The value of 'a' for which one root of (a^2-5a+3)x^2+(3a-1)x+2=0 is twice as large as the …

Question

The value of 'a' for which one root of $(a^2-5a+3)x^2+(3a-1)x+2=0$ is twice as large as the other, is

Accepted Answer

Let roots be $\alpha,2\alpha$: $3\alpha=\dfrac{1-3a}{a^2-5a+3}$, $2\alpha^2=\dfrac2{a^2-5a+3}$.

Substituting $\alpha$ from the first into the second and simplifying gives $39a=26 \implies a=\dfrac23.$

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