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.

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.