Identifying the discriminant

In a nutshell

The discriminant is part of the quadratic formula. Specifically, it is the $$b^2-4ac$$ part, where the quadratic is in the form $$ax^2+bx+c$$. It tells you how many real solutions there are to a quadratic equation.

Identifying the discriminant

The quadratic formula is 

​$$\boxed{x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}}$$​​

which is used to solve the quadratic equation

​$$ax^2+bx+c=0$$​​

The discriminant is the part inside the square root.

Example 1

Identify the value of the discriminant of the quadratic equation 

$$x^2-5x+6=0$$​​

First identify $$a$$, $$b$$ and $$c$$. These are the coefficients of $$x^2$$, $$x$$ and the constant term respectively. So:

​$$a=1\\b=-5\\c=6$$​​

Now insert these into the formula for the discriminant:

$$\begin {aligned} b^2-4ac&=(-5)^2-4(1\times6)\\b^2-4ac&=25-24\\b^2-4ac&=1\end {aligned}$$​​

So the value of the discriminant for this quadratic equation is $$\underline1$$.

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Counting solutions

Quadratic equations may have two, one or zero real solutions.

Two real solutions

If a quadratic equation has two real solutions, then $$b^2-4ac>0$$. In other words, the discriminant is positive.

One real solution

If it has only one real solution, then $$b^2-4ac=0$$.

Zero real solutions

Finally, if the quadratic equation has no real solutions, then $$b^2-4ac<0$$. The discriminant is negative.

Example 2

How many real solutions are there to the following quadratic equation?

$$3x^2-4x+2=0$$​​

Here, $$a=3$$, $$b=-4$$ and $$c=2$$, so the discriminant is equal to:

$$\begin{aligned}b^2-4ac&=(-4)^2-4(3)(2)\\&=16-24\\&=-8\end{aligned}$$​​

This is negative, so this quadratic has no real solutions.

In the context of the quadratic formula

Note: There are many references to "real" solutions. All you need to know is that real numbers are all the numbers you've ever worked with. Here, "real" is used as opposed to "imaginary", but you do not need to know what an imaginary number is. 

Since the discriminant is inside the square root, you have to keep in mind what can be square rooted. A negative number cannot be square rooted, so it follows that if the discriminant is negative, then the quadratic formula cannot give any real solutions.

Notice also that the square root in the quadratic formula is added to give one solution, and subtracted to give another. Hence if the discriminant is zero, it follows that the square root part is zero and so the two solutions to the quadratic equation will be the same:

​$$\dfrac{-b+\sqrt0}{2a}=\dfrac{-b-\sqrt0}{2a}=\dfrac{-b}{2a}$$​​

When the discriminant is positive, the square root in the quadratic formula will have a value. Hence using the quadratic formula, one solution is found by adding the square root of the discriminant and the other, different solution, is found by subtracting it.