Solving 59x2+82x+65 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:
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

An equation that can be written as bx+c=0 is called a linear equation. It has one unknown, x, and 2 constants, b and c. If this equation were also to include the square of x as an unknown, it would become a quadratic equation. A quadratic equation is an function that can be written as:
ax2 + bx + c = 0.
\ In this equation, a, b, and c are constants. X is a variable which is not known. A and b are referred to as coefficients. Also, a cannot equal to 0 in the equation ax2+bx+c=0.

Finding a solution to a quadratic equation may appear daunting, because both x and x2 are unknown. However, there are a number of methods for solving quadratic equations. One of the most widely used is the quadratic formula. This 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. Rarely, these two roots may be equal, producing one solution for x.

Quadratic equations have real-life applications. 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.

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

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