**The Discriminant of a Quadratic Equation**

Part 3

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:

The Discriminant of a Quadratic Equation – Part 1

The Discriminant of a Quadratic Equation – Part 2

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} + 8x + 7 = 0**. Then describe the nature of the roots of the equation, and describe the graph of the function

*y = x*^{2}*+*8

**x****+**

**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 = *b*^{2} – 4*ac* = 8^{2} – 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 (6^{2} = 36), the roots are actually rational.

The graph of the function *y = x ^{2} + *8

*x + 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:

*x*^{2} + 8*x *+ 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 ** ax^{2} + bx + 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 = ax**^{2}** + ****bx**** +*** c *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 ** 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**.

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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