When does $y=mx-5$ intersect $y=x^2-1$ twice?

When does $y=mx-5$ intersect $y=x^2-1$ twice?

Question

Find the range of values of $m$ for which the straight line $y=mx-5$ intersects the curve $y=x^2-1$ in two distinct points.

Suggestion

What effect will changing $m$ have on the line and the curve?

Use this applet to confirm your prediction and estimate an answer to the question.

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Solution

The line meets the curve where

$$mx-5=x^2-1,$$

which rearranges to

$$x^2-mx+4 = 0.$$

This has two distinct roots if the discriminant $m^2 -16$ is positive, that is if $m < -4$ or $m > 4$.

This can be written as $\big|m\big|>4$, using the modulus function.