How to Solve Using the Quadratic Formula

The quadratic formula solves any quadratic—even when factoring fails. Learn the exact steps, how to use the discriminant, and how to avoid sign mistakes.

📈 Algebra⏱️ ~16 min read📊 Intermediate

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The Quadratic Formula

Any quadratic equation can be rewritten in standard form: ax² + bx + c = 0. Once it's in that form, the solutions are:

x = (−b ± √(b² − 4ac)) / (2a)

This formula is powerful because it does not rely on guessing factors. It gives exact answers as fractions, radicals (square roots), or sometimes complex numbers.

For an additional worked-example style explanation, Math is Fun’s quadratic equation page is a quick read.

How to Use the Quadratic Formula (5 Steps)

Step 1: Move everything to one side

Your goal is ax² + bx + c = 0. If the equation is not equal to 0, subtract everything from one side.

Step 2: Identify a, b, and c

Carefully read off the coefficients. This is where most errors happen.

Quick reminders

• a is the number in front of x²
• b is the number in front of x
• c is the constant term
• If a term is missing, its coefficient is 0

Step 3: Compute the discriminant

The discriminant is D = b² − 4ac. It tells you how many real solutions exist.

D > 0

Two real solutions

D = 0

One real solution (double root)

D < 0

No real solutions (complex)

Step 4: Substitute into the formula

Plug a, b, c into x = (−b ± √(b² − 4ac)) / (2a). Use parentheses for b and for 2a to avoid sign mistakes.

Step 5: Simplify and check both answers

Simplify the radical when possible and reduce fractions. Then substitute both solutions back into the original equation to verify.

Worked Examples

Example 1: Two real solutions

Solve: x² - 5x + 6 = 0

a = 1, b = -5, c = 6

D = b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1

x = (−b ± √D)/(2a) = (5 ± 1)/2

x₁ = (5 + 1)/2 = 3

x₂ = (5 - 1)/2 = 2

Solutions: x = 2, x = 3

Example 2: Radical answers (doesn't factor nicely)

Solve: 2x² + 3x - 1 = 0

a = 2, b = 3, c = -1

D = 3² - 4(2)(-1) = 9 + 8 = 17

x = (−3 ± √17) / (4)

Solutions: x = (−3 + √17)/4 and x = (−3 − √17)/4

Example 3: One real solution (discriminant = 0)

Solve: x² + 6x + 9 = 0

a = 1, b = 6, c = 9

D = 6² - 4(1)(9) = 36 - 36 = 0

x = (−6 ± 0)/2 = -3

Solution: x = -3 (double root)

Example 4: No real solutions (discriminant < 0)

Solve: x² + 4x + 10 = 0

a = 1, b = 4, c = 10

D = 4² - 4(1)(10) = 16 - 40 = -24

D < 0 → no real solutions (solutions are complex)

Common Mistakes (and How to Avoid Them)

⚠️

Forgetting parentheses around b

If b = -5, then b² is (-5)² = 25 (not -25). Always use parentheses.

⚠️

Mixing up −b and b

The formula starts with −b. If b = 7, then −b = −7.

⚠️

Only computing one solution

The ± gives two answers unless the discriminant is 0.

When Should You Use the Quadratic Formula?

Use the quadratic formula when factoring is difficult or when you want a method that always works. It's especially useful for word problems, messy coefficients, and cases where you need exact answers.

If you want the full “choose the best method” overview, start here: how to solve a quadratic equation step by step.

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