📐 Quadratic Equation Calculator

Solving Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, written in standard form as ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. Quadratic equations describe parabolas when graphed and arise naturally in physics (projectile motion), engineering (structural optimization), economics (profit maximization), and statistics (least squares fitting). The fundamental goal is finding the values of x that make the equation true — called roots, zeros, or solutions.

The Quadratic Formula

The quadratic formula solves any quadratic equation directly from its coefficients:

x = (−b ± √(b² − 4ac)) / 2a

The "±" produces two possible solutions. When you add the square root, you get one root (x₁); when you subtract it, you get the other (x₂). Both solutions must be verified by substituting back into the original equation.

Derivation: The formula is derived by completing the square on the general form ax² + bx + c = 0. It is guaranteed to work for any values of a, b, and c (where a ≠ 0), making it the most reliable all-purpose solution method.

Understanding the Discriminant (b² − 4ac)

The expression under the square root, b² − 4ac, is called the discriminant (often written as Δ or D). It tells you the nature of the roots before you even calculate them:

• Δ > 0 (positive): Two distinct real roots. The parabola y = ax² + bx + c crosses the x-axis at two different points. Example: x² − 5x + 6 = 0 has Δ = 25 − 24 = 1 > 0, giving roots x = 3 and x = 2.
• Δ = 0: One repeated (double) root: x = −b/2a. The parabola touches the x-axis at exactly one point (its vertex). Example: x² − 4x + 4 = 0 has Δ = 16 − 16 = 0, giving root x = 2 (double).
• Δ < 0 (negative): Two complex conjugate roots of the form p ± qi, where p = −b/2a and q = √(−Δ)/2a. The parabola does not intersect the x-axis at any real point. Complex roots occur in pairs — if (p + qi) is a root, so is (p − qi).

Alternative Solution Methods

• Factoring: The fastest method when it works. Find two numbers that multiply to ac and add to b. Example: x² − 5x + 6 = 0 → (x−2)(x−3) = 0 → x = 2 or x = 3. Only works when the equation factors neatly over integers or rationals.
• Completing the Square: Rewrite the equation as a perfect square trinomial. Useful for deriving the vertex form of a parabola: y = a(x − h)² + k, where (h, k) is the vertex.
• Square Root Method: Works when b = 0 (i.e., ax² + c = 0). Rearrange to x² = −c/a and take the square root: x = ±√(−c/a).
• Graphing: Graph y = ax² + bx + c and identify x-intercepts. Good for visual intuition but not precise without technology.

The Vertex of the Parabola

Every quadratic function y = ax² + bx + c graphs as a parabola. The vertex is the maximum point (if a < 0, parabola opens down) or minimum point (if a > 0, parabola opens up). The x-coordinate of the vertex is h = −b/2a, and the y-coordinate is k = c − b²/4a. The vertex is important in optimization problems where you need to find the maximum or minimum value of a quadratic expression.

Real-World Applications

• Projectile motion: The height of a thrown object follows h(t) = −16t² + v₀t + h₀. Setting h(t) = 0 and solving finds when the object hits the ground.
• Area and fencing problems: "Maximize the area of a rectangular enclosure given a fixed perimeter" leads directly to a quadratic optimization.
• Break-even analysis: Profit = Revenue − Cost often produces quadratic equations when revenue or cost depends on quantity nonlinearly.
• Optics and lenses: The lens equation and mirror equations in physics produce quadratics when finding image distances.

Frequently Asked Questions

What if a = 0? If a = 0, the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula is undefined at a = 0 (division by zero). The solution is simply x = −c/b.

What do complex roots mean physically? In many physical problems (like projectile motion), complex roots mean the equation has no real solution — for example, a projectile that never reaches a certain height. In electrical engineering and signal processing, complex roots are meaningful and describe oscillatory behavior.

How do I verify my solution? Substitute each root back into the original equation. If the left side equals zero, the root is correct. For the equation x² − 5x + 6 = 0, testing x = 3: 9 − 15 + 6 = 0 ✓. Testing x = 2: 4 − 10 + 6 = 0 ✓.