How to Solve a Quadratic Equation

A ball is thrown into the air. When does it hit the ground? That question is a quadratic. To solve a quadratic equation, you find the values of x that make ax^2 + bx + c = 0 true. You can do this three ways: by factoring, with the quadratic formula, or by reading the discriminant to count the answers first. This guide walks through all three with worked examples you can check by hand.

What a Quadratic Equation Is

A quadratic equation is any equation you can write in the form ax^2 + bx + c = 0. The letters a, b, and c stand for numbers, and x is the unknown you want to find.

The one strict rule is that a cannot be 0. If a were 0, the x^2 term would vanish and you would have a plain line, not a curve. The squared term is what makes an equation quadratic.

Here a, b, and c are called the coefficients. In x^2 – 5x + 6 = 0, you have a = 1, b = -5, and c = 6. If you want a refresher on what that little 2 means, our sibling guide on exponents and scientific notation covers powers in depth.

Most quadratics have two solutions, which are also called roots. Some have one, and some have none in the real numbers. The methods below find them.

The x^2 term also gives every quadratic a curved graph called a parabola. The solutions are the exact spots where that curve touches or crosses the horizontal axis, which you will see pictured further down.

Method 1: Solve by Factoring

Factoring is often the quickest way when the numbers are friendly. The idea rests on one fact: if two things multiply to 0, at least one of them must be 0.

Let us solve x^2 – 5x + 6 = 0. You need two numbers that multiply to c (which is 6) and add to b (which is -5).

• Try -2 and -3. Multiply: (-2) * (-3) = 6. Correct.
• Add: (-2) + (-3) = -5. Correct.

So the equation factors into (x – 2)(x – 3) = 0. Now set each factor equal to 0. That gives x – 2 = 0, so x = 2, and x – 3 = 0, so x = 3.

The two solutions are x = 2 and x = 3. You can check either one: put x = 2 back in to get 4 – 10 + 6 = 0. It works.

Factoring is quickest when the numbers cooperate like this. If you cannot find two whole numbers that fit, the equation may not factor neatly, and that is your cue to switch to the formula below.

Method 2: The Quadratic Formula

Some quadratics do not factor with tidy whole numbers. The quadratic formula always works, no matter what a, b, and c are.

The formula is x = (-b +/- sqrt(b^2 – 4ac)) / (2a). The +/- symbol means you compute two answers: one with a plus, one with a minus.

Let us solve the same equation, x^2 – 5x + 6 = 0, so we can compare. Here a = 1, b = -5, and c = 6. Substitute carefully.

• -b = -(-5) = 5.
• b^2 – 4ac = (-5)^2 – 4(1)(6) = 25 – 24 = 1.
• sqrt(1) = 1, and 2a = 2(1) = 2.

Now x = (5 +/- 1) / 2. With the plus: (5 + 1) / 2 = 6 / 2 = 3. With the minus: (5 – 1) / 2 = 4 / 2 = 2.

The answers are x = 3 and x = 2, exactly what factoring gave. When a result is a fraction, our guide on how to simplify fractions helps you reduce it to lowest terms.

The Discriminant Tells You How Many Solutions

The discriminant is the part under the square root: b^2 – 4ac. Compute it first and you learn how many real solutions to expect before doing the rest.

• If b^2 – 4ac is greater than 0, there are two different real solutions.
• If b^2 – 4ac equals 0, there is exactly one real solution.
• If b^2 – 4ac is less than 0, there are no real solutions.

Our first example had a discriminant of 25 – 24 = 1, which is greater than 0. That is why it had two answers, 2 and 3.

Now try x^2 + 4x + 4 = 0, where a = 1, b = 4, c = 4. The discriminant is 4^2 – 4(1)(4) = 16 – 16 = 0. That signals one solution: x = -b / (2a) = -4 / 2 = -2.

Check it by putting x = -2 back in: (-2)^2 + 4(-2) + 4 = 4 – 8 + 4 = 0. The single root sits right where the parabola just touches the axis instead of crossing it. When the discriminant is negative, the roots are complex numbers, a topic beyond this guide.

Back to the Falling Ball

Remember the ball thrown into the air. Its height often follows a quadratic like h = -5t^2 + 10t, where h is height in meters and t is time in seconds. To find when it hits the ground, set the height to 0.

So we solve -5t^2 + 10t = 0. This one factors fast. Pull out the common factor -5t to get -5t(t – 2) = 0.

• Set -5t = 0, which gives t = 0. That is the moment of the throw.
• Set t – 2 = 0, which gives t = 2. That is when it lands.

The ball hits the ground at t = 2 seconds. Notice how the same factoring trick answered a real question. That is the point of solving quadratics: the two roots often mark two meaningful moments or positions.

The same shape shows up in areas, profits, and distances. Whenever a problem multiplies an unknown by itself, a quadratic is usually hiding inside, and these three methods pull the answer out.

Which Method to Use

All three tools solve the same problem, but each shines in a different spot. Use this quick guide to pick.

• Factoring: best when a, b, and c are small whole numbers that split cleanly. It is fast and needs no formula.
• Quadratic formula: best when factoring is hard or the answers are messy. It never fails on any quadratic.
• Discriminant first: best when you only need to know how many real answers exist, or want to check your work before solving.
• Completing the square: useful for finding the vertex form, though the formula usually gets the roots faster.

When you want a quick check or the numbers get ugly, let a tool do the arithmetic. Our Quadratic Equation Calculator solves any ax^2 + bx + c = 0 and shows the steps.

Frequently Asked Questions About Quadratic Equations

Sources

This article is for general education only. Always double-check the formulas and math for your own problem before you rely on the result. Reviewed for accuracy by Prof. Dr. Khalil Mudassar, PhD. Last updated September 11, 2026.