Struggling with quadratic equations? Our free online quadratic solver instantly finds the roots of any equation using the quadratic formula. Whether you're doing algebra homework, studying for exams, or solving real-world problems, this tool provides accurate solutions with step-by-step explanations. See the discriminant, vertex, and complete working - all in one place!
A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where a ≠ 0. These equations are fundamental in algebra and appear throughout mathematics, science, and engineering. The solutions to quadratic equations (called roots) can be found using the quadratic formula, factoring, completing the square, or graphing. Quadratic equations are unique because they always have exactly two solutions (which may be real and distinct, real and repeated, or complex conjugates).
Our quadratic solver provides: Solve any quadratic equation instantly. Step-by-step quadratic formula application. Discriminant calculation and interpretation. Real and complex root handling. Vertex coordinates. Axis of symmetry. Parabola direction. Multiple solution methods. Copy-to-clipboard functionality. Mobile-friendly design. No registration required. Free unlimited calculations.
The solver applies the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. First, it calculates the discriminant D = b² - 4ac. If D > 0: Two distinct real roots using the ± formula. If D = 0: One repeated root x = -b/(2a). If D < 0: Two complex roots with imaginary components. The tool also finds the vertex at (-b/2a, f(-b/2a)) and determines if the parabola opens up or down.
Education - Algebra homework and exam prep. Physics - Projectile motion problems. Engineering - Optimization and design. Economics - Profit/cost analysis. Architecture - Parabolic structures. Research - Mathematical modeling. Tutoring - Demonstrating concepts. Self-study - Learning algebra.
Our solver offers: Accuracy - correct solutions every time. Speed - instant results. Education - learn the quadratic formula. Steps - see detailed working. Complex Numbers - handle all cases. Convenience - no installation needed. Multiple Features - vertex, discriminant, and more. Cost - completely free.
Students learning algebra. Teachers demonstrating concepts. Tutors helping students. Engineers solving problems. Physicists calculating trajectories. Economists optimizing models. Anyone working with quadratic equations. Parents helping with homework.
Enter coefficient a (must not be zero). Enter coefficient b. Enter coefficient c. Click Solve Equation. View the roots, discriminant, and steps. Copy results as needed.
Check Form - ensure equation equals zero. Verify Coefficients - double-check signs. Understand Discriminant - interpret the results. Check Solutions - substitute back to verify. Learn Steps - understand the process.
Requires equation in standard form. Very large coefficients may affect precision. Complex roots shown in a + bi format. Assumes real coefficients.
A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The standard form is ax² + bx + c = 0, where a ≠ 0. The graph of a quadratic equation is a parabola. Quadratic equations appear in physics (projectile motion), engineering (optimization), economics (profit maximization), and many other real-world applications.
The quadratic formula is: x = (-b ± √(b² - 4ac)) / 2a. This formula gives the solutions (roots) to any quadratic equation in standard form. The term under the square root, b² - 4ac, is called the discriminant. The ± symbol means there are two solutions: one with addition and one with subtraction. This formula was known to ancient Babylonians and was rigorously proven by modern algebra.
The discriminant is D = b² - 4ac. It determines the nature of the roots: If D > 0: Two distinct real roots (parabola crosses x-axis twice). If D = 0: One repeated real root (parabola touches x-axis). If D < 0: Two complex conjugate roots (parabola doesn't cross x-axis). The discriminant also tells us if the roots are rational (perfect square discriminant) or irrational.
To solve by factoring: Write the equation in standard form ax² + bx + c = 0. Factor the quadratic expression into two binomials: (px + q)(rx + s) = 0. Set each factor equal to zero: px + q = 0 or rx + s = 0. Solve each linear equation for x. Example: x² - 5x + 6 = 0 factors to (x - 2)(x - 3) = 0, so x = 2 or x = 3.
Five main methods: 1) Quadratic Formula - works for all equations, 2) Factoring - when the quadratic factors nicely, 3) Completing the Square - rewrite in vertex form, 4) Graphing - find x-intercepts visually, 5) Square Root Property - when b = 0. Our solver uses the quadratic formula as it's universal and always works, showing detailed steps.
When the discriminant is negative (b² - 4ac < 0), the quadratic has no real solutions. Instead, it has two complex conjugate roots involving the imaginary unit i, where i² = -1. For example, if the formula gives √(-9), we write it as 3i. Complex roots always come in conjugate pairs: if a + bi is a root, then a - bi is also a root. These are important in electrical engineering and quantum physics.
Quadratic equations model many real situations: Physics - projectile motion (ball trajectory), Engineering - parabolic reflectors and bridges, Economics - profit maximization and supply/demand, Architecture - arch designs, Astronomy - planetary orbits (approximation), Sports - basketball free throw arcs, and Optimization - finding maximum or minimum values.
The vertex is the turning point of the parabola. For y = ax² + bx + c, the vertex x-coordinate is at x = -b/(2a). The y-coordinate is found by substituting this x back into the equation. The vertex represents: The maximum point if a < 0 (parabola opens downward), or the minimum point if a > 0 (parabola opens upward). This is crucial for optimization problems.
See also: Percentage Calculator, Average Calculator, Fibonacci Calculator.