Common Roots

IMPORTANT

Common Roots: Overview

This topic covers concepts, such as, Condition for the Common Roots, Common Root of Two Quadratic Equations, Quadratic Equations with Both Roots Common & Quadratic Equations with Exactly One Common Root etc.

Important Questions on Common Roots

EASY
IMPORTANT

If the quadratic equations x2-7x+3c=0 and x2+x-5c=0 have a common root, then for non-zero real value of c the sign of the expression x2-3x+c is

HARD
IMPORTANT

If the equation x2+2αx+α2-1=0 and x2+2βx+β2-1=0 have a common root α>β, then the value of the expression 2α2-4αβ-α-β+α-ββ24 is

HARD
IMPORTANT

Two quadratic equations x2-bx+6=0 and x2-6x+c=0 have a common root. If the remaining roots of the first and second equations are positive integers and are in the ration 3 :4 respectively, then the common root is

HARD
IMPORTANT

Two quadratic equations x2-9x+c=0 and x2-bx-18=0 have common roots. If the sum of the remaining roots of first and second equation is -2 then the common root is 

MEDIUM
IMPORTANT

If x=1 is a common root of the equations ax2+ax+3=0 and x2+x+b=0, then the value of ab is

EASY
IMPORTANT

If (x + k) is a common factor of (x2 + px + q) and (x2 + lx + m), then the value of k is:

EASY
IMPORTANT

The pair of equation x + 2y =  5 and  3x  6y =  1 will have _____.
 

HARD
IMPORTANT

a and b are roots of the quadratic equation x2+5x+d=0 and a and c are roots of the quadratic equation x2+6x+2d=0. If there is only one common root of the above two equations, then the possible value of d is: 

EASY
IMPORTANT

One of the value of k such that 4y2-3y+k=0 and 6y2-17y+12=0 have a common root, is

MEDIUM
IMPORTANT

The parabolas y = 2x2+5x-7 and y = 3x2+2x-5 intersect each other at two distinct points. What is the abscissa of one of these points?

MEDIUM
IMPORTANT

The equations x2-ax+b=0 and x2+bx-a=0 have a common root, then

EASY
IMPORTANT

If the equations x2+ax+1=0 and x2-x-a=0 have a real common root b , then the value of b is equal to

MEDIUM
IMPORTANT

Let f(x)=x3-3x+b and g(x)=x2+bx-3, where b is a real number. What is the sum of all possible values of b for which the equations f(x)=0 and g(x)=0 have a common root ?

EASY
IMPORTANT

If the equations x2+px+q=0 and x2+rx+s=0 have a common root, show that it must be ps-rqq-s or q-sr-p.

MEDIUM
IMPORTANT

The equations x2-cx+d=0 and x2-ax+b=0 have one common root and the 2nd equation has equal roots. Prove that 2b+d=ac.

MEDIUM
IMPORTANT

If the equation ax2+bx+c=0 and bx2+cx+a=0 (where acb2) have a common root, then show that, either a+b+c=0 or a=b=c.

EASY
IMPORTANT

If the quadratic equations x2+ax+b=0 and x2+bx+a=0 (ab) have a common root then find a+b.

HARD
IMPORTANT

If the equations x2+bx+ca=0 and x2+cx+ab=0 have exactly one non-zero common root, then prove that the other roots of the equations satisfy x2+ax+bc=0.

MEDIUM
IMPORTANT

If α, β are the roots of the equation x2+px+q=0 and γ, δ  are the roots of the equation x2+rx+s=0 evaluate (α-γ)(α-δ)(β-γ)(β-δ) in term of p, q, r, s Hence, show that (s-q)2=(r-p)(ps-rq) is the condition for the existence of a common root of the two equations.

MEDIUM
IMPORTANT

If the quadratic equations x2+ax+b=0 and x2+bx+a=0(ab) have a common root then find (a+b).