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).