Mathematico | Mathematico

Prove algebraically that, for any $m \in R$ , the straight line

y = m x - m + 2

intersects the parabola

y = 2 - 2 x + x^{2}

in two distinct points.

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

Hint

Consider the discriminant of

m x - m + 2 = 2 - 2 x + x^{2}

Solution

We consider the discriminant of

\begin{aligned} m x - m + 2 & = 2 - 2 x + x^{2} \\ x^{2} - (m + 2) x + m & = 0 \end{aligned}

which is

\begin{aligned} Δ & = (m + 2)^{2} - 4 m \\ = m^{2} + 4 m + 4 - 4 m \\ = m^{2} + 4 \end{aligned}

Clearly $m^{2} + 4 > 0$ for all choices of $m \in R$ , and so the equation must have two distinct solutions.