Quadratic EquationHard
Question
The quadratic equations x2 - 6x + a = 0 and x2 - cx + 6 = 0 have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then the common root is
Options
A.1
B.4
C.3
D.2
Solution
Let α and 4β be roots of x2 - 6x + a = 0 and α, 3β be the roots of x2 - cx + 6 = 0, then
α + 4β = 6 and 4αβ = a
α + 3β = c and 3αβ = 6.
We get αβ = 2 ⇒ a = 8
So the first equation is x2 - 6x + 8 = 0 ⇒ x = 2, 4
If α = 2 and 4β = 4 then 3β = 3
If α = 4 and 4β = 2, then 3β = 3/2 (non-integer)
∴ common root is x = 2.
α + 4β = 6 and 4αβ = a
α + 3β = c and 3αβ = 6.
We get αβ = 2 ⇒ a = 8
So the first equation is x2 - 6x + 8 = 0 ⇒ x = 2, 4
If α = 2 and 4β = 4 then 3β = 3
If α = 4 and 4β = 2, then 3β = 3/2 (non-integer)
∴ common root is x = 2.
Create a free account to view solution
View Solution FreeMore Quadratic Equation Questions
The quadratic equation with one root 2i is-...If α and β (α < β) are the roots of the equation x2 + bx + c = 0 where c < 0 < b , then...If α and β are the roots of x2 + px + q = 0 and α4, β4 are the root of x2 - rx + s = 0,then the equa...If a,b,c are positive real numbers, then the number of real roots of the equations ax2 + b |x| + c = 0 is-...A value of b for which the equations x2 + bx - 1 = 0 x2 + x + b = 0,have one root in common is...