Home
Class 12
MATHS
Consider two quadratic expressions f(x) ...

Consider two quadratic expressions `f(x) =ax^2+ bx + c and g (x)=ax^2+px+q,( a, b, c, p,q in R, b != p)` such that their discriminants are equal. If `f(x)= g(x)` has a root `x = alpha`, then

A

`alpha` will be AM of the roots of `f(x)=0` and `g(x)=0`

B

`alpha` will be AM of the roots of `f(x)=0`

C

`alpha` will be AM of the roots of `f(x)=0` or `g(x)=0`

D

`alpha` will be AM of the roots of `g(x)=0`

Text Solution

Verified by Experts

The correct Answer is:
A
Given `b^(2)-4ac=p^(2)-4aq`……….i
and `f(x)=g(x)`
`impliesax^(2)+bx+c=ax^(2)+px+q`
`implies(p-p)x=q-c`
`:.x=(q-c)/(b-p)=alpha` [given (ii)]
From Eq. (i) we get
`(b+p-)(b-p)+4a(q-c)=0`
`implies(p+p)(b-p)+4a alpha(b-p)=0` [from Eq. (ii)]
or `alpha=-((p+p))/(4a)[ :' b!=p]`
`=((-b/a)+(-p/a))/4` ltrbgt `=(["Sum of theroots of" (f(x)=0)+ "Sum of the roots of"(g(x)=0)])/4`
`=AM` of the roots of `f(x)=0`
and `g(x)=0`
Promotional Banner

Similar Questions

Explore conceptually related problems

ax^2 + bx + c = 0(a > 0), has two roots alpha and beta such alpha 2, then

For which of the following graphs of the quadratic expression f(x)=ax^(2)+bx+c , the product of abc is negative

Let f(x) = a x^2 + bx + c , where a, b, c in R, a!=0 . Suppose |f(x)| leq1, x in [0,1] , then

If p(q-r)x^2+q(r-p)x+r(p-q)=0 has equal roots, then;

If ax^(2)+ bx +c and bx ^(2) + ax + c have a common factor x +1 then show that c=0 and a =b.

The roots of the quadratic equation (a + b-2c)x^2- (2a-b-c) x + (a-2b + c) = 0 are

If P(x)=ax^2+bx+c , Q(x)=-ax^2+dx+c where ac!=0 then P(x).Q(x)=0 has

If f(x)=ax+b and g(x)=cx+d, then f(g(x))=g(f(x)) is equivalent to

In quadratic equation ax^(2)+bx+c=0 , if discriminant D=b^(2)-4ac , then roots of quadratic equation are:

In quadratic equation ax^(2)+bx+c=0 , if discriminant D=b^(2)-4ac , then roots of quadratic equation are: