### The Discriminant of a Quadratic Equation Part 3

Today I would like to continue our discussion of the discriminant of a quadratic equation. You can review parts 1 and 2 of this discussion here:

Recall that the discriminant of the quadratic equation ax2bx + 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 + 8+ 7 = 0. Then describe the nature of the roots of the equation, and describe the graph of the function y = x2 + 8x + 7.

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

Solution: In this question, we have a = 1, b = 8, and c = 7. So the discriminant is

Δ = b2 – 4ac = 82 – 4(1)(7) = 64 – 28 = 36.

Since the discriminant is positive, it follows that the two roots of the quadratic equation are distinct real numbers. Furthermore, since 36 is a perfect square (62 = 36), the roots are actually rational.

The graph of the function y = x2 + 8x + 7 is an upward facing parabola that intersects the x-axis at two points.

Notes: (1) In this example, we can easily find the two roots of the equation by factoring:

x2 + 8+ 7 = 0
(x + 1)(x + 7) =0
x + 1 = 0        or        x + 7 =0
x = –1       or       x = –7

So the two roots are –1 and –7.

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

(3) 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 = 7. It follows that the y-intercept of the parabola is the point (0,7).

### Negative Discriminants

Up until now, we’ve looked at the cases where the discriminant is 0 and positive. Today let’s talk about what happens if the discriminant is negative.

If the discriminant of the quadratic equation ax2bx + c = 0 is negative, then we wind up with a negative number under the square root in the quadratic formula. The square root of a negative number is an imaginary number. We therefore wind up with two complex solutions.

Graphically, if the discriminant is negative, the graph of the function y = ax2bx +is a parabola that does not intersect the x-axis.

An example is given by the yellow parabola in the image above.

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

I will post a solution to this problem tomorrow, and then discuss what happens if the determinant is negative. Feel free to post your own solutions in the comments.