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

A quadratic equation is any equation ax2 + bx + c = 0. In this equation, a, b, and c are constants. X is an unknown. A and b are called coefficients. Interestingly, a cannot be zero.

Solving a linear equation is relatively straightforward. Solving a quadratic equation is not as straightforward. However, any quadratic equation can quickly be solved using the quadratic formula. 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, the two roots may have the same value, producing one solution for x.

Why do we need to be able to solve quadratic equations? Quadratic equations are needed to calculate answers to many real-world problems. For example, to calculate whether a braking car can stop fast enough to avoid hitting something would require the use of s quadratic equation.

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.

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