Question

The gradient of the tangent to the curve
$y = p {x}^{2} - qx - r$
at the point (1, -2) is 1.

If the curve crosses the x-axis where x=2, find the values of p, q, and r. Find the other point of intersection. ​

Answer 1

Because the curve crosses the $x$-axis at $x=2$, we know the point (2, 0) lies on the curve, so that

$0=4p-2q-r$

Tangents to the curve have slope $y'$:

$y'=2px-q$

and at the point (1, -2), the slope is 1, so that

$1=2p-q$

This also tells us the point (1, -2) is on the curve, so that

$-2=p-q-r$

Solve for $p,q,r$; you should get

$p=1,q=1,r=2$

so the equation of the curve is

$y=x^2-x-2$

Factorizing this yields

$y=(x-2)(x+1)$

which means $x=-1$ is a root and the curve intersects the $x$-axis at the point (-1, 0).