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

A quadratic equation is any equation that can be written as:
ax2 + bx + c = 0,
where x is unknown. A, b, and c are constants. The constants a and b, are referred to as coefficients. Interestingly, a cannot be equal to 0. If a=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.

Calculating a solution to a quadratic equation may seem challenging. Fortunately, you have this handy-dandy quadratic equation calculator. All kidding aside, quadratic equations can be reliably 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. Depending on the values of a, b, and c, these two roots may be equal, meaning there will only be one solution for x.

You may be asking yourself, "Why is this stuff so important?" Quadratic equations are needed to find answers to many real-world problems. For example, to calculate how an object will rise and fall due to Earth's gravity would require the use of s quadratic equation.

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

We hope you find this quadratic equation solver useful. We encourage you to plug in different values for a, b, and c. But we totally understand if you just want to use it to find the answers you're looking for. Thank you for your interest in Quadratic-Equation-Calculator.com.

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