**The Discriminant of a Quadratic Equation**

Part 4

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 ** ax^{2} + bx + c = 0 **is the quantity

**Δ**defined by

**Δ = b^{2} – 4ac**

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

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

**Example: **Find the discriminant of ** x^{2} + 4x + 8 = 0**. Then describe the nature of the roots of the equation, and describe the graph of the function

*y = x*^{2}+*4*

**x****+**

**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 = *b*^{2} – 4*ac* = 4^{2} – 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 = x*^{2} + 4*x *+ 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 + 2*i* and –2 – 2*i.*

(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*=

*x*

^{2}

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**. *e*^{2} + 4*df* < 0

**B.** *e*^{2} – 4*df* < 0

**C. ***d*^{2} – 4e*f* < 0

**D. ***e*^{2} – 4*df* > 0

**E. ***d*^{2} + 4e*f* > 0

I will post a solution to this problem tomorrow. Feel free to post your own solutions in the comments.

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