Quadratic Formula Calculator

The calculator below solves the quadratic equation of

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In algebra, a quadratic equation is any polynomial equation of the second degree with the following form:

ax² + bx + c = 0

In this expression, x is the unknown variable, a serves as the quadratic coefficient, b the linear coefficient, and c the constant term. The symbols a, b and c are known numbers that multiply the respective powers of x. Crucially, a must not be zero; otherwise the relation would be linear, not quadratic. A quadratic can be tackled in several ways – factoring, applying the quadratic formula, completing the square, or graphing. Here we concentrate on the quadratic formula and briefly show how completing the square leads to it. The formula and its derivation appear below.

Derivation of the Quadratic Formula

ax2 + bx + c = 0	Multiply by   1 a
x2 +  b a x +  c a  = 0
x2 +  b a x = –  c a

From this point, it is possible to complete the square using the relationship that:

x² + bx + c = (x - h)² + k

Continuing the derivation using this relationship:

x2 +  b a x + ( b 2a )2 = –  c a  + ( b 2a )2	Simplify
(x +  b 2a )2 = –  4ac 4a2  +  b2 4a2  =  b2 - 4ac 4a2	Square root both sides
x +  b 2a  = ±  √b2 - 4ac 2a	Solve for x
x =  -b ± √b2 - 4ac 2a

Recall that the ± exists as a function of computing a square root, making both positive and negative roots solutions of the quadratic equation. The x values found through the quadratic formula are roots of the quadratic equation that represent the x values where any parabola crosses the x-axis. Furthermore, the quadratic formula also provides the axis of symmetry of the parabola. This is demonstrated by the graph provided below. Note that the quadratic formula actually has many real-world applications, such as calculating areas, projectile trajectories, and speed, among others.

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