How to Solve a Quadratic Equation Step by Step
You can solve a quadratic equation, any equation in the form ax² + bx + c = 0, three ways: factoring, completing the square, or the quadratic formula. Factoring is fastest when the equation factors cleanly, while the quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), works for every quadratic. For example, x² + 5x + 6 = 0 factors to (x + 2)(x + 3) = 0, which gives x = -2 and x = -3.
A quadratic equation is one step up from a linear equation: the variable is squared, and that small change means it can have two answers instead of one. Quadratics show up all over algebra, from areas and projectile motion to graphing parabolas, so being able to solve them reliably matters.
There are three standard methods, and the good news is you do not need all three every time. This guide walks through each one with a worked example, explains the discriminant that tells you how many solutions to expect, and shows how to pick the right method quickly.
What Is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation, which means the highest power of the variable is 2. Its standard form is:
ax² + bx + c = 0
Here a, b, and c are numbers (the coefficients), and a cannot be 0, because without the x² term it would just be a linear equation. Graphed, a quadratic is always a parabola, a U-shaped curve.
These are quadratic equations:
- x² + 5x + 6 = 0
- 2x² + 5x - 3 = 0
- x² + 6x - 7 = 0
A quadratic can have two real solutions, one repeated solution, or no real solutions at all. The methods below find them.
The Three Ways to Solve a Quadratic Equation
There are three reliable methods:
- Factoring: fastest when the equation factors into whole numbers, but not every quadratic does.
- Completing the square: always works, and it is how the quadratic formula is derived.
- The quadratic formula: always works, and it is the go-to when factoring is not obvious.
Method 1: Solving by Factoring
Factoring is the quickest method when the quadratic breaks cleanly into two linear expressions.
- Write it in standard form. Make sure the equation reads ax² + bx + c = 0.
- Factor the expression. Find two binomials (px + q)(rx + s) that multiply back to give ax² + bx + c.
- Set each factor to zero. By the zero product property, if two things multiply to zero, at least one of them is zero. So set px + q = 0 and rx + s = 0.
- Solve each factor for x. That gives your two solutions.
Example: solve x² + 5x + 6 = 0
Find two numbers that multiply to 6 (the constant) and add to 5 (the middle coefficient): 2 and 3.
- Factor: (x + 2)(x + 3) = 0
- Set each factor to zero: x + 2 = 0 and x + 3 = 0
- Solve: x = -2 and x = -3
Check: for x = -2, (-2)² + 5(-2) + 6 = 4 - 10 + 6 = 0. Correct.
Method 2: Completing the Square
Completing the square rewrites the equation so one side is a perfect square. It always works, and it is worth knowing because it is the logic behind the quadratic formula.
- Move the constant to the right side. Rewrite ax² + bx + c = 0 as ax² + bx = -c.
- Make the leading coefficient 1. If a is not already 1, divide every term by a.
- Complete the square. Take half of the x coefficient, square it, and add that to both sides.
- Factor the perfect square. The left side now factors into (x + half the x coefficient)².
- Take the square root of both sides and solve for x. Remember the plus-or-minus.
Example: solve x² + 6x - 7 = 0
- Move the constant: x² + 6x = 7
- Half of 6 is 3, and 3 squared is 9, so add 9 to both sides: x² + 6x + 9 = 7 + 9
- Factor the left side: (x + 3)² = 16
- Take the square root: x + 3 = ±4
- Solve: x = -3 ± 4, which gives x = 1 and x = -7
Check: for x = 1, 1² + 6(1) - 7 = 1 + 6 - 7 = 0. Correct.
Method 3: The Quadratic Formula
The quadratic formula solves any quadratic in standard form, whether or not it factors. It is the method to reach for when factoring is not obvious.
x = (-b ± √(b² - 4ac)) / (2a)
- Identify a, b, and c from the standard form ax² + bx + c = 0.
- Substitute those values into the formula.
- Compute the discriminant, the part under the square root, b² - 4ac. It tells you how many real solutions to expect (more on this below).
- Simplify and solve for both the plus and the minus.
Example: solve 2x² + 5x - 3 = 0
- Identify coefficients: a = 2, b = 5, c = -3
- Substitute: x = (-5 ± √(5² - 4(2)(-3))) / (2(2))
- Simplify under the root: x = (-5 ± √(25 + 24)) / 4 = (-5 ± √49) / 4 = (-5 ± 7) / 4
- Solve both signs: x = (-5 + 7) / 4 = 2/4 = 1/2, and x = (-5 - 7) / 4 = -12/4 = -3
Check: for x = -3, 2(-3)² + 5(-3) - 3 = 18 - 15 - 3 = 0. Correct.
The Discriminant: How Many Solutions Are There?
The discriminant is the expression under the square root in the quadratic formula: b² - 4ac. Before solving, it tells you how many real solutions a quadratic has.
- If b² - 4ac is greater than 0: two different real solutions. (In 2x² + 5x - 3 = 0 above, the discriminant is 49, so there are two.)
- If b² - 4ac equals 0: exactly one real solution, a repeated root. For example, x² - 6x + 9 = 0 has discriminant 36 - 36 = 0, and it factors to (x - 3)² = 0, so x = 3 is the only solution.
- If b² - 4ac is less than 0: no real solutions (the two solutions are complex). For example, x² + x + 1 = 0 has discriminant 1 - 4 = -3, so it has no real answer.
Checking the discriminant first can save you time, and it stops you from hunting for a real answer that is not there.
Which Method Should You Use?
All three methods give the same answer, so the choice is about speed:
- Try factoring first if the numbers are small and it factors in a few seconds. It is the fastest route when it works.
- Use the quadratic formula whenever factoring is not obvious. It never fails, which makes it the safest default.
- Reach for completing the square when you specifically need it, for example to put a parabola in vertex form. For simply finding the roots, the formula is usually quicker.
A simple rule: if you can factor it almost immediately, factor it. Otherwise, go straight to the quadratic formula.
Common Mistakes to Avoid
- Not writing standard form first. The equation must equal zero before you factor or apply the formula.
- Sign errors in the formula. Watch the -4ac term closely when c is negative, since two negatives make a positive.
- Dropping the plus-or-minus. A quadratic usually has two solutions, so keep both the plus and the minus when you take a square root.
- Dividing incorrectly when completing the square. If a is not 1, every term has to be divided by a, not just some of them.
- Assuming there are always two real answers. Check the discriminant first, because the equation may have one repeated root or none.
Frequently Asked Questions
What is a quadratic equation? A quadratic equation is a second-degree equation in the form ax² + bx + c = 0, where a is not 0. The highest power of the variable is 2, and its graph is a parabola. It can have two real solutions, one repeated solution, or none.
What are the three ways to solve a quadratic equation? The three standard methods are factoring, completing the square, and the quadratic formula. Factoring is fastest when the equation factors cleanly, while completing the square and the quadratic formula work for every quadratic.
What is the quadratic formula? The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a). It solves any quadratic written in standard form ax² + bx + c = 0, whether or not it can be factored.
How do you know how many solutions a quadratic equation has? Use the discriminant, b² - 4ac. If it is greater than 0 there are two real solutions, if it equals 0 there is one repeated solution, and if it is less than 0 there are no real solutions.
Which method is easiest for solving a quadratic equation? Factoring is the fastest when the equation factors with whole numbers. When it does not factor cleanly, the quadratic formula is the most reliable choice because it always works.