[Common Math 1 Part 14] The Discriminant and the Number of Roots

In the previous post, we derived the quadratic formula and noticed that one part of it stands out:

$$b^2-4ac.$$

Now we focus on that value itself and use it to read the number and type of roots at a glance.

Understand how the discriminant determines the number and type of roots of a quadratic equation, and connect that idea back to the quadratic formula.

The flow is simple.

• The quadratic formula contains $\sqrt{b^2-4ac}$.
• So the sign of $b^2-4ac$ controls the shape of the roots.
• That value $b^2-4ac$ is called the discriminant.
• Once we know the discriminant, we can predict the roots much faster.

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1. Where Does the Discriminant Come From?

For the quadratic equation

$$ax^2+bx+c=0 \qquad (a\neq 0),$$

the quadratic formula is

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.$$

The part that changes the nature of the roots is not the denominator $2a$, but

$$\sqrt{b^2-4ac}.$$

That is because

• if the value inside the square root is positive, the square root is a real number
• if it is 0, the square root becomes 0
• if it is negative, the real-number square root does not exist and complex numbers appear

So we define

$$D=b^2-4ac$$

and call it the discriminant.

So from this point on, we simply write $D=b^2-4ac$.

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2. When the Discriminant Is Positive: Two Distinct Real Roots

First consider the case $D>0$.

If the discriminant is positive, then $\sqrt{D}$ is a nonzero real number. So in the quadratic formula,

$$x=\frac{-b+\sqrt{D}}{2a}, \qquad x=\frac{-b-\sqrt{D}}{2a}$$

give two different real values.

Therefore the equation has two distinct real roots.

2-1. Example

$$x^2-5x+6=0$$

has

$$a=1, \qquad b=-5, \qquad c=6,$$

so

$$D=b^2-4ac=25-24=1.$$

Since $D>0$, the equation has two distinct real roots. Indeed, the roots are $x=2$ and $x=3$.

The important point is that even before solving completely, we already know the equation must have two different real roots.

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3. When the Discriminant Is Zero: A Repeated Root

Now consider the case $D=0$.

Then $\sqrt{D}=0$, so the quadratic formula becomes

$$x=\frac{-b\pm0}{2a}=\frac{-b}{2a}.$$

The symbol $\pm$ is still there in form, but both choices give the same value. This is called a repeated root. More precisely, it is one root with multiplicity 2.

3-1. Example

$$x^2-4x+4=0$$

has

$$a=1, \qquad b=-4, \qquad c=4,$$

so

$$D=b^2-4ac=16-16=0.$$

Therefore the equation has a repeated root. Indeed,

$$(x-2)^2=0,$$

so the only root is $x=2$.

It is helpful to think of this as the same value appearing twice in the algebraic structure.

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4. When the Discriminant Is Negative: Two Complex Conjugate Roots

Finally consider the case $D<0$.

If the discriminant is negative, then $\sqrt{D}$ is not a real number. But in the complex number system, we can still continue the formula.

So there is no real root, but there are still roots in the complex number system.

So the equation still has roots, but they are two complex conjugate roots.

4-1. Example

$$x^2-4x+5=0$$

has

$$a=1, \qquad b=-4, \qquad c=5,$$

so

$$D=b^2-4ac=16-20=-4.$$

Since $D<0$, there are no real roots, but there are complex roots.

Here,

$$\sqrt{-4}=\sqrt{4\cdot(-1)}=2\sqrt{-1}=2i,$$

so

$$x=\frac{4\pm\sqrt{-4}}{2}=\frac{4\pm2i}{2}=2\pm i.$$

The two roots are $2+i$ and $2-i$. Because the quadratic formula uses $\pm$, the real part stays the same while the imaginary part changes sign. That is why the two roots form a conjugate pair.

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5. What Can We Read Immediately from the Discriminant?

The biggest advantage of the discriminant is that we do not need to finish every calculation before learning the nature of the roots.

From a graph point of view, it also tells us whether the parabola crosses the $x$-axis twice, touches it once, or misses it in the real plane.

We can summarize it as follows:

$$\begin{array}{c|c} \text{Discriminant } D & \text{Type of roots} \\ \hline D>0 & \text{two distinct real roots} \\ D=0 & \text{one repeated root} \\ D<0 & \text{two complex conjugate roots} \end{array}$$

This table is not just something to memorize. It is a compressed version of what the quadratic formula is already telling us.

If you step through the three cases on the plane below, it becomes much easier to connect the sign of the discriminant with the graph and the type of roots at the same time.

5-1. Why Is It Called the Discriminant?

Because this single value distinguishes

• whether the roots are real or complex
• and, if real, whether there are two different roots or one repeated root

So the discriminant does not directly give the roots. It tells us their character in advance.

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6. Common Mistakes

The discriminant is short, so it is easy to calculate carelessly.

6-1. Computing $b^2$ Incorrectly

If $b=-3$, then

$$b^2=(-3)^2=9,$$

not $-9$.

6-2. Forgetting That We Subtract All of $4ac$

In $b^2-4ac$, we subtract the entire quantity $4ac$.

For example, if $a=2$ and $c=5$, then

$$4ac=4\cdot2\cdot5=40,$$

not something like $8+5$.

6-3. Saying "There Is No Root" When $D<0$

The correct statement is that there is no real root. There may still be roots in the complex number system.

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7. Key Takeaways

• The discriminant is $D=b^2-4ac$.
• If $D>0$, there are two distinct real roots.
• If $D=0$, there is a repeated root.
• If $D<0$, there are no real roots, but there are two complex conjugate roots.
• The discriminant is a tool for judging the number and type of roots before solving fully.

In the next post, we will connect the roots of a quadratic equation to the graph of a quadratic function, especially how the number of intersections with the $x$-axis relates to the discriminant.

One-line conclusion:

The discriminant is not just a shortcut. It is the key signal inside the quadratic formula that tells us the number and type of roots in advance.