Solving 55x2+-33x+25 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
Reset

You entered:
55x2+-33x+25=0.

There are no solutions in the real number domain.
There are two complex solutions: x = 0.3 + 0.60377599699347i, and x = 0.3 - 0.60377599699347i,
where i is the imaginary unit.

Here's how we found that solution:

You entered the following equation:
(1)           55x2+-33x+25=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)           \(x=--33\pm\frac{\sqrt{-33^2-4*55*25}}{2*55}\)

which simplifies to:
(4)           \(x=--33\pm\frac{\sqrt{1089-5500}}{110}\)

Now, note that b2-4ac is a negative number. Specifically in our case, 1089 - 5500 = -4411.
(5)           \(x=--33\pm\frac{\sqrt{-4411}}{110}\)

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)           \(x=--33\pm\frac{66.415359669281i}{110}\)

This equation further simplifies to:
(7)           \(x=-\frac{--33}{110}\pm0.60377599699347i\)

Solving for x, we find two solutions which are both complex numbers:
x = 0.3 + 0.60377599699347i
  and
x = 0.3 - 0.60377599699347i

Both of these solutions are complex numbers.
These are the two solutions that will satisfy the equation 55x2+-33x+25=0.






Notes

What is a quadratic equation? A quadratic equation is an function
ax2 + bx + c = 0,
where x is a variable which is not known, and a, b, and c are constants. A and b are called coefficients. Further, a cannot equal to zero. Otherwise, the equation ceases to be a quadratic equation, and becomes a linear equation.

Solving a linear equation is fairly straightforward. Solving a quadratic equation is less straightforward. Fortunately, you have this handy-dandy quadratic equation solver. All kidding aside, quadratic equations can be quickly solved using the quadratic formula, which is the same technique used by this quadratic equation solver. Try it, and it will explain each of the steps to you. This is the quadratic formula:


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, the two roots may have the same value, producing one solution for x.

There are many uses for quadratic equations. Quadratic equations are needed to compute answers to many real-world problems. The geometry of a parablolic mirror is one example of an application of quadratic equations.

The quadratic equation calculator on this website uses the quadratic formula to solve your quadratic equations, and this is a reliable and relatively simple way to do it. But there are other ways to solve a quadratic equation, such as completing the square or factoring.

We hope you find this quadratic equation solver useful. We encourage you to try it with different values, and to read the explanation for how to reach your answer. But we totally understand if you just want to use it to find the answers you're looking for. Thank you for using Quadratic-Equation-Calculator.com.

click here for a random example of a quadratic equation.