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
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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

An equation that can be written as bx+c=0 is called a linear equation. It has one unknown, x, and 2 constants, b and c. If this equation were also to include the square of x as an unknown, it would become a quadratic equation. A quadratic equation is any function that takes the form:
ax2 + bx + c = 0,
where x is an unknown, and a, b, and c are constants. A and b are referred to as coefficients. Also, it should be noted that a cannot equal to zero.

Finding a solution to a quadratic equation can appear challenging. However, any quadratic equation can quickly be solved using the quadratic formula. Here is the quadratic formula:


Solving a quadratic equation will always result in 2 solutions for x. These solutions are called roots. These roots may both be real numbers or, they may both be complex numbers. Under extraordinary circumstances, 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 laws of motion is one example of an application of quadratic equations.

In our equation, a cannot be zero. However, b can be zero, and so can c.

We hope you find this quadratic equation calculator useful. We encourage you to plug in different values for a, b, and c. But, if you just want to use it to calculate the answers to your quadratic equations, that's cool too. Thank you for using Quadratic-Equation-Calculator.com.

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