Quadratic Formula Calculator
Solve quadratic equations using the quadratic formula. Find roots, vertex, and axis of symmetry.
A quadratic equation calculator finds the roots of ax² + bx + c = 0 using the quadratic formula, including complex roots.
Examples
Two real roots
Repeated root
Complex roots
Frequently Asked Questions
What does the discriminant tell us?
What is the vertex of a parabola?
What is the axis of symmetry?
Quick Tips
- •Double-check your inputs — small errors lead to incorrect results.
- •Check the discriminant first to know whether roots are real, repeated, or complex.
- •Use decimal form for fractions when entering values.
A quadratic equation calculator finds the roots of ax² + bx + c = 0 using the quadratic formula, including complex roots.
How to Use This Calculator
Enter the coefficients a, b, and c for the equation ax² + bx + c = 0. The calculator will find the roots using the quadratic formula, determine the discriminant, vertex, and axis of symmetry.
Understanding the Formula
x = (-b ± √(b² - 4ac)) / 2a. The discriminant Δ = b² - 4ac determines the nature of the roots.
Examples
Two real roots
x² - 5x + 6 = 0: Δ = 25-24 = 1 > 0, roots are x = 2 and x = 3
Repeated root
x² - 4x + 4 = 0: Δ = 16-16 = 0, root is x = 2 (repeated)
Complex roots
x² + x + 1 = 0: Δ = 1-4 = -3 < 0, roots are complex: -0.5 ± 0.866i
Frequently Asked Questions
What does the discriminant tell us?
If Δ > 0, there are two distinct real roots. If Δ = 0, there is one repeated root. If Δ < 0, there are two complex conjugate roots.
What is the vertex of a parabola?
The vertex is the highest or lowest point of the parabola y = ax² + bx + c. It occurs at x = -b/(2a) with y = a(-b/(2a))² + b(-b/(2a)) + c.
What is the axis of symmetry?
The axis of symmetry is a vertical line x = -b/(2a) that divides the parabola into two mirror images.
Assumptions & Limitations
- Assumes the coefficient a is not zero; otherwise the equation is linear.
- Assumes exact input values; rounding in inputs propagates to results.
- Results may show floating-point approximations for irrational numbers.