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 as: ax2 + bx + c = 0, where a, b, and c are constants. X is an unknown. A and b are called coefficients. Interestingly, a cannot be 0 in the equation ax2+bx+c=0.

Calculating a solution to a quadratic equation may appear daunting, because both x and x2 are unknown. Fortunately, there are a number of methods for solving quadratic equations. One of the most widely used is the quadratic formula. The quadratic formula is:


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, both roots may have the same value, meaning there will only be one solution for x.

Quadratic equations have real-life applications. Quadratic equations are needed to find answers in many real-world fields, including physics, pharmacokinetics and architecture.

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

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