Quadratic Equation Solver
The Quadratic Equation Solver helps you instantly find the roots of second-degree polynomials in the form ax² + bx + c = 0.
Whether you are checking homework or engineering a curve, this tool calculates the discriminant and provides accurate solutions for:
- Real and Distinct Roots
- Real and Equal Roots
- Complex (Imaginary) Roots
Why use this solver?
Manual calculation is prone to sign errors (especially with negative b or c values).
This calculator automates the Quadratic Formula to ensure precision without the tedious math.
The Quadratic Formula
To solve for x, we use the standard Quadratic Formula derived from completing the square:
The term inside the square root, b² - 4ac, is called the Discriminant. It tells us the "nature" of the roots (how many solutions exist and if they are real or complex).
How the Discriminant Determines Roots
The value of the discriminant (Δ) decides the outcome of the equation.
| Discriminant Value (Δ) | Nature of Roots | Example Graph Behavior |
|---|---|---|
| Positive (> 0) | 2 Real, Distinct Roots | Graph cuts the x-axis at two points. |
| Zero (= 0) | 1 Real, Repeated Root | Graph touches the x-axis at exactly one point (vertex). |
| Negative (< 0) | 2 Complex Roots (Imaginary) | Graph never touches the x-axis (floats above/below). |
How to Use This Solver Correctly
The calculator requires your equation to be in Standard Form before you extract the coefficients (a, b, and c).
⚠️ Important: Set Equation to Zero
If your equation looks like 2x² = 5x - 3, you cannot enter a=2, b=5, c=-3. The math will be wrong.
You must rearrange it first:
Move everything to the left side so the right side equals zero:
2x² - 5x + 3 = 0
- a = 2
- b = -5 (Don't forget the negative sign!)
- c = 3
Missing Terms?
- If
xis missing (e.g.,x² - 9 = 0), then b = 0. - If the constant is missing (e.g.,
x² + 4x = 0), then c = 0.
Common Quadratic Examples
Here is how the formula handles different types of equations.
Values: a=1, b=-5, c=6
Discriminant (Δ): b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1
Since Δ > 0, we have two distinct real roots.
Result: x = 3, x = 2
Values: a=1, b=-4, c=4
Discriminant (Δ): b² - 4ac = (-4)² - 4(1)(4) = 16 - 16 = 0
Since Δ = 0, the graph touches the x-axis at exactly one point.
Result: x = 2
Values: a=1, b=1, c=1
Discriminant (Δ): 1² - 4(1)(1) = 1 - 4 = -3
Since Δ < 0, there are no real solutions, only complex numbers involving i.
Result: x = -0.5 ± 0.866i
Where is this used in real life?
Quadratic equations aren't just for exams. They model any scenario involving gravity, area, or curves.
🚀 Projectile Motion
Calculating the trajectory of a ball, a rocket, or a water fountain jet.
The equation h(t) = -gt² + vt + h calculates when an object will hit the ground.
💰 Profit Optimization
Businesses use quadratic functions to determine the ideal price point to maximize revenue. The "vertex" of the parabola represents the maximum profit.
Limitations: What This Solver Can & Cannot Do
While this calculator is highly accurate for engineering and standard homework, math has nuances you should be aware of.
- Decimal vs. Radical Form: This tool outputs answers in decimal format (e.g.,
1.414). If your professor requires "Exact Form" (e.g.,√2) or fractions, you will need to convert the decimal manually. - Imaginary Notation: Complex roots are displayed using the standard
inotation (e.g.,2 + 3i).
Precision Note (Floating Point Math)
Like all digital calculators (including JavaScript engines), extremely large or small coefficients (e.g., 0.00000005x²) may encounter slight rounding errors due to floating-point arithmetic.
For standard textbook problems, this is negligible.
Frequently Asked Questions
Yes. You can enter negative numbers for a, b, or c.
Example: For x² - 5x - 6 = 0, enter -5 for b and -6 for c.
"NaN" stands for "Not a Number." This usually happens if you:
- Entered text instead of numbers.
- Left the a value as 0 (which makes it linear, not quadratic).
- Entered symbols like "√" or "/" directly into the input.
Currently, this tool focuses on finding the Roots (x-intercepts).
However, you can find the x-coordinate of the vertex manually using the formula x = -b / 2a.
The letter i represents an Imaginary Number. This happens when the discriminant is negative (the graph never touches the x-axis). It is a valid mathematical solution in the complex plane.
Disclaimer: The Quadratic Equation Solver on Countimator.com is provided for educational and verification purposes only. While we utilize standard double-precision floating-point arithmetic (IEEE 754) to ensure accuracy, extreme inputs (very large or very small coefficients) may be subject to minor rounding errors inherent to digital computing. This tool should not be used for mission-critical engineering calculations without secondary verification.