example solved question for each?

Answer 1

for 1, 2,3,  consider the following equation
x² + 6x + 5 = 0

1) x² + 6x + 5 = 0
since factors of 5 are 5, 1 and 5+1 = 6
   x² + 6x + 5
= x² + x + 5x + 5
= x(x+1)+5(x+1)
= (x+5)(x+1)=0
hence either x+5=0 ⇒x = -5 or x+1 = 0⇒x = -1

2) x² + 6x + 5 = 0
= x² + 6x + 5
= x² + 6x + 9 - 4 
= (x+3)² - 4 =0
(x+3)² = 4
x+3 = (+/-)2 ⇒ x = -1 or -5

3) x² + 6x + 5 = 0 comparing with ax² + bx + c = 0⇒ a=1, b=6, c=5
the x=(-b(+/-)√(b²-4ac))/2a
so x = (-(6)(+/-)√(6²-4*1*5))/2*1 = -3(+/-)2
∴ x = -1 or -5

4) if a, b, c are irrational numbers then 
the roots are usually irrational
and consider the expression  b²-4ac and if a, b, c are rational numbers and
i) b²-4ac>0 and a perfect square then the roots are rational numbers
ii) b²-4ac>0 and not a perfect square then the roots are irrational numbers
iii) b²-4ac=0 and the roots are same
iv) b²-4ac<0 then the roots are imaginary numbers