a) The roots of the equation 3x2 + kx + 96 =0 are both positive and one is twice as large as the other Calculate the value of each root and find k
b) Given that p2 +q2 = 13 and that pq = 6, construct the quadratic equation whose roots are p2 and q2. Find all possible values of p.
Answer:
a)
Let a be the first root, and 2a be the second root (twice as large)
Therefore, (x-2a) (x-a) = 0 is the equation for the roots.
Expanding
x2 - 3ax + 2a2 = 0
3x2 - 9ax + 6a2 = 0
Comparing with 3x2 + kx + 96 = 0,
6a2 = 96
thus,
a is 4 or -4 (reject because a is positive)
So the value of the roots are 4 and 8 (a and 2a)
Solving for k, k = 9a = -36
b)
For a quadratic equation whose roots are p2 and q2,
(x-p²)(x-q²) = 0
x² - (p²+q²)x + p²q² = 0
x² - 13x + 36 = 0
The equation can be factorised to
x² - 13x + 36 = 0 ⇒ (x - 4)(x - 9) = 0
thus,
p² = 4 or p² = 9
p = ±2 or ±3
