Mathematics, Statistics & Geometry

Quadratic Formula Calculator

Solve second-degree algebraic polynomials perfectly. Evaluates the discriminant to find distinct real, repeated, or complex roots.

Discriminant (Δ)
1
Root 1 (Real)2
Root 2 (Real)1
Calculation StepsEquation: 1x² - 3x + 2 = 0 1. Calculate Discriminant (Δ): Δ = b² - 4ac Δ = (-3)² - 4(1)(2) = 1 Δ > 0, so there are two distinct real roots. x = (-b ± √Δ) / 2a x₁ = (3 + 1.0000) / 2 = 2.0000 x₂ = (3 - 1.0000) / 2 = 1.0000

Calculated locally in your browser. Fast, secure, and private.

The Universal Algebraic Key

The Quadratic Formula Calculator solves any second-degree polynomial instantly. By deeply analyzing the Discriminant (Δ), it accurately returns distinct real roots, repeated roots, or complex conjugate pairs.

x=−b±b2−4ac2a\begin{aligned} x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \end{aligned}

Where:
x=
The calculated solutions (zeroes) of the equation
a, b, c=
The known values from the standard form ax² + bx + c = 0
b2−4acb^2 - 4ac=
The term inside the root that dictates the type of solutions

Completing the Square

The Quadratic Formula isn't just magic; it is the mathematical result of taking the standard equation ax2+bx+c=0ax^2 + bx + c = 0 and performing an algebraic technique called 'Completing the Square' to isolate xx.

Because it is a mathematically perfect derivation, it never fails. While factoring only works on 'nice' numbers, the Quadratic Formula will plow through massive decimals and negative square roots to find the exact mathematical truth.

Real-World Applications

  • Kinematics: Calculating the exact trajectory and impact time of a ballistic object under the acceleration of gravity (1/2at2+vt+d1/2 at^2 + vt + d).
  • Optimization: Businesses use quadratics to find their maximum profit curve, where price increases eventually start reducing total sales volume.
  • Electrical Engineering: Solving complex RLC circuit impedance where the resonance creates a secondary degree polynomial.

Frequently Asked Questions

It is the universal 'skeleton key' of algebra. It can solve absolutely any second-degree polynomial equation, even ones that cannot be factored cleanly.

The Discriminant (Δ) is the part of the formula under the square root: b² - 4ac. It 'discriminates' or tells you exactly what kind of answers you are going to get.

If Δ < 0, you are taking the square root of a negative number. This results in two 'Complex' or 'Imaginary' roots containing the letter 'i'.

If Δ = 0, the entire square root disappears. You are left with exactly one, single real root (the vertex touches the x-axis perfectly).

Because a parabola is U-shaped, it usually crosses the X-axis in two different places. The plus/minus calculates both the left and right crossing points simultaneously.

Related Calculators in Mathematics, Statistics & Geometry

Angle Between Vectors Calculator

Try It Now

ANOVA Calculator

Try It Now

Antilog Calculator

Try It Now

Arc Length of Curve Calculator

Try It Now

Area of Annulus Calculator

Try It Now

Area of Circle Calculator

Try It Now