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. What is a quadratic equation? A quadratic equation is any function that can take the form: ax2 + bx + c = 0. In this equation, a, b, and c are constants. X is unknown. The constants a and b, are referred to as coefficients. Also, a cannot equal to zero.

Solving a linear equation is relatively straightforward. Solving a quadratic equation requires more work. 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. Here is the quadratic formula:


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. Rarely, the two roots may be the same, resulting in one solution for x.

Quadratic equations are more than just mathematical chores we have to endure. Quadratic equations are needed to compute answers to many real-world problems. The laws of motion is one example of an application of quadratic equations.

As mentioned above, in the equation ax2+bx+c=0, a cannot be zero. If a were 0, then ax2 = 0x2 = 0 for any value of x, so our equation becomes 0 + bx + c = 0, which is the same as bx + c = 0, which is no longer a quadratic equation. In fact, bx + c = 0 is a linear equation, which is much simpler to solve than a quadratic equation.

We this quadratic equation calculator 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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