At what value of q does the equation have roots, one of which is 2.5 times larger than the other?

So, we are given a quadratic equation 5x ^ 2 – 14x + q = 0 and the first step is to divide both of its parts by 5:

x ^ 2 – 2.8x + 0.2q = 0;

We now apply Vieta’s theorem in it and write the system:

x1 + x2 = 2.8;

x1 * x2 = 0.2q.

Based on the condition that one of the roots is 2.5 times larger than the other, we write the equality:

x2 = 2.5×1.

Let us calculate the value of the roots from the first equation of the system:

x1 + 2.5×1 = 2.8;

3.5×1 = 2.8;

x1 = 2.8: 3.5;

x1 = 0.8.

Then x2 = 2.5 * 0.8 = 2.

Looking for q from the second equation:

0.2q = 0.8 * 2;

0.2q = 1.6;

q = 1.6: 0.2;

q = 8.