About this Quadratic Solver
A quadratic equation is a second-degree polynomial of the form ax² + bx + c = 0. These equations appear frequently in physics (projectile motion), engineering, and geometry. This solver finds the values of 'x' (roots) where the equation equals zero.
Who this calculator is for
- Students solving quadratic equations in algebra or pre-calculus.
- Tutors and teachers checking discriminants and root types quickly.
- Anyone graphing parabolas or solving for x-intercepts.
When to use it
- Use this calculator when you have a quadratic in standard form and need the roots.
- Use it to see whether the equation has two real roots, one repeated root, or no real roots.
- Use it to check homework, graphing, or word-problem equations that reduce to ax² + bx + c = 0.
How to use this calculator
- Ensure your equation is in standard form: ax² + bx + c = 0.
- Enter the coefficient a (must be non-zero).
- Enter the coefficient b.
- Enter the constant c.
- The tool calculates the two roots ($x_1, x_2$) and the discriminant.
Key assumptions
- The equation is already rearranged into standard form ax² + bx + c = 0.
- The coefficient a is not zero, so the equation is truly quadratic.
- This calculator returns real roots only; negative discriminants are reported as NaN.
Accuracy and testing
Review formula sources, default test cases, assumptions, unit handling, rounding rules, and changelog for this calculator.
How this calculator works
It uses the famous Quadratic Formula:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
The term inside the square root ($b^2 - 4ac$) is called the Discriminant ($\\Delta$).
Inputs explained
a
The coefficient of $x^2$. Controls the width and direction of the parabola.
b
The coefficient of $x$. Shifts the parabola axis.
c
The constant term. The y-intercept of the parabola.
Understanding your results
The behavior depends on the Discriminant:
* Positive (>0): Two distinct real roots. The parabola crosses the x-axis twice.
* Zero (=0): One real root. The vertex touches the x-axis.
* Negative (<0): No real roots (Complex/Imaginary). The parabola never touches the x-axis.
Example scenarios
Two Real Roots
$x^2 - 3x + 2 = 0$. Roots: 1, 2.
One Real Root
$x^2 + 2x + 1 = 0$. Root: -1.
Complex Roots
$x^2 + 1 = 0$. Roots: NaN (Imaginary).
Key Features
- Quadratic Formula
- Discriminant Calculation
- Real Root Finding
Tips & Best Practices
- If 'a' is negative, the parabola opens downward (like a frown). If positive, it opens upward (like a smile).
- This tool currently calculates Real roots only. If the result is NaN, the roots are imaginary.
Glossary of Terms
- Root
- A value of x that makes the equation equal to zero. Also known as a zero or x-intercept.
- Discriminant
- The value $b^2 - 4ac$. It determines the nature of the roots.
- Vertex
- The peak or valley of the parabola.