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The line y=2x2 intersects the parabola

y=2x2+11x+c

at two distinct points. One of those points is (5,12) and the other is (p,q).

Find the value of

128c(p3+q3)
Hint

(5,12) lies on the parabola, so it satisfies its equation. Substitute the point into the equation to find c.

Solution

Given that (5,12) lies on the parabola, we know that

12=2(5)2+11(5)+c12=5055+cc=7

so the equation of the parabola is

y=2x2+11x7

The points of intersection are the solutions of

2x2=2x2+11x72x2+9x5=0(2x1)(x+5)=0x{12,5}

So p=12 and

q=2p2=12=1