About this calculator
A quadratic equation is a quadratic polynomial equation of the form ax² + bx + c = 0 (where a ≠ 0). The root formula x = (−b ± √(b²−4ac)) / (2a) always gives a solution.
The discriminant (Δ = b²−4ac) determines the properties of the roots: when Δ > 0 there are two distinct real roots; when Δ = 0 there is one repeated real root; when Δ < 0 the root is a complex (imaginary) number.
Quadratic equations appear in projectile motion, profit maximization, bridge engineering, lens optics, and numerous areas of physics and economics. Our solver shows you each step to help you understand and master the solution.
What it calculates
The quadratic equation calculator solves ax^2 + bx + c = 0 for real or complex roots and identifies how many solutions exist.
Formula
x = (-b ± sqrt(b^2 - 4ac)) / (2a). The discriminant D = b^2 - 4ac determines the root type.
Inputs
- Quadratic coefficient a, where a cannot be 0.
- Linear coefficient b.
- Constant term c.
Example
| Equation | Discriminant | Result |
|---|---|---|
| x^2 - 5x + 6 = 0 | 1 | x = 2, 3 |
| x^2 + 2x + 1 = 0 | 0 | x = -1 |
| x^2 + 1 = 0 | -4 | x = ±i |
How to interpret the result
D > 0 gives two distinct real roots, D = 0 gives one repeated root, and D < 0 gives a complex conjugate pair.
Common mistakes
- a cannot be 0.
- Watch the -b and 2a parts of the formula.
- A negative discriminant has no real roots.
How to use
Using the quadratic equation calculator is very easy. First, organize the equation into the standard form ax²+bx+c=0, and determine the values of the coefficients a, b, and c. Note that a cannot be 0 (otherwise it is not a quadratic equation). Then, enter the values of a, b, and c in the corresponding input boxes, which can be positive, negative, or zero.
For example, to solve the equation x²-5x+6=0, enter a=1, b=-5, and c=6. After clicking "Solve", the system will display: The discriminant Δ=(-5)²-4×1×6=25-24=1>0, the equation has two unequal real roots. x₁=[5+√1]/(2×1)=3, x₂=[5-√1]/(2×1)=2. Verification: 3²-5×3+6=0, 2²-5×2+6=0, correct.
If the equation has complex roots, such as x²+2x+5=0, enter a=1, b=2, c=5. The discriminant Δ=4-20=-16<0, the equation has two conjugate complex roots: x₁=(-2+4i)/2=-1+2i, x₂=(-2-4i)/2=-1-2i. The calculator also displays an image of a parabola to help you understand the geometric significance of the roots.
Main features
This quadratic equation calculator has comprehensive and professional functions. Standard root-finding formulas are used to ensure accurate calculation results. Supports the calculation of real roots and complex roots, and automatically determines the type of root (two unequal real roots, multiple roots, and conjugate complex roots). Provide calculation and analysis of the discriminant Δ to help understand the properties of roots.
Detailed solution steps are shown, including discriminant calculation, root formula substitution, and simplification process. It is suitable for learning and use. Supports multiple solution methods: root formula method, combination method, factorization method (when decomposable). Provides root verification function, substituting the obtained roots into the original equation for verification.
Draw the image of the parabola y=ax²+bx+c, mark the vertex, axis of symmetry, and intersection with the coordinate axis to visually demonstrate the geometric significance of the roots of the equation. Supports equations whose coefficients are fractions, decimals, and negative numbers. The interface is clear, the input is simple, and the results are displayed in real time. Completely free and suitable for all devices.
Use cases
The Quadratic Equation Calculator is very useful in multiple scenarios. In mathematics learning, students use calculators to verify homework answers and check whether manual calculations are correct. Understand the application of the root formula and the meaning of the discriminant by reviewing the detailed steps. In mathematics for high school entrance examination and college entrance examination, quadratic equations are a compulsory subject.
In physics, many problems involve quadratic equations. For example, the trajectory equation of projectile motion is h=-gt²/2+v₀t+h₀. To find the landing time of the object, you need to solve the quadratic equation. The displacement formula s=v₀t+at²/2 for uniform linear motion is also a quadratic equation. Quadratic equations are also commonly used in circuit analysis, vibration problems, etc.
In engineering design, optimization problems are often transformed into quadratic equations. For example, find the optimal solution to maximize profits and minimize costs. Calculation of arch structures and parabolic shapes in architectural design. Quadratic equations are also used in supply and demand balance and cost-benefit analysis in economics.
In daily life, problems such as calculating area and distance may involve quadratic equations. For example, given the perimeter and area of a rectangle, find the length and width. Calculate investment returns, loan repayments and other financial issues. Calculation of parabolic trajectories, collision detection, etc. in game development.