How it works
The quadratic formula is a universal method to find the roots (solutions) of any quadratic equation. Instead of factoring or completing the square, you substitute the coefficients a, b, and c directly into the formula and evaluate.
The calculator performs three key steps:
- Calculates the discriminant (b² − 4ac) to determine the nature of the roots
- Applies the quadratic formula to find both solutions
- Displays results as real numbers or complex numbers, depending on the discriminant
The formula
x = (−b ± √(b² − 4ac)) / 2a
Worked example
Solve: 2x² + 7x + 3 = 0
First, identify the coefficients:
- a = 2
- b = 7
- c = 3
Step 1: Calculate the discriminant
Discriminant = b² − 4ac = 7² − 4(2)(3) = 49 − 24 = 25
Since 25 > 0, we expect two distinct real roots.
Step 2: Apply the quadratic formula
x = (−7 ± √25) / (2 × 2) = (−7 ± 5) / 4
Step 3: Solve for both roots
Using the + sign: x₁ = (−7 + 5) / 4 = −2 / 4 = −0.5
Using the − sign: x₂ = (−7 − 5) / 4 = −12 / 4 = −3
Verification:
- For x = −0.5: 2(−0.5)² + 7(−0.5) + 3 = 0.5 − 3.5 + 3 = 0 ✓
- For x = −3: 2(−3)² + 7(−3) + 3 = 18 − 21 + 3 = 0 ✓
Example with complex roots
Solve: x² + 2x + 5 = 0
Coefficients: a = 1, b = 2, c = 5
Discriminant = 2² − 4(1)(5) = 4 − 20 = −16
Since the discriminant is negative, roots are complex:
x = (−2 ± √(−16)) / 2 = (−2 ± 4i) / 2 = −1 + 2i and −1 − 2i
These are complex conjugates. Both are valid solutions, even though they're not real numbers.
Common mistakes
Sign errors with b: The formula uses −b, not b. If b is already negative, −b becomes positive. Double-check the sign before substituting.
Forgetting the ±: Many people calculate only one root. Always compute both the + and − versions unless the discriminant is zero (which gives a repeated root).
Arithmetic under the square root: The discriminant calculation involves squaring b and multiplying 4ac carefully. A small error here cascades into wrong roots.
Mishandling complex numbers: If you get a negative discriminant, don't assume "no solution." Complex roots are valid mathematical solutions. Express them as a + bi format.
Dividing by 2a, not 2: The entire numerator (−b ± √discriminant) divides by 2a, not just the square root. Use parentheses to keep track: (−b ± √...) / (2a).