QUADRATIC EQUATION - NATURE OF ROOTS
QUADRATIC EQUATION - NATURE OF ROOTS
a.
Consider the quadratice quation ax2 +bx+c=0 wherea,b,c belongs to R & a != 0then
(i) D > 0 i.e roots are real & distinct (unequal).
(ii) D = 0 i.e roots are real & coincident (equal)
(iii) D < 0 i.e roots are imaginary.
(iv) If p + i q is one root of a quadratic equation, then the other root must be the conjugate
p–i q & vice versa. (p,q belongs R & 
).
b.
Consider the quadratice quation ax2 +bx+c=0 wherea,b,c belongs to Q & a != 0 then
(i) If D is a perfect square, then roots are rational.
(ii) If 
is one root in this case,(where p is rational & q is a surd) then other root will be 
Example :
If the coefficient of the quadratic equation are rational & the coefficient of x2 is 1, then find the equation one of whose roots is 
Solution :
We know that
Irrational roots always occur in conjugational pairs
Hence if one root is 
the other root will be 
equation is
Example :
Find all the integral values of a for which the quadratic equation (x – a)(x – 10) + 1 = 0 has integral roots.
Splution :
Here the equation is x2 – (a + 10)x + 10a + 1 = 0. Since integral roots will always be rational it means D should be a perfect square.
From(i)
D=a2 –20a+96.
i.e D=(a–10)(a-10) –4 i.e 4=(a–10)(a-10) –D
If D is a perfect square it means we want difference of two perfect square as 4 which is possible only when
(a–10)(a-10) =4 and D=0.
i.e (a–10) = ±2 so a= 12, 8