# Using the Discriminant Worksheets

We use the quadratic formula to find the x-intercepts of quadratic equations when they present in the form: ax^{2} + bx + c = 0. This mean that when we use this formula the solution will tell us where the graphed form of this equation crosses the x-incept(s). Sometimes that line will cross the x-intercept twice, once, or even never at all. You can quickly assess how many times a particular equation crosses that axis by looking at the value that is created in a particular part of the quadratic formula. We call the portion of the quadratic formula found under the square root (b^{2} - 4ac) the discriminant. Depending on whether the end value of the discriminant is positive, negative, or zero will indicate to you how many times the graph of the equation crosses the x-intercept(s). In the selection of worksheets and lessons below we will show you how to determine and interpret the values of the discriminant.

### Aligned Standard: HSA-REI.B.4b

- Finding the Discriminant Step-by-step Lesson- This is an often overlooked, yet helpful concept for all math students.
- Guided Lesson - You want to know how overlooked this concept; MS Word marks the word "Discriminant" as a misspelling in all contexts.
- Guided Lesson Explanation - Breaking down parts of the Quadratic formula does make it seem less obtrusive.
- Practice Worksheet - It would be nice if they had more standards that broke down a formula so deeply.
- Matching Worksheet - Match the quadratic to the value of it's simplified discriminant.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

This skill use to just apply to trigonometry, but we see it pop up with algebra more often now.

- Homework 1 - Find the value of the discriminant by breaking down what appears under the radical.
- Homework 2 - The discriminant of a quadratic equation (ax
^{2}+ bx +c = 0) is b^{2}- 4ac. It is the expression under the radial in the quadratic formula. - Homework 3 - Four problems that are totally broken down for you.

### Practice Worksheets

Kids often ask what the discriminant actually is. I usually just say it's your answer to a quadratic equation.

- Practice 1 - Apply the formula and run away with it. There is plenty of space to work with here.
- Practice 2 - How many moves can you do it in?
- Practice 3 - The discriminant can also tell you the nature of the possible solutions.

### Math Skill Quizzes

See if you can spot the pattern I have going on in these.

- Quiz 1 - When a discriminant has a positive value there are two solutions and the as a result two x-intercepts.
- Quiz 2 - When it is zero, there is only one solution and one x-intercept.
- Quiz 3 - When it is negative there are no solutions and no x-intercepts.

### What Is a Discriminant?

(In Math) Remember the quadratic formula? For any quadratic equation in the form of:

ax^{2} + bx + c =0

The discriminant is the part of the quadratic formula that is present under the square root, which is b^{2} - 4ac. The discriminant can be a positive number, a negative number, or a zero. It determines the number of possible solutions to the given quadratic equation.

How the discriminant presents itself can tell you the types of solution(s) that you can expect. A positive discriminant shows that a quadratic equation has two distinct real number solutions. A negative discriminant shows that neither of the two solutions is a real number. A zero value for the discriminant shows that a quadratic equation has repeated a real number solution.

**EXAMPLE** - Suppose, you are asked to find out how many solutions the following quadratic equation has: 2x^{2} + 5x - 1 = 0

From the above equation, we can put together these values for the variable to be placed in the quadratic formula: a= 2, b= 5, c= -1. When we put these values in the discriminant, we get:

b^{2} - 4ac, (5)^{2} - 4(2)(-1), 25 - (-8), 25 + 8 = 33, 33 is a positive number, so it means that the quadratic has two solutions. This means that if you were to graph this equation it would cross the x-axis at two points.

### Why is This Value Important to Us?

The nature of this value is important for us to understand the different possible and types of solutions when working with quadratic equations. This is a great way to check your final answer. When working with these problems we can solve them in one of four different ways. Factoring can be used to condense and expand problem sets. We can use the complete the square method when we are dealing with complicated equations. We can also use the square root method to collect all the exponentials and place them to one side of the equation. If the quadratic is in standard form, we can simply use the quadratic formula to solve it. Which method we use to process these calculations is all dictated by understanding the spirit of the discriminant.