**The Discriminant of a Quadratic Equation**

Part 1

Part 1

Recall that the quadratic equation ** ax^{2} + bx + c = 0 **can be solved using the quadratic formula:

For more information on the quadratic formula, see the following posts:

Quadratic Formula – Part 1 Quadratic Formula – Part 2

Before we go on, you may also want to review the information on square roots that can be found here:

The **discriminant** of the quadratic equation ** ax^{2} + bx + c = 0 **is the quantity

**Δ**defined by

**Δ = b^{2} – 4ac**

In other words, the discriminant is simply the expression that appears under the square root in the quadratic formula.

Although computing the discriminant of a quadratic equation does not give the roots (solutions) of the equation, it does give us a lot of information about the nature of the roots and the graph of the equation.

For example, if the discriminant of the quadratic equation ** ax^{2} + bx + c = 0 **is 0, then the quadratic formula simplifies to

** x = –b/2a**,

and we see that there is just one solution.

If the coefficients ** a** and

**are integers, then the unique solution will be a rational number.**

*b*Graphically, this means that the graph of the function * y = ax^{2} + bx + c *is a parabola that intersects the

*x*-axis at one point.

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

**Note: **The discriminant **does not** tell us if the parabola opens upwards or downwards. However, this is easy to determine simply by looking at the value of ** a**.

If *a* > 0 (i.e. *a* is a positive number), then the parabola opens upwards.

If *a* < 0 (i.e. *a* is a negative number), then the parabola opens downwards.

So, the red parabola in the image above has an equation of the form *y = ax ^{2} + bx + c* where

*a*< 0 and Δ = 0

**Example: **Find the discriminant of ** x^{2} + 6x + 9 = 0**. Then describe the nature of the roots of the equation, and describe the graph of the function

**y = x**^{2}+**6**

**x +****9**.

I will post a solution to this problem tomorrow, and then discuss other possibilities for the determinant. Feel free to post your own solutions in the comments.

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