The Discriminant And Three Cases

Using The Quadratic Formula Through Examples

The quadratic formula is a useful formula for solving x-intercepts of quadratic equations in the form of

The quadratic formula (with ) is:

It is preferable to use the quadratic formula when factoring techniques do not work.

For many (highschool) students, it is not expected to know how to come up with the quadratic formula. This proof is more of a reference to help you see where this formula comes from.

The coefficient that is with is . This is divided by 2 gives us . The square of yields or . Adding and subtracting by gives us:

We then factor the first three terms. That is, we factor into .

The distributive law is applied such that is expanded to the first two terms.

Next, we simplify the term.

Because we want to find x-intercepts, we set and solve for .

The final line is the quadratic formula or the value such that it makes in .

The Discriminant And Three Cases

Notice how in the quadratic formula there is a square root part after the plus and minus sign (). The part inside the square root () is called the discriminant.

An important property of square roots is that square roots take on numbers which are at least 0 (non-negative). A negative number inside the square root is undefined (in the real numbers).

We look at three cases for the discriminant and what each case means.

If then there would be 2 distinct solutions for (or x-intercepts) in the equation .\

If then there would be one value for in the equation .

If , we would have a negative value inside the square root. The square root of a negative value is undefined. There would be no real-numbered values for in the equation .

Using The Quadratic Formula Through Examples

The quadratic formula can be applied to any quadratic equation in the form . It does not really matter whether the quadratic form can be factored or not.

Example One

Given the quadratic equation , what are the x-intercepts?

From , we have , and . Using these values, the quadratic formula is as follows:

The x-intercepts (when ) for are (or 0.618034) and (or -1.618034). You can choose to use exact values or using the decimal of the answers (with a calculator).

Example Two

Apply the quadratic formula to find x-intercepts for the equation .

Here, we have , and .

The x-intercepts are (or 1.780776) and (or -0.2807764).

Example Three

Show that the equation has no x-intercepts.

You can use the quadratic formula right away here. However, it is a bit easier and faster to check the discriminant .

Since the discriminant is negative, we have a negative inside the square root of the quadratic formula. The equation has no x-intercepts.