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Question

If one root of the quadratic equation kx2 – 5x + 2 = 0 is 4 times the other, find k.

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Solution

Given: kx2 – 5x + 2 = 0
On comparing this equation with ax2 + bx + c = 0, we get:
a = k, b = –5 and c = 2
Let α and β be the roots of the quadratic equation kx2 – 5x + 2 = 0. Then,
α = 4 β
We know that α + β = –ba and αβ = ca.
From α + β = –ba, we get:
4 β + β = --5k = 5k
5β = 5k
β = 1k …(1)
From αβ = ca, we get:
4 β ×β = 2k
4 β2 = 2k

β2 = 12k
From the equation (1), we get:

1k2 = 12k1k2 = 12k k2 = 2k k2 - 2k = 0 k (k-2) = 0 k = 0 or 2

Now, on substituting k = 0 in the given equation, we observe that the coefficient of x2 becomes zero, which is not possible for a quadratic equation. Thus, k = 0 is not possible.
Therefore, k = 2.

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