How to use Quadratic Equation Solver
The quadratic equation solver solves any equation of the form ax² + bx + c = 0 using the quadratic formula. Enter the coefficients a, b, and c and it returns the discriminant, tells you whether the roots are two real, one repeated, or two complex values, and displays each root — including complex conjugates written in a + bi form. It is an instant, private way to check algebra homework or engineering calculations.
- Enter the coefficient a (it must not be zero).
- Enter b and c — leave a field blank to treat it as zero.
- Press Solve to compute the discriminant and roots.
- Read the root type and each root, then copy the result if needed.
Your data never leaves your device — 100% private processing.
Reading the discriminant
The discriminant, b²−4ac, is the part of the quadratic formula under the square root, and it controls everything about the roots. When it is positive the square root is a real number, giving two different real solutions where the parabola crosses the x-axis. When it is exactly zero the parabola just touches the axis, giving a single repeated root. When it is negative the square root involves √(−1) = i, producing a pair of complex conjugate roots that share the same real part but have opposite imaginary parts. Checking the discriminant first tells you what kind of answer to expect.
| Discriminant | Roots | Graph |
|---|---|---|
| > 0 | Two distinct real | Crosses x-axis twice |
| = 0 | One repeated real | Touches x-axis once |
| < 0 | Two complex conjugates | Never touches x-axis |
Complex roots explained
When the discriminant is negative there is no real number whose square is negative, so the roots use the imaginary unit i, defined by i² = −1. The two roots come out as a + bi and a − bi, a conjugate pair, where the shared real part is −b ÷ 2a and the imaginary part is √(−discriminant) ÷ 2a. These complex roots are perfectly valid solutions — they simply mean the parabola never crosses the real x-axis. Complex roots appear constantly in electrical engineering, signal processing, and control theory, so the solver formats them clearly rather than hiding them.
Worked examples
Two real roots
Inputs: a=1, b=−5, c=6
Result: x = 2 and x = 3
One repeated root
Inputs: a=1, b=−4, c=4
Result: x = 2
Complex roots
Inputs: a=1, b=0, c=1
Result: x = 0 + 1i and 0 − 1i
Glossary
- Quadratic equation
- A polynomial equation of degree two, written ax² + bx + c = 0 with a ≠ 0.
- Discriminant
- The expression b²−4ac that determines the number and type of roots.
- Root
- A value of x that makes the equation equal to zero; also called a solution or zero.
- Complex conjugate
- A pair of complex numbers a + bi and a − bi with the same real part and opposite imaginary parts.
Related reading
Frequently Asked Questions
Why use Quadratic Equation Solver?
- Solves any quadratic using the quadratic formula with exact arithmetic
- Reports the discriminant and classifies the root type automatically
- Handles complex roots and displays them cleanly in a + bi form
- Validates that a is non-zero so you never get a misleading result
Common use cases
- Check the roots of a quadratic for an algebra or precalculus assignment
- Find where a projectile’s height equation crosses zero in a physics problem
- Solve for break-even points in a quadratic cost or revenue model
- Verify factoring by confirming the roots match your hand calculation
Get weekly tool tips & updates
New tools, power-user tips, and productivity hacks — delivered free every Friday.
No spam, ever. Unsubscribe with one click.
Related Calculators
Scientific Calculator
Free online scientific calculator with trigonometric functions, logarithms, powers, and constants. Supports keyboard input and calculation history.
Scientific Notation Converter
Convert numbers between standard decimal form and scientific notation in both directions, with full expanded digits for very large or tiny values. Free and private.
Percentage Calculator
Calculate percentages instantly. Find what X% of a number is, what percentage X is of Y, or calculate percentage change.
Prime Number Checker
Check whether any whole number is prime, see its prime factorization if it is not, and list every prime up to it. Free and instant in your browser.
BMI Calculator
Calculate your Body Mass Index (BMI) instantly. Supports metric (kg/cm) and imperial (lbs/ft+in) units. Free online BMI calculator.
Age Calculator
Calculate your exact age in years, months, and days. Find your next birthday countdown and day of birth. Free online age calculator.
Explore all Calculators.