Share:
How It Works
The quadratic formula x = (−b ± √(b² − 4ac)) / 2a solves any equation of the form ax² + bx + c = 0. Enter the three coefficients and the calculator determines the discriminant, root type, and both solutions, including complex roots when the discriminant is negative.
Example Problem
Solve 2x² + 3x − 5 = 0:
- Discriminant = 3² − 4(2)(−5) = 9 + 40 = 49.
- x₁ = (−3 + 7) / 4 = 1
- x₂ = (−3 − 7) / 4 = −2.5
Frequently Asked Questions
What does the discriminant tell you about a quadratic equation?
The discriminant b² − 4ac reveals the root type. If it is positive, there are two real roots. If zero, one repeated root. If negative, two complex conjugate roots. For 49 the roots are real and distinct.
How to solve a quadratic equation without the formula?
You can factor the expression (if the roots are rational), complete the square, or graph the parabola. The quadratic formula works for all cases, including irrational and complex roots.
What are complex roots?
Complex roots occur when the discriminant is negative. They come in conjugate pairs like 2 + 3i and 2 − 3i. The parabola does not cross the x-axis in this case.
Why can't the coefficient a be zero?
If a = 0 the equation becomes linear (bx + c = 0), not quadratic. A quadratic must have an x² term, so a must be non-zero.
Related Calculators
- Line Equation Calculator — find slope, distance, and slope-intercept form.
- Logarithm Calculator — solve log equations for any base.
- Trigonometry Calculator — compute trig functions and inverses.
- Circle Equation Calculator — solve equations involving x² + y² = r².
- Natural Log Calculator — solve exponential and logarithmic equations.