The Discriminant of a Quadratic Equation

The Discriminant of a Quadratic Equation
Part 2

Last week, I began discussing the discriminant of a quadratic equation. You can review that post here: The Discriminant of a Quadratic Equation – Part 1

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 + 6x + 9 = 0. Then describe the nature of the roots of the equation, and describe the graph of the function y = x² + 6x + 9.

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

Solution: In this question, we have a = 1, b = 6, and c = 9. So the discriminant is

Δ = b² – 4ac = 6² – 4(1)(9) = 36 – 36 = 0.

It follows that the roots of the quadratic equation are equal (in other words, there is really just one root) and rational (a fraction, where the numerator and denominator are both integers).

The graph of the function y = x² + 6x + 9 is an upward facing parabola that intersects the x-axis at one point.

Notes: (1) The unique rational root of this quadratic equation is

x = –b/2a = –6/2 = –3.

This means that the only x-intercept of the parabola is the point (–3,0). In this case, this point also happens to be the vertex of the parabola.

(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 = 9. It follows that the y-intercept of the parabola is the point (0,9).

(4) Now that we know the x-intercepts, y-intercept, and vertex of the parabola, and we know that the parabola opens upwards, it’s very easy to sketch the graph. I leave it to the reader to draw a nice sketch.

Positive Discriminants

So far, we’ve just looked at the case where the discriminant is 0. Today let’s talk about what happens if the discriminant is positive.

If the discriminant of the quadratic equation ax² + bx + c = 0 is positive, then we wind up with a positive number under the square root in the quadratic formula. The square root of a positive number is a real number. We therefore wind up with the two real solutions

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

If the discriminant also happens to be a perfect square (such as 1, 4, 9, 16, etc.), then both solutions will be rational numbers.

Graphically, if the discriminant is positive, the graph of the function y = ax² + bx + c is a parabola that intersects the x-axis at two points.

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

Example: Find the discriminant of x² + 8x + 7 = 0. Then describe the nature of the roots of the equation, and describe the graph of the function y = x² + 8x + 7.

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.

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