**The Discriminant of a Quadratic Equation**

Part 5

Part 5

Today I’d like to solve the problem about discriminants from the last post. You can review parts 1, 2, 3, and 4 of our discussion on the discriminant 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

The Discriminant of a Quadratic Equation – Part 4

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

**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

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

**Solution: **By the second equation, we have *y* = *x*^{2}, so we can replace *y* by *x*^{2 }in the first equation to get

*dx* + *ex*^{2} = *f*

Writing this quadratic equation in general form gives the following.

*ex*^{2} + *dx –* *f* = 0

So, we have *a* = *e*, *b* = *d*, and *c* = *–* *f* . Thus, the discriminant is

Δ = *b*^{2} – 4*ac = d*^{2} – 4*e*(*–* *f*) = *d*^{2} + 4*e**f*

We want the quadratic equation to have two real solutions. Therefore, the discriminant must be positive. So, we must have

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

This is choice **E**.

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