Quadratic Formula Calculator
The calculator below solves the quadratic equation ax2 + bx + c = 0.
Fractional values such as 3/4 can be used.
Quadratic Equations and the Quadratic Formula
In algebra, a quadratic equation is any polynomial equation of the second degree, with the form:
ax2 + bx + c = 0
where x is the unknown, a is the quadratic coefficient, b the linear coefficient, and c the constant. The coefficients are known numbers; a cannot be 0, or the equation would be linear rather than quadratic. A quadratic equation can be solved in several ways — factoring, the quadratic formula, completing the square, or graphing. Only the quadratic formula, and the basics of completing the square (which its derivation uses), are covered here. The formula is:
x = −b ± √b2 − 4ac2a
Derivation of the Quadratic Formula
Start from ax2 + bx + c = 0 and multiply through by 1a to make the first coefficient 1:
x2 + bax + ca = 0 → x2 + bax = −ca
From this point the square can be completed using the relationship x2 + bx + c = (x − h)2 + k. Add b2a squared to both sides:
x2 + bax + (b2a)2 = −ca + (b2a)2
Simplifying the left side into a squared bracket and combining the right side over the denominator 4a2 (Simplify):
(x + b2a)2 = b2 − 4ac4a2
Take the square root of both sides (Square root both sides), then solve for x (Solve for x):
x + b2a = ±√b2 − 4ac2a → x = −b ± √b2 − 4ac2a
Recall that the ± exists as a function of computing a square root, making both the positive and negative roots solutions of the quadratic equation. The x values found through the quadratic formula are the roots of the equation — the points where its parabola crosses the x-axis. The formula also provides the axis of symmetry of the parabola, x = −b/(2a). The quadratic formula has many real-world applications, such as calculating areas, projectile trajectories, and speed, among others.
Frequently Asked Questions
How do you solve a quadratic equation with the quadratic formula?
Write the equation as ax2 + bx + c = 0, identify a, b, and c, and substitute them into x = (−b ± √(b2 − 4ac)) / 2a. The plus and minus give the two roots; if the discriminant b2 − 4ac is zero there is one repeated root, and if it is negative the roots are imaginary.
What does the discriminant tell you?
The discriminant D = b2 − 4ac determines the nature of the roots: if D > 0 the equation has two distinct real roots, if D = 0 it has one repeated real root, and if D < 0 it has two imaginary (complex) roots.
What if a = 0 in a quadratic equation?
If a = 0 the equation is no longer quadratic — it is the linear equation bx + c = 0, whose solution is x = −c/b. This calculator handles that case automatically.
Can the coefficients be fractions?
Yes. Fractional values such as 3/4 can be entered for any of a, b, and c, and the steps keep the coefficients as exact fractions until the final decimal roots.