Solving 41x2+18x+18 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:
41x2+18x+18=0.

There are no solutions in the real number domain.
There are two complex solutions: x = -0.21951219512195 + 0.62517100575494i, and x = -0.21951219512195 - 0.62517100575494i,
where i is the imaginary unit.

Here's how we found that solution:

You entered the following equation:
(1)           41x2+18x+18=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)           \(x=-18\pm\frac{\sqrt{18^2-4*41*18}}{2*41}\)

which simplifies to:
(4)           \(x=-18\pm\frac{\sqrt{324-2952}}{82}\)

Now, note that b2-4ac is a negative number. Specifically in our case, 324 - 2952 = -2628.
(5)           \(x=-18\pm\frac{\sqrt{-2628}}{82}\)

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)           \(x=-18\pm\frac{51.264022471905i}{82}\)

This equation further simplifies to:
(7)           \(x=-\frac{-18}{82}\pm0.62517100575494i\)

Solving for x, we find two solutions which are both complex numbers:
x = -0.21951219512195 + 0.62517100575494i
  and
x = -0.21951219512195 - 0.62517100575494i

Both of these solutions are complex numbers.
These are the two solutions that will satisfy the equation 41x2+18x+18=0.






Notes

What is a quadratic equation? A quadratic equation is an function that can be written in the form:
ax2 + bx + c = 0,
where x is unknown, and a, b, and c are constants. A and b are called coefficients. Also, it is worth noting that a cannot equal to 0. If a is equal to 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.

Solving a linear equation is pretty basic. Solving a quadratic equation is less straightforward. However, you have this handy-dandy quadratic equation calculator. All kidding aside, 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. 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. Rarely, these two roots may be the same, resulting in one solution for x.

Who cares? Why do we care about qudratic equations? Quadratic equations are needed to calculate answers in many real-world fields, including engineering, pharmacokinetics and business.

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 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.

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