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The line $y = m x$ intersects with the parabola

y = x^{2} + 1

Find the possible values of $m .$

Use the table below to find your answer based on your solution set.

Hint

The equation

x^{2} + 1 = m x

has at least one solution.

Solution

The equation

\begin{aligned} x^{2} + 1 & = m x \\ x^{2} - m x + 1 & = 0 \end{aligned}

has at least one solution, so

\begin{aligned} b^{2} - 4 a c & \geq 0 \\ (- m)^{2} - 4 (1) (1) & \geq 0 \\ m^{2} - 4 \geq 0 \end{aligned}

Noting that the roots are $m \in {- 2, 2}$ , we make a sketch

parabola

We see that the solution is

m \leq - 2 or m \geq 2