3 Methods for Solving Quadratic Equations- Step-by-Step Guide

What Is a Quadratic Equation?

A quadratic equation is any equation in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The goal is always to find the values of x that make the equation true. These solutions are called roots.

You will encounter these in algebra classes, standardized tests, and real-world applications like physics and engineering. There are three main ways to solve them. Each has strengths and weaknesses. Here they are.

Method 1: Factoring

Factoring is the fastest method when it works. The problem is it only works when the numbers cooperate.

When to Use Factoring

• The equation has small, integer coefficients
• You can find two numbers that multiply to give c and add to give b
• The equation is already set equal to zero

Step-by-Step Process

Example: x² + 5x + 6 = 0

Step 1: Identify a, b, and c. Here a = 1, b = 5, c = 6.

Step 2: Find two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3.

Step 3: Rewrite the middle term using those numbers: x² + 2x + 3x + 6 = 0.

Step 4: Factor by grouping. (x² + 2x) + (3x + 6) = 0 becomes x(x + 2) + 3(x + 2) = 0.

Step 5: Factor out the common binomial: (x + 2)(x + 3) = 0.

Step 6: Set each factor to zero: x + 2 = 0 or x + 3 = 0.

Solutions: x = -2 and x = -3

When Factoring Fails

Not every quadratic factors neatly. If you cannot find integer factors, move to the quadratic formula. Do not waste time forcing a factorization that does not exist.

Method 2: Quadratic Formula

The quadratic formula works for every quadratic equation. No exceptions. This is your backup plan and often your first plan.

The Formula

x = (-b ± √(b² - 4ac)) / 2a

The expression under the square root, b² - 4ac, is called the discriminant. It tells you what kind of solutions to expect.

How the Discriminant Works

Step-by-Step Process

Example: 2x² + 7x + 3 = 0

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

Step 2: Plug into the formula: x = (-7 ± √(7² - 4(2)(3))) / 2(2).

Step 3: Calculate inside the square root: 49 - 24 = 25. √25 = 5.

Step 4: Solve both versions: x = (-7 + 5) / 4 = -2/4 = -1/2. x = (-7 - 5) / 4 = -12/4 = -3.

Solutions: x = -1/2 and x = -3

Why This Method Wins

You can solve any quadratic with this formula. It takes more steps than factoring, but it never leaves you stuck. Memorize it. It will save you.

Method 3: Completing the Square

Completing the square converts the equation into a form you can solve by taking square roots. It is the method behind the quadratic formula itself.

When to Use It

• When the coefficient of x² is 1 (or can be made 1)
• When you need to convert a quadratic to vertex form
• When working with conic sections or graphing

Step-by-Step Process

Example: x² + 6x + 5 = 0

Step 1: Move the constant to the other side: x² + 6x = -5.

Step 2: Take half of the coefficient of x, square it, and add to both sides. Half of 6 is 3. 3² = 9. Add 9 to both sides: x² + 6x + 9 = -5 + 9.

Step 3: The left side is now a perfect square: (x + 3)² = 4.

Step 4: Take the square root of both sides: x + 3 = ±2.

Step 5: Solve for x: x = -3 + 2 = -1 or x = -3 - 2 = -5.

Solutions: x = -1 and x = -5

What Happens with a ≠ 1

Factor out the coefficient of x² first. For example, with 2x² + 8x + 6 = 0, divide everything by 2 to get x² + 4x + 3 = 0, then proceed with the steps above.

Which Method Should You Use?

Use factoring when you can spot the numbers quickly. Use the quadratic formula for everything else. Learn completing the square because it shows up in higher math and gives you a deeper understanding of how quadratics work.

Common Mistakes to Avoid

• Forgetting to set the equation to zero before factoring
• Losing the negative sign when moving terms across the equals sign
• Messy arithmetic with the quadratic formula, especially under the square root
• Forgetting the ± when taking square roots
• Dividing incorrectly when completing the square with a coefficient other than 1

Getting Started: Quick Practice Routine

1. Start with 10 problems that factor cleanly. Do them by hand, no calculator.
2. Solve the same 10 using the quadratic formula. Compare your answers.
3. Pick 5 problems that do not factor nicely and solve only with the formula.
4. Master completing the square with 5 problems where a = 1.
5. Mix all three methods on a timed quiz until you can switch between them without hesitation.

That is it. These three methods cover every quadratic you will encounter. Pick the right tool for the job and move on.