Solving 35x2+-83x+60 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
Reset

You entered:
35x2+-83x+60=0.

There are no solutions in the real number domain.
There are two complex solutions: x = 1.1857142857143 + 0.55530833501648i, and x = 1.1857142857143 - 0.55530833501648i,
where i is the imaginary unit.

Here's how we found that solution:

You entered the following equation:
(1)           35x2+-83x+60=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)           \(x=--83\pm\frac{\sqrt{-83^2-4*35*60}}{2*35}\)

which simplifies to:
(4)           \(x=--83\pm\frac{\sqrt{6889-8400}}{70}\)

Now, note that b2-4ac is a negative number. Specifically in our case, 6889 - 8400 = -1511.
(5)           \(x=--83\pm\frac{\sqrt{-1511}}{70}\)

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)           \(x=--83\pm\frac{38.871583451154i}{70}\)

This equation further simplifies to:
(7)           \(x=-\frac{--83}{70}\pm0.55530833501648i\)

Solving for x, we find two solutions which are both complex numbers:
x = 1.1857142857143 + 0.55530833501648i
  and
x = 1.1857142857143 - 0.55530833501648i

Both of these solutions are complex numbers.
These are the two solutions that will satisfy the equation 35x2+-83x+60=0.






Notes

What is a quadratic equation? Any equation that can be written in 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. Furthermore, it is worth pointing out that a cannot be equal to 0. 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. However, you have this handy-dandy quadratic equation calculator. All kidding aside, quadratic equations can be always 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 written:


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, both roots may be equal, meaning there will only be one solution for x.

Quadratic equations are an important part of mathematics. Quadratic equations are needed to compute answers in many real-world fields, including physics, biology and business.

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

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 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.

click here for a random example of a quadratic equation.