Solving 59x2+82x+65 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:
59x2+82x+65=0.

There are no solutions in the real number domain.
There are two complex solutions: x = -0.69491525423729 + 0.78663060243202i, and x = -0.69491525423729 - 0.78663060243202i,
where i is the imaginary unit.

Here's how we found that solution:

You entered the following equation:
(1)           59x2+82x+65=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)           \(x=-82\pm\frac{\sqrt{82^2-4*59*65}}{2*59}\)

which simplifies to:
(4)           \(x=-82\pm\frac{\sqrt{6724-15340}}{118}\)

Now, note that b2-4ac is a negative number. Specifically in our case, 6724 - 15340 = -8616.
(5)           \(x=-82\pm\frac{\sqrt{-8616}}{118}\)

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)           \(x=-82\pm\frac{92.822411086978i}{118}\)

This equation further simplifies to:
(7)           \(x=-\frac{-82}{118}\pm0.78663060243202i\)

Solving for x, we find two solutions which are both complex numbers:
x = -0.69491525423729 + 0.78663060243202i
  and
x = -0.69491525423729 - 0.78663060243202i

Both of these solutions are complex numbers.
These are the two solutions that will satisfy the equation 59x2+82x+65=0.






Notes

A quadratic equation is an function that can be written as: ax2 + bx + c = 0. In this equation, x is an unknown, and a, b, and c are constants. A and b are referred to as coefficients. Further, it is worth noting that a cannot be equal to zero. Otherwise, the equation ceases to be a quadratic equation, and becomes a linear equation.

Finding a solution to a quadratic equation may appear challenging. Fortunately, there are a number of methods for solving quadratic equations. One of the most widely used is the quadratic formula. Here is the quadratic formula:


Solving a quadratic equation will always result in 2 solutions for x. These solutions are called roots. These roots may both be real numbers or, they may both be complex numbers. Under extraordinary circumstances, these two roots may have the same value, producing one solution for x.

Quadratic equations are an important part of mathematics. Quadratic equations are needed to calculate answers to many real-world problems. For example, to compute how an object will rise and fall due to Earth's gravity would require the use of s quadratic equation.

Because equations can be rearranged without losing their meaning, sometimes you may see an equation that isn't written exactly this way, but it's still a quadratic equation. For example, you probably know that
ax2 + bx + c = 0 means exactly the same thing as 0 = c + bx + ax2. They're just written differently.
Here are some other examples of ways to write the quadratic equation. They all mean the same thing:
  (1)     \(ax^2+bx=d\), where d = -c
  (2)     \(x^2+bx-d=e\), where a=1 and d=e-c
  (3)     \(ax^2=ex+d\), where d=-c and e=-b
  (4)     \(\frac{x^2}{f}-d=ex\), where d=-c and e=-b and \(f=\frac{1}{a}\)
Look at each of the examples above. Do you understand why they are still quadratic equations, and how they can be rearranged to look like our familiar formula?


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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