Question

(iii) If alfa and beta the roots of the quadratic equation 3x2 + kx + 8 = 0 and alfa/beta=2/3
then
find the value of k.​

Answer 1

$\alpha :\beta =2:3$

$3\alpha=2\beta$

∴ $\beta =\frac{3}{2} \alpha$

I'm going to use relation between roots and coefficients.

Sum of two roots is $\alpha +\beta$, but we know $\beta =\frac{3}{2} \alpha$.

$\alpha +\beta =\frac{5}{2} \alpha$

We can express the multiplication of two roots, which is $\alpha \beta$, into : $\frac{3}{2} \alpha ^2$.

At the same time, $\alpha \beta = \frac{8}{3}$.

$\frac{3}{2} \alpha ^2=\frac{8}{3}$

$\alpha ^2=16$

∴ $\alpha = \pm 4$

And $-\frac{k}{3} =\frac{5}{2} \times ( \pm 4)$.

$k=-3\times \frac{5}{2} \times ( \pm 4)$

∴ $k=\mp 30$

$k$ is equal to 30 or -30.