Solving 83x2+-51x+44 using the Quadratic Formula

A free quadratic equation calculator that shows and explains each step in solving your quadratic equation.

For your equation of the form "ax2 + bx + c = 0," enter the values for a, b, and c:

= 0

You entered:

There are no solutions in the real number domain.
There are two complex solutions: x = 0.30722891566265 + 0.66009914051486i, and x = 0.30722891566265 - 0.66009914051486i,
where i is the imaginary unit.

Here's how we found that solution:

You entered the following equation:
(1)           83x2+-51x+44=0.

For any quadratic equation ax2 + bx + c = 0, one can solve for x using the following equation, which is known as the quadratic formula:

In the form above, you specified values for the variables a, b, and c. Plugging those values into Eqn. 1, we get:
(3)           \(x=--51\pm\frac{\sqrt{-51^2-4*83*44}}{2*83}\)

which simplifies to:
(4)           \(x=--51\pm\frac{\sqrt{2601-14608}}{166}\)

Now, note that b2-4ac is a negative number. Specifically in our case, 2601 - 14608 = -12007.
(5)           \(x=--51\pm\frac{\sqrt{-12007}}{166}\)

This means that our solution will require finding the square root of a negative number. There is no real number solution for this, so our solution will be a complex number (that is, it will involve the imaginary number i, defined as the square root of -1.).
Let's calculate the square root:
(6)           \(x=--51\pm\frac{109.57645732547i}{166}\)

This equation further simplifies to:
(7)           \(x=-\frac{--51}{166}\pm0.66009914051486i\)

Solving for x, we find two solutions which are both complex numbers:
x = 0.30722891566265 + 0.66009914051486i
x = 0.30722891566265 - 0.66009914051486i

Both of these solutions are complex numbers.
These are the two solutions that will satisfy the equation 83x2+-51x+44=0.


A quadratic equation is an function ax2 + bx + c = 0. In this equation, a, b, and c are constants. X is unknown. The constants a and b are called coefficients. Also, a cannot be equal to 0 in the equation ax2+bx+c=0. Otherwise, the equation ceases to be a quadratic equation, and becomes a linear equation.

Compared to solving a linear equation, solving a quadratic equation is a more complicated task. Fortunately, any quadratic equation can always be solved using the quadratic formula. The quadratic formula is:

When you compute a solution to a quadratic equation, you will always find 2 values for x, called "roots". These roots may both be real numbers or, they may both be complex numbers. Depending on the values of a, b, and c, both roots may have the same value, resulting in one solution for x.

You may be asking yourself, "Why is this stuff so important?" Quadratic equations are needed to calculate answers to many real-world problems. The acceleration of an object as it falls to earth is one example of an application of quadratic equations.

The term "quadratic" comes from the Latin word quadratum, which means "square." Why? Because what defines a quadratic equation is the inclusion of some variable squared. In our equation above, the term x2 (x squared) is what makes this equation quadratic.

We hope you find this quadratic equation solver useful. We encourage you to plug in different values for a, b, and c. But we totally understand if you just want to use it to find the answers you're looking for. Thank you for your interest in

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