HSA.REI.C.7 Tutor Intake — Linear-Quadratic Systems

A line and a parabola are graphed in the same coordinate plane.

How many intersection points can they have?

When solving a linear-quadratic system by substitution, which substitution
step is correct?

A student solves a linear-quadratic system and finds $x = 1$ and $x = -3$.
They stop here.

What step has the student missed?

Solve the system:
$y = x + 2$
$y = x^2$

Substitute $y = x + 2$ into the second equation:
$x + 2 = x^2 \Rightarrow x^2 - x - 2 = 0 \Rightarrow (x - $   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   $)(x +$   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   $) = 0$

So $x = $   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   or $x = $   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲  

The system below has two solutions. How many real solutions does the
system have when the discriminant of the resulting quadratic is $-4$?

(Enter the number of real solutions.)