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Prove algebraically that, for any mR, the straight line

y=mxm+2

intersects the parabola

y=22x+x2

in two distinct points.

(Proofs based on graphical methods will gain no credit.)

Hint

Consider the discriminant of

mxm+2=22x+x2
Solution

We consider the discriminant of

mxm+2=22x+x2x2(m+2)x+m=0

which is

Δ=(m+2)24m=m2+4m+44m=m2+4

Clearly m2+4>0 for all choices of mR, and so the equation must have two distinct solutions.