Quadratic Calculator: Solve Quadratic Equations with Steps
Quadratic equations appear everywhere in mathematics, physics, engineering, and economics. The standard form ax² + bx + c = 0 can represent projectile motion, profit optimization, and area problems. Our Quadratic Calculator solves for x using the quadratic formula, completing the square, and factoring methods, showing step-by-step solutions and plotting the parabola. Use our Quadratic Formula Calculator to estimate your costs.
Understanding Quadratic Equations
A quadratic equation has the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola. If a > 0, the parabola opens upward. If a < 0, it opens downward. The solutions (roots) are where the parabola crosses the x-axis.
The quadratic formula x = [-b ± √(b² - 4ac)] / (2a) works for all quadratic equations. The discriminant b² - 4ac determines the nature of the roots: if positive, two real roots; if zero, one repeated root; if negative, two complex roots.
Using the Quadratic Calculator
Enter the coefficients a, b, and c. The calculator solves for x using the quadratic formula and shows the discriminant, the nature of the roots, and the step-by-step solution. It also shows the factored form (if factorable), the vertex coordinates, the axis of symmetry, and a graph of the parabola.
The calculator handles real and complex roots, displaying complex solutions in a + bi form. You can also enter a quadratic expression to be factored, or enter the roots to find the original equation.
Solution Methods
Quadratic Formula
The universal method: x = [-b ± √(b² - 4ac)] / (2a). Always works, regardless of whether the equation is factorable. Best for equations where factoring is not obvious.
Factoring
If the quadratic can be written as (x + p)(x + q) = 0, then x = -p and x = -q. Only works when the discriminant is a perfect square. Example: x² + 5x + 6 factors to (x + 2)(x + 3) = 0, giving roots x = -2 and x = -3.
Completing the Square
Rewrite ax² + bx + c = 0 as a(x + h)² + k = 0, then solve. This method reveals the vertex form and is the basis for deriving the quadratic formula. Useful for understanding the parabola properties.
Graphing
Find the x-intercepts of the parabola. The roots are where the curve crosses the x-axis. This visual method helps verify algebraic solutions.
Parabola Properties
- Vertex: The highest or lowest point of the parabola. Located at x = -b/(2a), y = f(-b/(2a)).
- Axis of symmetry: The vertical line x = -b/(2a) through the vertex.
- Y-intercept: Where x = 0, y = c.
- Direction: Opens upward if a > 0 (vertex is minimum), downward if a < 0 (vertex is maximum).
- Width: Smaller |a| values produce wider parabolas; larger |a| values produce narrower ones.
Real-World Example
A company's profit function is P(x) = -2x² + 80x - 600, where x is the number of units produced. Find the break-even points (where profit = 0):
- Equation: -2x² + 80x - 600 = 0
- Divide by -2: x² - 40x + 300 = 0
- Quadratic formula: x = [40 ± √(1600 - 1200)] / 2 = [40 ± √400] / 2 = [40 ± 20] / 2
- Roots: x = (40 - 20)/2 = 10, and x = (40 + 20)/2 = 30
- Interpretation: Profit is positive between 10 and 30 units. Maximum profit at x = -b/(2a) = -80/(2×-2) = 20 units, with P(20) = -2(400) + 80(20) - 600 = -800 + 1,600 - 600 = $200.
The discriminant b² - 4ac = 1600 - 1200 = 400 > 0, confirming two distinct real roots.
Start Calculating
Use our Quadratic Calculator below to solve any quadratic equation with step-by-step solutions. Also check our Graphing Calculator for visualizing functions and our Pythagorean Calculator for geometry problems.
Reading the Discriminant: Three Cases
Before solving, compute the discriminant b² - 4ac to know what to expect. Take x² + 5x + 6 = 0: here a = 1, b = 5, c = 6, so the discriminant is 25 - 24 = 1. Because 1 is a perfect square, the equation factors cleanly as (x + 2)(x + 3) = 0, giving the two real roots x = -2 and x = -3. Now try x² - 6x + 9 = 0: the discriminant is 36 - 36 = 0, so there is exactly one repeated root, x = 3, and the parabola only touches the x-axis at its vertex. Finally, consider x² + 2x + 5 = 0: the discriminant is 4 - 20 = -16, which is negative, so the parabola never crosses the x-axis and the solutions are the complex numbers x = -1 ± 2i. In the calculator, the discriminant line is the fastest way to sanity-check your result — a positive value means two separate x-intercepts, zero means a single tangent point, and a negative value means a pair of complex roots displayed in a + bi form. Checking this line before reading the final roots is a good habit for students, since it tells you in advance whether the answer should contain two numbers, one number, or a pair with i, and it catches entry mistakes in a, b, or c.
Related Calculators
Use our Quadratic Calculator Guide together with related tools such as Area Calculator, Combination Calculator, Derivative Calculator to plan more accurately. Each calculator runs instantly in your browser with step-by-step guidance.
Written by the CalcMaster Pro Editorial Team — financial, health, and DIY tools reviewed for accuracy. All calculators run on standard, widely accepted formulas. Always confirm final numbers with a qualified professional for decisions that require official figures.