Solve quadratic equations in the form ax² + bx + c = 0. This calculator provides both real and complex solutions.
The Quadratic Equation Solver is a powerful online tool designed to quickly and accurately solve any quadratic equation in the form ax² + bx + c = 0. Whether you're a student, engineer, or anyone dealing with algebra, this solver provides step-by-step results including real and complex roots.
A quadratic equation is a second-degree polynomial equation involving the variable x. It is written in the standard form:
ax² + bx + c = 0
The most common method for solving a quadratic equation is the quadratic formula:
x = [-b ± √(b² - 4ac)] / (2a)
This formula gives the roots (solutions) of the quadratic equation. The part under the square root, b² - 4ac, is called the discriminant.
Equation: x² - 5x + 6 = 0
Equation: x² - 4x + 4 = 0
Equation: x² + 2x + 5 = 0
The discriminant (D = b² - 4ac) helps determine the nature of the roots:
The graph of a quadratic equation is a parabola. It opens upward if a > 0 and downward if a < 0. The x-intercepts represent the real roots, and the vertex is the minimum or maximum point.
Vertex (h, k):
h = -b / 2a
k = f(h)
The vertex provides useful insight into optimization problems.
If the quadratic equation can be factored easily, you can solve it faster without using the formula. For example:
x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0 → x = -2, -3
The solver supports decimal and fractional inputs for a, b, and c. Make sure to enter values carefully, especially when using parentheses or negative signs.
Yes, if the discriminant is negative, the roots will be complex numbers (i.e., involve the imaginary unit i).
Yes, you can enter any real number including decimals or fractions.
Absolutely. The tool is mobile-optimized and works seamlessly across devices.
Yes, this calculator is designed to help with academic algebra, homework, and competitive exams.
import cmath
def solve_quadratic(a, b, c):
d = cmath.sqrt(b**2 - 4*a*c)
x1 = (-b + d)/(2*a)
x2 = (-b - d)/(2*a)
return x1, x2
Free Quadratic Equation Solver to find roots of any equation in the form ax² + bx + c = 0. Supports real and complex solutions. Fast, accurate, and easy to use.
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The Quadratic Equation Solver offers a fast, convenient, and reliable way to solve quadratic equations. With support for both real and imaginary solutions, it’s ideal for algebra students, teachers, and professionals alike. Whether you're solving problems by hand or checking answers, this calculator ensures accuracy every time.
Quadratic equations aren't just confined to the classroom—they’re everywhere in real life. Understanding how to apply them gives you a strong edge in various careers and scenarios.
In physics, quadratic equations describe the path of any object thrown in the air, forming a parabolic trajectory. For example, the equation h(t) = -16t² + vt + s models vertical motion where:
Solving this quadratic helps find when an object hits the ground or reaches maximum height.
Businesses often use quadratic equations in profit modeling. For instance, if revenue and costs are expressed as functions of output, the break-even point can be found by solving a quadratic:
Profit = Revenue - Cost = ax² + bx + c
Engineers use quadratic functions to calculate load distributions, beam strengths, and cable tensions. Architects use parabolic arches in bridges and buildings, which require quadratic solutions for structural balance.
Quadratic Bezier curves are used in vector-based graphic software and font design. Solving quadratic equations allows precise drawing and animation.
Farmers may use quadratic models to maximize crop yield based on area and cost constraints. Solving for the vertex helps determine the best input values.
An equation is quadratic if:
Example: 2x² - 3x + 7 = 0 → quadratic
x - 4 = 0 → not quadratic (linear)
If you plot y = ax² + bx + c on a coordinate system, the x-values where the graph intersects the x-axis are the roots of the equation. These points are the solutions to ax² + bx + c = 0.
Besides the standard form, quadratics can be expressed in other forms:
y = a(x - h)² + k
Easier to find the vertex (h, k) of the parabola
y = a(x - r₁)(x - r₂)
Makes it easy to identify roots r₁ and r₂
If you need to solve a quadratic without a calculator, completing the square is a helpful technique:
Example: x² + 6x + 5 = 0
Pair this solver with a graphing calculator to visualize how coefficients affect the shape and roots of the parabola. Change values of a, b, and c to see shifts, stretches, and flips in real-time.
1. Area Problem: A rectangle has an area of 60 m². Its length is (x + 5) and width is (x - 3). Find x.
Solution: (x + 5)(x - 3) = 60 → x² + 2x - 15 = 60 → x² + 2x - 75 = 0
Use the calculator to solve x² + 2x - 75 = 0
2. Falling Object: A ball is thrown upward with a velocity of 20 m/s. Its height in meters is modeled by h(t) = -5t² + 20t. When does the ball reach the ground?
Solve: -5t² + 20t = 0 → t(-5t + 20) = 0 → t = 0 or 4 seconds
Whether you're analyzing motion, modeling profits, or solving complex math problems, the Quadratic Equation Solver provides everything you need to solve with confidence. It’s fast, accurate, and perfect for algebra students and professionals alike. Try different values to explore how quadratics behave and enhance your problem-solving skills today.