[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.

In particular, it is often more efficient to check the discriminant first than to push the quadratic formula all the way through.

• If we use the quadratic formula immediately, we must carry out both computations in $-b\pm\sqrt{D}$.
• But if we only want the type of roots, then computing $D$ alone is enough.
• So the discriminant is the fast way to read the structure before solving for exact values.

For example, in $x^2-5x+6=0$, once we see that $D=1$, we already know the equation has two distinct real roots. At that stage, we do not need to finish the entire quadratic-formula calculation yet.

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.

5-1. Quick Sign-Based Tips

Not every quadratic can be settled from coefficient signs alone. Still, a few cases can be read immediately before any full calculation.

• If $a$ and $c$ have opposite signs, then $ac<0$, so $D=b^2-4ac$ is automatically positive. Therefore the equation has two distinct real roots.
• If $b=0$ and $a$ and $c$ have the same sign, then $D=-4ac<0$, so there is no real root.
• If $b=0$ and $a$ and $c$ have opposite signs, then $D=-4ac>0$, so there are two distinct real roots.

These are not replacements for the discriminant. They are quick observations that help us see the discriminant faster.

5-2. This Classification Is Cleanest When $a,b,c$ Are Real

The usual summary

• $D>0$ : two real roots
• $D=0$ : a repeated root
• $D<0$ : two nonreal roots

is the standard version used when $a,b,c$ are real numbers.

In that setting,

• if $D<0$, the nonreal roots come as a conjugate pair
• if $D=0$, the repeated root is always real

But once the coefficients contain imaginary parts, the story changes a little. Then reading the discriminant by its sign no longer works in the same way.

For example,

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

expands to

$$x^2-2ix-1=0$$

and its discriminant is

$$D=(-2i)^2-4\cdot1\cdot(-1)=0.$$

So this is a repeated root, but the repeated root is not real. It is the nonreal repeated root $x=i$.

So in the school-level setting, it is safest to read the discriminant under the usual assumption that $a,b,c$ are real.

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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. Practice

Use the checkpoint below to practice reading root types from the discriminant.

8. 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.