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:

a
 
x2
 
+
b
 
x
 
+
c
 
= 0
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You entered:
83x2+-51x+44=0.

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:
(2)          

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
  and
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.






Notes

A quadratic equation is any 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 zero in the equation ax2+bx+c=0. If a equals 0, then ax2=0, and the equation becomes 0+bx+c=0, or bx+c=0. The equation bx+c=0 is a linear equation, and not a quadratic equation.

Calculating a solution to a quadratic equation can seem challenging. However, you have this handy-dandy quadratic equation calculator. Acutally, quadratic equations can be always solved using the quadratic formula, which is the same technique used by this quadratic equation calculator. Try it, and it will explain each of the steps to you. Here is the quadratic formula:


Since there are always 2 solutions to a square root (one negative, one positive), solving the quadratic equation results in 2 values for x. The two solutions for x (which may be positive or negative, real or complex) are called roots. Depending on the values of a, b, and c, these two roots may be equal, resulting in one solution for x.

So what? Why do we care about qudratic equations? Quadratic equations are needed to find 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 quadratic formula has been known for centuries. Brahmagupta, a mathematician from India, first described the quadratic formula as a means to calculate solutions to quadratic equations in the 7th Century AD.

We hope you find this quadratic equation calculator 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 Quadratic-Equation-Calculator.com.

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