Quadratic Formula Calculator
Solve quadratic equations using the quadratic formula.
Quadratic Equation: ax² + bx + c = 0
Coefficient of x² (cannot be 0)
Coefficient of x
Constant term
Equation:
1x² -5x + 6 = 0
Solutions
Discriminant (Δ)
1.0000
Solution Type
Two Real Solutions
x₁ (First Root)
3.0000
x₂ (Second Root)
2.0000
Vertex
(2.5000, -0.2500)
Minimum point
About This Calculator
The Quadratic Formula Calculator solves quadratic equations of the form ax² + bx + c = 0 using the quadratic formula. It finds all solutions (roots), calculates the discriminant to determine the nature of solutions, and finds the vertex of the parabola.
The quadratic formula is one of the most important formulas in algebra. It can solve any quadratic equation and tells you whether the solutions are real or complex based on the discriminant value.
Formula & Calculation Method
Quadratic Formula:
Standard Form: ax² + bx + c = 0
Quadratic Formula:
x = (-b ± √(b² - 4ac)) / (2a)
Discriminant: Δ = b² - 4ac
Solution Types:
• Δ > 0: Two real solutions
• Δ = 0: One real solution (repeated root)
• Δ < 0: Two complex solutions
Vertex:
x = -b/(2a), y = f(-b/(2a))
The discriminant (b² - 4ac) determines the nature of the roots. If positive, there are two distinct real roots. If zero, there's one repeated real root. If negative, there are two complex conjugate roots.
How to Use This Calculator
- Enter Coefficient a: Input the coefficient of x² (cannot be zero).
- Enter Coefficient b: Input the coefficient of x.
- Enter Coefficient c: Input the constant term.
- View Solutions: See the roots, discriminant, solution type, and vertex.
Frequently Asked Questions
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a), which solves any quadratic equation of the form ax² + bx + c = 0. It always gives the solutions (roots) of the equation.
What does the discriminant tell us?
The discriminant (b² - 4ac) determines the nature of the roots: positive means two real solutions, zero means one repeated real solution, and negative means two complex solutions.
Can a be zero?
No, if a = 0, the equation is linear (not quadratic) and should be solved using linear equation methods. The quadratic formula requires a ≠ 0.
What is the vertex of a parabola?
The vertex is the highest or lowest point of the parabola. For ax² + bx + c, the vertex is at x = -b/(2a). If a > 0, it's a minimum; if a < 0, it's a maximum.
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Calculator by
Prof. Emily Watson, PhD
PhD in Applied MathematicsStatistics Certification