# 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