Quadratic Equation Calculator – Find Real and Complex Roots
Solve quadratic equations online using the quadratic formula. Enter coefficients a, b, and c to find the roots, discriminant, and solution type.
Use this quadratic equation calculator to solve equations in the form ax² + bx + c = 0. Enter the three coefficients to calculate the discriminant and find two distinct real roots, one repeated real root, or two complex roots.
How to Use the Quadratic Equation Calculator
Identify coefficients a, b, and c in your equation. Enter the coefficient of x² as a, the coefficient of x as b, and the constant term as c. If the x term or constant term is missing, enter 0 for its coefficient.
The Quadratic Formula
The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a). It solves every quadratic equation with real-number coefficients as long as a is not zero.
What Is the Discriminant?
The discriminant is b² - 4ac, the expression inside the square root of the quadratic formula. Its sign determines the number and type of roots.
Positive, Zero, and Negative Discriminants
A positive discriminant gives two distinct real roots. A discriminant of zero gives one repeated real root. A negative discriminant gives two nonreal complex conjugate roots.
Real and Complex Quadratic Roots
When the discriminant is negative, the square root contains a negative number. The calculator displays the resulting complex roots using i as the imaginary unit, where i² = -1.
Why Coefficient a Cannot Be Zero
A quadratic equation must contain an x² term. If a equals zero, that term disappears and the equation becomes linear or constant rather than quadratic.
Missing Terms in a Quadratic Equation
A missing term has a coefficient of zero. For example, x² - 9 = 0 has a = 1, b = 0, and c = -9. The equation 2x² + 3x = 0 has a = 2, b = 3, and c = 0.
Numerical Root Results
The calculator evaluates roots numerically and displays values with up to 10 decimal places. Irrational and complex roots may therefore appear as rounded approximations rather than exact radical expressions.
How Is the Result Calculated?
x = (-b ± √(b² - 4ac)) / (2a)
Quadratic Equation Calculator Example
For x² - 5x + 6 = 0, enter a = 1, b = -5, and c = 6. The discriminant is (-5)² - 4(1)(6) = 1, so the equation has two distinct real roots: x₁ = 3 and x₂ = 2.
Frequently Asked Questions
What is a quadratic equation?
A quadratic equation is an equation that can be written as ax² + bx + c = 0, where a, b, and c are constants and a is not zero.
How do I solve a quadratic equation?
Enter coefficients a, b, and c. The calculator evaluates the discriminant and applies the quadratic formula to find the roots.
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a). The plus and minus signs produce the two roots, which may be equal.
What is the discriminant?
The discriminant is b² - 4ac. It determines whether the equation has two distinct real roots, one repeated real root, or two complex roots.
What happens when the discriminant is positive?
A positive discriminant produces two distinct real roots.
What happens when the discriminant is zero?
Both quadratic-formula solutions are equal, so the equation has one repeated real root with multiplicity 2.
What happens when the discriminant is negative?
The equation has two nonreal complex conjugate roots, which the calculator displays using i.
Can this calculator find complex roots?
Yes. It calculates and displays complex roots when the discriminant is negative.
What should I enter if the x term is missing?
Enter 0 for coefficient b. For example, x² - 9 = 0 uses a = 1, b = 0, and c = -9.
What should I enter if the constant term is missing?
Enter 0 for coefficient c. For example, 2x² + 3x = 0 uses a = 2, b = 3, and c = 0.
Can coefficient a be zero?
No. If a is zero, the x² term disappears and the equation is not quadratic.
Are the calculated roots exact?
Integer and terminating-decimal roots may display exactly. Irrational and complex roots are numerical values rounded to a maximum of 10 decimal places.
Is the quadratic equation calculator free?
Yes. It is free to use and does not require an account.