The Discriminant of a Quadratic Equation

The Discriminant of a Quadratic Equation
Part 4

Let’s continue talking about the discriminant of a quadratic equation. You can review parts 1, 2 and 3 of this discussion here:

The Discriminant of a Quadratic Equation – Part 1
The Discriminant of a Quadratic Equation – Part 2
The Discriminant of a Quadratic Equation – Part 3

Recall that the discriminant of the quadratic equation ax² + bx + c = 0 is the quantity Δ defined by

Δ = b² – 4ac

That is, the discriminant is simply the expression that appears under the square root in the quadratic formula.

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Last week, I asked you to solve the following problem:

Example: Find the discriminant of x² + 4x + 8 = 0. Then describe the nature of the roots of the equation, and describe the graph of the function y = x² + 4x + 8.

Make sure to try this problem yourself before reading the following solution.

Solution: In this question, we have a = 1, b = 4, and c = 8. So the discriminant is

Δ = b² – 4ac = 4² – 4(1)(8) = 16 – 32 = –16.

Since the discriminant is negative, it follows that the two roots of the quadratic equation are distinct complex numbers.

The graph of the function y = x² + 4x + 8 is an upward facing parabola that does not intersect the x-axis.

Notes: (1) In this example, we can find the two complex solutions of the quadratic equation by using the quadratic formula or by completing the square. I leave it as an exercise to show that the two solutions are –2 + 2i and –2 – 2i.

(2) Notice that the two solutions are complex conjugates of each other. This will always happen when the discriminant is negative. For more information on complex conjugates, see the following post: Complex Numbers – Division

(3) We know that the parabola opens upwards because a = 1 > 0.

(4) It’s very easy to also find the y-intercept of the parabola. We simply substitute 0 in for x into the equation. So we get y = 8. It follows that the y-intercept of the parabola is the point (0,8).

Now try the following problem which is similar to a problem found on a recent ACT:

Example: You are given the following system of equations.

dx + ey = f
y = x²

where d, e, and f are integers. For which of the following will there be more than one (x, y) solution, with real-number coordinates for the system?

A. e² + 4df < 0
B. e² – 4df < 0
C. d² – 4ef < 0
D. e² – 4df > 0
E. d² + 4ef > 0