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

What is a quadratic equation? Any equation that can take the form:
ax2 + bx + c = 0,
where x is a variable which is not known, and a, b, and c are constants. A and b are called coefficients. Further, a cannot be zero. Otherwise, the equation ceases to be a quadratic equation, and becomes a linear equation.

Compared to solving a linear equation, solving a quadratic equation requires a few more steps. Fortunately, you have this handy-dandy quadratic equation solver. All kidding aside, quadratic equations can be quickly solved using the quadratic formula, which is the same technique used by this quadratic equation solver. Try it, and it will explain each of the steps to you. The quadratic formula is written:


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. Depending on the values of a, b, and c, both roots may be equal, resulting in one solution for x.

Why do we care about qudratic equations? Quadratic equations are needed to compute answers to many real-world problems. The contour of a parablolic dish antenna is one example of an application of quadratic equations.

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.

We hope you find this quadratic equation calculator useful. We hope the explanations showing how you can solve the equation yourself are educational and helpful. 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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