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Try an Example
Pick a scenario to see how the calculator works, then adjust the values
Standard Form
Classic quadratic x² − 5x + 6 = 0 with two integer roots (x = 2 and x = 3)
Key values: a = 1 · b = −5 · c = 6
Physics Problem
Projectile height equation −4.9t² + 20t + 1.5 = 0 (when does the ball hit the ground?)
Key values: a = −4.9 · b = 20 · c = 1.5
Complex Roots
Equation x² + 2x + 5 = 0 with no real solutions (discriminant < 0)
Key values: a = 1 · b = 2 · c = 5
This calculator is also known as Quadratic Equation Calculator.
Read the complete guideUnderstanding Your Results
The calculator outputs everything you need to fully understand a quadratic equation: the roots tell you where the parabola crosses the x-axis, the vertex tells you the minimum or maximum point, and the different equation forms each highlight different properties.
Examples
Finding Garden Dimensions
A rectangular garden has length 3m more than its width, and an area of 108 m². Setting up the equation w² + 3w - 108 = 0 to find the width.
The discriminant is 441, a perfect square. The roots are w = 9 and w = -12. Since width cannot be negative, w = 9m and length = 12m.
Key takeaway: Real-world problems often yield two mathematical solutions, but physical constraints determine which solution is valid.
Making the Most of Your Results
Use the comprehensive output effectively:
- Compare standard, vertex, and factored forms to see the same equation from different perspectives
- Use the parabola graph to visualize where roots, vertex, and y-intercept appear
- Check Vieta's formulas (sum and product of roots) as a quick verification
- The vertex form is most useful for graphing and finding maximum/minimum values
- The factored form directly reveals the roots of the equation
Frequently Asked Questions about Quadratic Equation Calculator
What is a quadratic equation?
A quadratic equation is a polynomial equation of degree 2, written in standard form as ax² + bx + c = 0, where a, b, and c are constants and a is not zero. The graph of a quadratic function is a parabola.
What does the discriminant tell you?
The discriminant (b² - 4ac) reveals the nature of the roots: positive means two real roots, zero means one repeated root, and negative means two complex conjugate roots.
How many solutions can a quadratic equation have?
A quadratic equation always has exactly two solutions (by the Fundamental Theorem of Algebra), but they may be: two distinct real numbers, one repeated real number, or two complex conjugate numbers.
Specialized Calculators
Choose from 5 specialized versions of this calculator, each optimized for specific use cases and calculation methods.
Calculation Focus
2 CalculatorsSolving Method
2 CalculatorsRelated Calculators
6 CalculatorsMore Math calculators