How to Solve Quadratic Equations: 4 Methods Explained Step-by-Step

If you’ve ever stared at an equation like ax² + bx + c = 0 and wondered where to even start, you’re not alone. Learning how to solve quadratic equations is one of the most common algebra challenges students face — but the good news is there isn’t just one way to do it. There are 4 methods to solve quadratic equations, and once you know when to use each one, solving quadratics step by step becomes second nature.

In this guide, we’ll walk through every method — from the easiest way to solve quadratic equations (factoring) to the most universal one (the quadratic formula) — with clear examples at every step.

Quick Answer: What are the 4 methods of solving quadratic equations? Factoring, Completing the Square, the Quadratic Formula, and Graphing.

Method 1: Factoring

Factoring is often the easiest way to solve quadratic equations — but it only works cleanly when the equation factors into whole numbers.

Steps:

1. Write the equation in standard form: ax² + bx + c = 0
2. Find two numbers that multiply to a×c and add to b
3. Rewrite the middle term using those two numbers
4. Factor by grouping
5. Set each factor equal to zero and solve for x

Example: Solve x² + 5x + 6 = 0
Find two numbers that multiply to 6 and add to 5 → 2 and 3
(x + 2)(x + 3) = 0
x = −2 or x = −3

Method 2: Completing the Square

When an equation doesn’t factor nicely, completing the square turns it into a perfect square trinomial so you can solve it by taking a square root.

Steps:

1. Move the constant term to the right side
2. Divide every term by a if a ≠ 1
3. Add (b/2)² to both sides
4. Write the left side as a squared binomial
5. Take the square root of both sides and solve for x

Example: Solve x² + 6x − 7 = 0
x² + 6x = 7
x² + 6x + 9 = 7 + 9
(x + 3)² = 16
x + 3 = ±4 → x = 1 or x = −7

Method 3: The Quadratic Formula

This is the one method that works for every quadratic equation, no matter how messy — which is why it’s the most reliable way of how to find roots of a quadratic equation when other methods fail. If you’re wondering how to find roots of a quadratic equation without guessing, this formula is your answer.

x = (−b ± √(b² − 4ac)) / 2a

Example: Solve 2x² + 3x − 2 = 0 (a=2, b=3, c=−2)
x = (−3 ± √(9 + 16)) / 4
x = (−3 ± 5) / 4
x = 0.5 or x = −2

Once you’ve worked through the formula by hand, it’s worth double-checking your answer instantly with a quadratic equation solver before moving on to more complex problems.

Method 4: Graphing

Graphing shows you exactly where the parabola crosses the x-axis — those crossing points are your solutions.

Steps:

1. Plot y = ax² + bx + c as a parabola
2. Identify where the curve crosses the x-axis
3. Those x-values are the roots

Graphing is more visual than exact, so it’s best for understanding behavior rather than pinpoint answers — pair it with the quadratic formula for exact roots.

How to Solve Quadratic Equations: Which Method Should You Use?

• Equation factors into whole numbers → Factoring
• Need to derive vertex form → Completing the Square
• Equation doesn’t factor easily / any equation → Quadratic Formula
• Want a visual understanding → Graphing

Factoring vs quadratic formula comes down to speed vs reliability — factoring is faster when it works, but the quadratic formula never fails. That’s the core trade-off in factoring vs quadratic formula: one is quick but limited, the other is slower but universal.

So when to use the quadratic formula instead of factoring? Anytime the numbers look messy, involve fractions or decimals, or you simply want a guaranteed method that works every time. In short, when to use the quadratic formula instead of factoring comes down to how “clean” the equation looks.