Factoring is one of the fastest ways to solve a quadratic equation, and it works whenever the equation can be split into two simple factors. Instead of memorizing a formula, you reverse the multiplication you already know. You can check your work with the free tools on Calc Solver.
What Factoring Means
A quadratic equation in standard form looks like ax^2 + bx + c = 0. Factoring means rewriting it as two brackets multiplied together, like (x + 2)(x + 3) = 0. When two things multiply to zero, at least one of them must be zero, and that fact gives you the answers.
Step 1: Write the Equation in Standard Form
Move every term to one side so the other side is zero. For example, x^2 + 5x = -6 becomes x^2 + 5x + 6 = 0. If the x^2 coefficient is not 1, factor it out first or keep it in mind for the next step.
Step 2: Find Two Numbers That Fit
Look for two numbers that multiply to c (the constant term) and add to b (the x coefficient). For x^2 + 5x + 6 = 0, you need two numbers that multiply to 6 and add to 5. The numbers 2 and 3 fit, since 2 times 3 is 6 and 2 plus 3 is 5.
Step 3: Write the Factored Form
Use those two numbers to write the brackets: (x + 2)(x + 3) = 0. Multiply them back out to check: x^2 + 3x + 2x + 6 = x^2 + 5x + 6. The check matches, so the factoring is correct.
Step 4: Use the Zero Product Property
Set each bracket equal to zero and solve: x + 2 = 0 gives x = -2, and x + 3 = 0 gives x = -3. So the solutions are x = -2 and x = -3. Put each value back into the original equation to confirm it works.
When Factoring Does Not Work
Not every quadratic factors neatly with whole numbers. If you cannot find two numbers that fit after a careful try, use the quadratic formula instead. Factoring is a shortcut, not a rule, and knowing when to switch methods is part of the skill.
Practice Problems to Try
Solve x^2 + 7x + 12 = 0, then x^2 – 9x + 20 = 0, then x^2 – 4 = 0. For each one, find the two numbers, write the brackets, and check both answers in the original equation.