Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 using the quadratic formula — with the discriminant, roots and full step-by-step working shown.
Quadratic formula calculator
Step by step calculations
Enter values for a, b and c and press Solve to see the step by step calculations.
Quadratic Formula Calculator Explained
Core concepts, classroom applications, and plain-English explanations for all audiences.
When you'd actually need this
A ball kicked upward follows h = −5t² + 20t + 1. Set h = 0 and the quadratic formula tells you exactly when it hits the ground. Fencing a rectangular garden where perimeter is fixed but area is maximised — that's a quadratic. A business's profit curve crosses zero at the break-even point — found by solving a quadratic. Any situation that rises then falls (or falls then rises) is likely quadratic.
Worked example
"Solve x² − 5x + 6 = 0."
a = 1, b = −5, c = 6
Discriminant: (−5)² − 4(1)(6) = 25 − 24 = 1
√1 = 1
x = (5 + 1)/2 = 3
x = (5 − 1)/2 = 2
Result: x = 2 or x = 3
Check: (x−2)(x−3) = x²−5x+6 ✓
Common mistake to avoid
- ✗ Forgetting the ±: the formula gives two roots — one using + and one using −. A single answer is only correct when the discriminant is zero (a repeated root). Always check both, then decide which makes physical sense for the context.
What is the Quadratic Formula?
A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually 'x') is 2. It takes the standard form: ax² + bx + c = 0, where 'a', 'b', and 'c' are coefficients, and 'a' cannot be zero.
The quadratic formula is a powerful tool used to find the values of 'x' (also known as the roots or solutions) that satisfy any quadratic equation. It's particularly useful when factorising the equation is difficult or impossible. The formula is: x = (-b ± √(b² - 4ac)) / 2a.
How to Use the Quadratic Formula Calculator
Identify a, b, and c
Make sure your quadratic equation is in the standard form ax² + bx + c = 0. Then, identify the values of the coefficients 'a', 'b', and 'c'. Remember to include their signs.
Input the Values
Enter the numerical values for 'a', 'b', and 'c' into the respective fields in the calculator. For example, if your equation is x² - 5x + 6 = 0, you would enter a=1, b=-5, and c=6.
Get Your Solutions
The calculator will instantly apply the quadratic formula, show you the discriminant, and provide the roots of the equation. It will also explain the nature of the roots (two real, one repeated, or no real roots).
Common mistakes to avoid
- x = -5 ± √(5² - 4(1)(6)) / 2(1) for x² - 5x + 6 = 0:A common mistake is forgetting the negative sign for 'b' when substituting into '-b'. The formula is -b, so if b is -5, then -b is -(-5) = 5. x = -(-5) ± √((-5)² - 4(1)(6)) / 2(1)
- x = -b ± √(b² - 4ac) / 2a (only the square root term is divided by 2a):The entire numerator, including the '-b' term, must be divided by '2a'. This is a common error in calculation. x = (-b ± √(b² - 4ac)) / 2a (the whole expression is divided by 2a)
- Calculating b² - 4ac as 5² - 4(1)(-6) = 25 - 24 = 1 for x² + 5x - 6 = 0:Careless calculation of the discriminant, especially with negative 'c' values. Here, -4(1)(-6) should be +24, not -24. b² - 4ac = 5² - 4(1)(-6) = 25 + 24 = 49
Questions students ask
1. What is the quadratic formula?
2. What is the discriminant and what does it tell us?
• If D > 0 (positive), there are two distinct real roots.
• If D = 0, there is exactly one real root (also called a repeated root).
• If D < 0 (negative), there are no real roots (instead, there are two complex conjugate roots).