What is the solution of a quadratic equation?

The solution of a quadratic equation ax² + bx + c = 0 is found using the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). The values of x that satisfy the equation are called roots or zeros of the quadratic equation.

How many solutions does a quadratic equation have?

A quadratic equation can have 0, 1, or 2 real solutions depending on the discriminant (D = b² - 4ac). If D > 0: two distinct real roots. If D = 0: one repeated real root. If D < 0: two complex (imaginary) roots.

What is the discriminant in a quadratic equation?

The discriminant is the expression b² - 4ac inside the square root of the quadratic formula. It determines the nature of the roots without fully solving the equation.

Math · Algebra

Solution of Quadratic Equation

Find all roots of ax² + bx + c = 0 using the quadratic formula — with step-by-step working, discriminant analysis, and a live parabola graph.

Current Equation

x² + 5x + 6 = 0

Enter Coefficients

a (x²)

≠ 0

b (x)

c (const)

Quick Examples

Solution Results

Discriminant D = b² − 4ac

D<0 (Complex) D=0 (Repeated) D>0 (Real)

The Quadratic Formula

x = (-b ± √(b²−4ac)) / 2a

The quadratic formula is the universal method for finding the solution of a quadratic equation. Unlike factoring, it works for every quadratic — even when roots are irrational or complex.

Discriminant (D = b²−4ac)

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D > 0 — Two Real Distinct Roots

Parabola crosses x-axis at two distinct points

D = 0 — One Repeated Real Root

Parabola touches x-axis at exactly one point (vertex)

D < 0 — Two Complex Roots

Parabola does not cross x-axis — roots involve √−1

How to Find the Solution of a Quadratic Equation

Identify Coefficients

Write the equation in standard form ax²+bx+c=0 and identify a, b, and c.

Calculate Discriminant

Compute D = b² − 4ac. This determines how many and what kind of solutions exist.

Apply the Formula

Substitute into x = (−b ± √D) / 2a to find both roots simultaneously.

Verify Your Answer

Substitute x back into the original equation. If it equals 0, your solution is correct!

Frequently Asked Questions

The solution of a quadratic equation ax² + bx + c = 0 consists of the values of x (called roots) that make the equation true. Every quadratic equation has exactly two solutions in the complex number system, given by the quadratic formula: x = (−b ± √(b²−4ac)) / 2a.

Yes! When the discriminant D = b² − 4ac is negative, the quadratic equation has no real solutions. Instead, it has two complex conjugate roots of the form p + qi and p − qi, where i = √−1. On the graph, the parabola does not intersect the x-axis at all.

The solutions of a quadratic equation are the x-coordinates where the parabola y = ax²+bx+c crosses the x-axis (y=0). If it crosses at 2 points → two real roots. Touches at 1 point → one repeated root. Doesn't cross → complex roots. This visual connection is why graphing is a powerful tool in algebra.

By Vieta's formulas: Sum of roots (x₁ + x₂) = −b/a, and Product of roots (x₁ × x₂) = c/a. These relationships let you find the sum and product without fully solving the equation — very useful for verification!

Understanding the Solution of Quadratic Equations

A quadratic equation is any polynomial equation of degree 2, written in standard form as ax² + bx + c = 0, where a ≠ 0. The solution of a quadratic equation—also called its roots or zeros—are the values of x that satisfy this equation.

The most reliable method is the quadratic formula, derived by completing the square on the general form. It guarantees solutions for every quadratic, making it indispensable in algebra, physics, engineering, and economics. For example, projectile motion equations, profit maximization models, and signal processing all involve finding the solution of quadratic equations.

Our CalcSuit solver not only gives you the numerical answer but walks you through each step of the derivation, plots the parabola visually, and verifies the result—making it an ideal resource for students studying algebra and for professionals who need rapid, reliable computation.