The Discriminant Explained: How Many Solutions Does a Quadratic Have?

The discriminant is the small expression hiding under the square root in the quadratic formula. Working it out first tells you how many solutions to expect, what kind they are, and whether the quadratic can be factored.

Quick idea

For ax² + bx + c = 0, the discriminant is b² − 4ac. Positive means two real solutions, zero means one repeated solution, and negative means no real solutions.

Quadratic formula solver

Type the coefficients of ax² + bx + c = 0. You get the discriminant, both solutions (exact and decimal), and every step.

abc

Two real solutions: x ≈ 3 and x ≈ 2

$$x=3\ \text{or}\ x=2$$

1. Step 1.Start from standard form and read off a, b, and c (signs included).

   $$\begin{gathered}x^2-5x+6=0\\ a=1,\ b=-5,\ c=6\end{gathered}$$

2. Step 2.Substitute every value into the quadratic formula, keeping negatives in parentheses.

   $$x=\frac{-(-5)\pm\sqrt{(-5)^2-4(1)(6)}}{2(1)}$$

3. Step 3.Work out the discriminant, the part under the square root.

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

4. Step 4.The discriminant is positive, so there are two different real solutions.

   $$x=\frac{5\pm\sqrt{1}}{2}=\frac{5\pm 1}{2}$$

5. Step 5.Split the ± into a plus version and a minus version.

   $$\begin{aligned}x_1&=\frac{5+1}{2}\approx 3\\ x_2&=\frac{5-1}{2}\approx 2\end{aligned}$$

6. Step 6.Exact form, fully simplified:

   $$x=3\ \text{or}\ x=2$$

7. Step 7.Bonus: the vertex of the parabola sits halfway between the roots, at x = −b/(2a).

   $$\left(2.5,\ -0.25\right)$$

Why the discriminant works

The discriminant is the part of the quadratic formula under the square root:

• If it is positive, the square root is a real number, and the $\pm$ produces two different answers.
• If it is zero, $\pm 0$ changes nothing, so both answers are the same.
• If it is negative, there is no real square root, so there are no real answers.

Steps

1. Write the equation in standard form, ax² + bx + c = 0.
2. Identify a, b, and c, keeping their signs.
3. Calculate b² − 4ac carefully, using parentheses for negative values.
4. Read the sign: positive gives two real roots, zero gives one, negative gives none.

Worked example

How many real solutions does 3x² − 4x − 2 = 0 have?

Full solution

Step-by-step walkthrough

Step 1

Here a = 3, b = −4, and c = −2. Substitute them into b² − 4ac.

Step 2

(−4)² is 16, and −4(3)(−2) is +24 because two negatives multiply to a positive.

Step 3

The discriminant is 40, which is positive, so there are two real solutions. Because 40 is not a perfect square, they are irrational.

More worked examples

Example 2: A perfect-square discriminant

Describe the solutions of $2x^2-7x+3=0$.

Step 1. Identify a = 2, b = −7, c = 3 and compute.

Step 2. 25 is positive and a perfect square, so there are two rational solutions — and the quadratic factors.

Answer: Two rational solutions: $x=\tfrac{1}{2}$ and $x=3$.

Example 3: A negative discriminant

Describe the solutions of $x^2+x+1=0$.

Step 1. Identify a = 1, b = 1, c = 1 and compute.

Answer: The discriminant is negative, so there are no real solutions. The parabola sits entirely above the x-axis.

Example 4: Finding k for exactly one solution

For what values of $k$ does $x^2+kx+9=0$ have exactly one real solution?

Step 1. One solution means the discriminant equals 0.

Step 2. Solve for k.

Answer: $k=6$ or $k=-6$, giving $(x+3)^2=0$ and $(x-3)^2=0$.

What the discriminant tells you about the graph

The real solutions of $ax^2+bx+c=0$ are where the parabola $y=ax^2+bx+c$ meets the x-axis.

Discriminant	Graph
$b^2-4ac>0$	Crosses the x-axis at two points
$b^2-4ac=0$	Touches the x-axis once, at the vertex
$b^2-4ac<0$	Stays completely above or below the x-axis

Perfect squares tell you whether it factors

When $a$, $b$, and $c$ are integers, a discriminant that is a perfect square (0, 1, 4, 9, 16, 25, …) means the solutions are rational and the quadratic factors over the integers. Any other positive value means the solutions are irrational, so factoring will not work and you should use the formula.

Common mistakes to avoid

• The discriminant is b² − 4ac, not the square root of it. You only decide the number of solutions from its sign, so there is no need to take a square root at this stage.
• Writing $-b^2$ instead of $b^2$. The discriminant starts with $b^2$, which is never negative.
• Forgetting that $-4ac$ flips sign when $a$ and $c$ have opposite signs.
• Calculating the discriminant before the equation is in standard form.

Practice problems

1. How many real solutions does x² + 4x + 4 = 0 have?
2. How many real solutions does x² + x + 1 = 0 have?
3. Find the discriminant of 2x² − 7x + 3 = 0 and describe the solutions.

Answers

1. Discriminant 0: one repeated solution (x = −2)
2. Discriminant −3: no real solutions
3. Discriminant 25: two rational solutions (x = 3 and x = 1/2)

Frequently asked questions

What is the discriminant of a quadratic?

For ax² + bx + c = 0, the discriminant is b² − 4ac. Its sign tells you how many real solutions the equation has.

What does a discriminant of zero mean?

A discriminant of 0 means the quadratic has exactly one real solution, called a repeated or double root. The parabola touches the x-axis at its vertex.

Can the discriminant tell me the actual solutions?

Not by itself. It tells you how many solutions there are and what kind they are. To find them, put it into the quadratic formula: x = (−b ± √discriminant) / (2a).

Ask PennPaper to explain it live

If the steps make sense but you still feel stuck, start a PennPaper lesson and ask the tutor to draw the problem on the whiteboard. Seeing the symbols move step by step is often what makes the concept click.

Related guides

• The Quadratic Formula Explained
• How to Factor Quadratics
• Vertex Form and How to Graph a Parabola