Quadratic Formula Calculator

Quadratic formula calculator at a glance#

A quadratic formula calculator solves any equation of the form ax^2 + bx + c = 0 for x. It uses the quadratic formula, x = (-b plus or minus the square root of (b^2 - 4ac)) divided by 2a. You enter the three coefficients a, b and c, and it returns the roots along with the discriminant.

What is a worked quadratic formula example?#

Take x^2 - 5x + 6 = 0, so a = 1, b = -5 and c = 6. The discriminant b^2 - 4ac is (-5)^2 - 4 times 1 times 6, which is 25 - 24 = 1. The square root of 1 is 1, so x = (5 plus or minus 1) divided by 2. That gives x = (5 + 1) / 2 = 3 and x = (5 - 1) / 2 = 2. Both check out, since 3 + 2 = 5 and 3 times 2 = 6.

What does the discriminant tell you?#

The discriminant is the part under the square root, b^2 - 4ac, and it sets the number and type of roots before you finish solving.

For 2x^2 + 3x - 2 = 0, the discriminant is 3^2 - 4 times 2 times -2 = 9 + 16 = 25, which is positive, so the roots are real: x = 0.5 and x = -2.

Enter your values of a, b and c in the calculator above for the exact roots and the discriminant, including complex roots when the discriminant is negative. The coefficient a cannot be zero, otherwise the equation is linear rather than quadratic.

How to use the calculator#

Enter the three coefficients a, b and c from your equation written as ax^2 + bx + c = 0. The calculator returns the discriminant and the roots, including complex roots when the discriminant is negative. The value a cannot be zero, because then the equation is linear, not quadratic.

More worked examples#

Two complex roots#

For x^2 + 2x + 5 = 0, the discriminant is 2^2 - 4 times 1 times 5 = 4 - 20 = -16. Because it is negative, the roots are complex: x = (-2 plus or minus the square root of -16) / 2 = -1 plus or minus 2i.

One repeated root#

For 3x^2 + 6x + 3 = 0, the discriminant is 6^2 - 4 times 3 times 3 = 36 - 36 = 0. A zero discriminant means one repeated root: x = -6 / (2 times 3) = -1.

Where the formula comes from#

The quadratic formula is derived by completing the square on ax^2 + bx + c = 0. You divide through by a, move the constant across, add the square of b/(2a) to both sides to form a perfect square, then take the square root. That isolates x and gives x = (-b plus or minus the square root of (b^2 - 4ac)) / 2a.

Frequently asked questions#

What is a quadratic equation?#

A polynomial equation of the second degree, written ax^2 + bx + c = 0, where a, b and c are constants and a is not zero. The highest power of x is 2.

Can you show a full quadratic formula example?#

For 2x^2 + 5x - 3 = 0, a = 2, b = 5, c = -3. The discriminant is 5^2 - 4 times 2 times -3 = 25 + 24 = 49, and its square root is 7. So x = (-5 plus or minus 7) / 4, giving x = 0.5 and x = -3.

What if the discriminant is zero?#

There is exactly one real root, repeated. It is x = -b / 2a, since the square root term drops out.

What is the difference between a polynomial and a quadratic equation?#

A polynomial equation can be any degree. A quadratic is specifically degree two, so the highest term is x^2. Every quadratic is a polynomial, but not every polynomial is quadratic.

Where are quadratic equations used?#

In physics for projectile motion, in engineering for structural and optimization problems, and in economics for cost and revenue models. The Indian mathematician Brahmagupta described a general solution method around 628 AD.