Discriminant Of A Quadratic Equation

What Is the Discriminant of a Quadratic Equation?

The discriminant of a quadratic equation is the expression found inside the square root of the quadratic formula:

Its name comes from the verb "to discriminate," which means to tell the difference between things. In algebra, the discriminant's job is to tell us the difference between the possible types of answers, or roots, we can get for a quadratic equation. By calculating just this one value, you can instantly determine whether the equation has two distinct real solutions, exactly one real solution, or no real solutions at all (instead having two complex solutions). It's a powerful shortcut that helps you understand the nature of a quadratic equation before you even start solving it.

First, a Quick Review of Quadratic Equations

Before we can master the discriminant, let's quickly refresh our memory on quadratic equations and the famous formula used to solve them. A quadratic equation is any equation that can be written in the standard form:

In this form:

• $x$ is the variable.
• $a$, $b$, and $c$ are coefficients, which are known numbers.
• The coefficient $a$ cannot be zero (if it were, it wouldn't be a quadratic equation!).

When we want to find the values of $x$ that make this equation true, we are looking for the "roots" or "solutions" of the equation. While factoring can sometimes work, the most reliable method for solving any quadratic equation is the quadratic formula:

Look closely at what's under the square root symbol: $b^2-4ac$. That's our hero, the discriminant! Its value directly impacts the outcome of this entire formula.

What Does the Value of the Discriminant Tell Us?

The magic of the discriminant lies in its sign. Whether it's positive, zero, or negative determines the number and type of solutions the quadratic equation has. Let's break down the three possible cases.

Case 1: The Discriminant is Positive ($b^2-4ac>0$)

If the discriminant is a positive number, the quadratic equation will have two distinct real solutions. Why? In the quadratic formula, we have $\pm \sqrt{\text{discriminant}}$. If the discriminant is positive, its square root is a real number. This means we will perform two different calculations: one where we add the square root ($-b+\sqrt{...}$) and one where we subtract it ($-b-\sqrt{...}$). This gives us two different, or distinct, real answers for $x$.

Case 2: The Discriminant is Zero ($b^2-4ac=0$)

If the discriminant is exactly zero, the quadratic equation will have exactly one real solution. This is sometimes called a repeated root or a double root. The reason is simple. The $\pm \sqrt{...}$ part of the formula becomes $\pm \sqrt{0}$, which is just $\pm 0$. Adding zero and subtracting zero gives you the same result. So, the formula simplifies to $x=\frac{-b}{2a}$, yielding only one unique solution.

Case 3: The Discriminant is Negative ($b^2-4ac<0$)

If the discriminant is a negative number, the quadratic equation will have no real solutions. In the real number system, which is what you primarily work with in Algebra 1, you cannot take the square root of a negative number. Since the discriminant is under the square root, a negative value creates an impossible situation. Therefore, there are no real numbers for $x$ that can solve the equation. (Later in your math journey, you'll learn about imaginary and complex numbers, which provide a way to express these solutions, but for now, we say there are no real solutions.)

Summary Table

Here is a handy table to summarize the three cases:

Putting It All Together: Step-by-Step Examples

Let's work through some examples from start to finish. For each one, we will calculate the discriminant and then state the number and type of solutions.

How Does the Discriminant Relate to the Graph of a Parabola?

The solutions (or roots) of a quadratic equation are the x-intercepts of its graph, which is a U-shaped curve called a parabola. The discriminant gives us a powerful clue about how this parabola looks without ever having to graph it!

• If $D>0$ (two real solutions), it means the parabola crosses the x-axis at two distinct points.
• If $D=0$ (one real solution), it means the vertex (the lowest or highest point) of the parabola touches the x-axis at exactly one point.
• If $D<0$ (no real solutions), it means the parabola completely misses the x-axis. It will be entirely above or entirely below the x-axis, never crossing it.

Thinking about the graph is a great way to build a visual understanding of what the discriminant is actually telling you. It connects the abstract algebra to a concrete geometric picture.

What Are Some Common Mistakes to Avoid?

Using the discriminant is straightforward, but a few common slip-ups can lead to the wrong answer. Be sure to watch out for these:

1. Forgetting to Set the Equation to Zero: The discriminant formula $b^2-4ac$ only works when the equation is in standard form $a{x}^2+bx+c=0$. Always rearrange the equation first, like we did in Example 3.
2. Sign Errors with 'b': When you square a negative value for $b$, the result is always positive. For example, if $b=-5$, then $b^2=(-5{)}^2=25$, not $-25$. Always use parentheses: $(-5{)}^2$.
3. Sign Errors with '-4ac': This is the most common mistake. Pay close attention to the signs of $a$ and $c$. If either $a$ or $c$ is negative, the $-4ac$ term will become positive. For example, if $a=1$ and $c=-2$, then $-4ac=-4(1)(-2)=+8$.
4. Misidentifying a, b, or c: Double-check that you have correctly identified the coefficients. If a term is missing, its coefficient is zero. For example, in $x^2-9=0$, $a=1$, $b=0$, and $c=-9$.
5. Confusing the Discriminant with the Solution: Remember, the discriminant tells you how many solutions there are, not what the solutions are. A discriminant of $25$ doesn't mean the answer is $25$; it means there are two real solutions.

Quick Summary: Discriminant at a Glance

Need a quick refresher? Here's everything you need to know in a nutshell.

• What is it? The discriminant is a part of the quadratic formula.
• The Formula: $D=b^2-4ac$
• The Three Rules:
  1. If $D>0$ (positive) $\to$ Two distinct real solutions.
  2. If $D=0$ (zero) $\to$ One repeated real solution.
  3. If $D<0$ (negative) $\to$ No real solutions.
• The First Step: Always make sure your equation is in the form $a{x}^2+bx+c=0$ before you begin!