Solving 41x2+18x+18 using the Quadratic Formula

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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 any function that can be written in the form:
ax2 + bx + c = 0.
\ In this equation, a, b, and c are constants. X is unknown. The constants a and b are called coefficients. Interestingly, a cannot be zero. Otherwise, the equation ceases to be a quadratic equation, and becomes a linear equation.

Solving a linear equation is pretty simple. Solving a quadratic equation is less simple. Fortunately, any quadratic equation can reliably be solved using 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. Under extraordinary circumstances, the two roots may have the same value, meaning there will only be one solution for x.

Quadratic equations are more than just mathematical mumbo-jumbo Quadratic equations are needed to find answers to many real-world problems. The geometry of a parablolic dish antenna 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 this quadratic equation solver is useful to you. We encourage you to try it with different values, and to read the explanation for how to reach your answer. 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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