Quadratic Formula Calculator
Solve ax² + bx + c = 0
ax² + bx + c = 0
Quadratic Formula
x = (−b ± √(b² − 4ac)) / (2a)
The quadratic formula solves equations of the form ax² + bx + c = 0, where a ≠ 0.
The ± symbol means there are typically two solutions: one using + and one using −.
Examples
Example 1: x² − 5x + 6 = 0
a = 1, b = −5, c = 6
Δ = (−5)² − 4(1)(6) = 25 − 24 = 1
x = (5 ± √1) / 2 = (5 ± 1) / 2
x₁ = 3, x₂ = 2
Example 2: x² − 4x + 4 = 0
a = 1, b = −4, c = 4
Δ = 16 − 16 = 0
x = 2 (one repeated solution)
Example 3: x² + x + 1 = 0
a = 1, b = 1, c = 1
Δ = 1 − 4 = −3
x = −0.5 ± 0.866i (complex solutions)
About Quadratic Equations
A quadratic equation is a polynomial equation of degree 2. The graph of a quadratic equation is a parabola. The quadratic formula provides a systematic method to find the roots (solutions) of any quadratic equation.
The Discriminant (Δ = b² − 4ac)
- Δ > 0: Two distinct real solutions
- Δ = 0: One repeated real solution (vertex touches x-axis)
- Δ < 0: Two complex conjugate solutions (parabola doesn't cross x-axis)
Real-World Applications
- Physics: Projectile motion, acceleration
- Engineering: Optimization problems
- Finance: Profit/loss models
- Architecture: Parabolic arches and structures
- Sports: Trajectory of balls
How to Use This Calculator
Enter Coefficient a
Type the coefficient of the x² term (a). This cannot be zero, as the equation must be quadratic.
Enter Coefficient b
Type the coefficient of the x term (b). This can be zero if there's no x term.
Enter Constant c
Type the constant term (c). This can be zero if there's no constant.
Click Solve Equation
Press "Solve Equation" to find the roots (solutions) of the quadratic equation.
View Solutions
See the solutions (roots), discriminant analysis, vertex, and factored form (if applicable).
About Quadratic Formula Calculator
The Quadratic Formula Calculator solves quadratic equations of the form ax² + bx + c = 0 using the quadratic formula. This tool provides not just the solutions, but also discriminant analysis, vertex coordinates, and helpful information about the nature of the solutions.
When to Use This Calculator
- Algebra Homework: Solve quadratic equations quickly and accurately
- Factoring Verification: Check if factored forms are correct
- Discriminant Analysis: Determine number and type of solutions
- Vertex Finding: Find maximum or minimum points of parabolas
- Complex Solutions: Handle equations with no real solutions
- Exam Preparation: Practice solving quadratic equations efficiently
Why Use Our Calculator?
- ✅ Shows Discriminant: Analyzes the nature of solutions
- ✅ Finds Vertex: Calculates parabola turning point
- ✅ Handles Complex: Works with complex number solutions
- ✅ Step-by-Step Display: Shows discriminant calculation
- ✅ 100% Accurate: Precise mathematical calculations
- ✅ Completely Free: No registration required
Understanding the Discriminant
The discriminant (Δ = b² - 4ac) tells you about the nature of the solutions:
- Δ > 0: Two distinct real solutions (parabola crosses x-axis twice)
- Δ = 0: One repeated real solution (parabola touches x-axis at vertex)
- Δ < 0: Two complex conjugate solutions (parabola doesn't cross x-axis)
Real-World Applications
Physics: Projectile motion equations are quadratic. The formula finds when objects hit the ground or reach maximum height.
Business: Profit functions are often quadratic. The vertex shows maximum profit.
Engineering: Many optimization problems involve quadratic functions solved with this formula.
Tips for Using This Calculator
- Coefficient a cannot be zero (then it's not quadratic)
- Pay attention to signs - negative coefficients affect results
- Check the discriminant to understand solution types
- The vertex shows the parabola's maximum or minimum point
- Complex solutions come in conjugate pairs (a + bi, a - bi)
- Use this calculator to verify hand-calculated solutions
Frequently Asked Questions
What if a = 0?
If a = 0, the equation is not quadratic—it's linear (bx + c = 0). The quadratic formula only works when a ≠ 0.
What are complex solutions?
When the discriminant is negative, solutions involve the imaginary unit i (where i² = −1). Complex solutions come in conjugate pairs: a + bi and a − bi.
Can I factor instead of using the formula?
Yes, if the quadratic factors nicely. But the quadratic formula works for ALL quadratic equations, even those that don't factor easily.
What is the vertex of a parabola?
The vertex is the turning point (minimum or maximum) of the parabola. Its x-coordinate is −b/(2a), and you find y by substituting this x back into the equation.
Why are there two solutions?
The ± in the formula gives two values. A parabola can cross the x-axis at two points, one point (vertex), or not at all (complex solutions).