Solving 66x2+-38x+6 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:
66x2+-38x+6=0.

There are no solutions in the real number domain.
There are two complex solutions: x = 0.28787878787879 + 0.089637572471206i, and x = 0.28787878787879 - 0.089637572471206i,
where i is the imaginary unit.

Here's how we found that solution:

You entered the following equation:
(1)           66x2+-38x+6=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)           \(x=--38\pm\frac{\sqrt{-38^2-4*66*6}}{2*66}\)

which simplifies to:
(4)           \(x=--38\pm\frac{\sqrt{1444-1584}}{132}\)

Now, note that b2-4ac is a negative number. Specifically in our case, 1444 - 1584 = -140.
(5)           \(x=--38\pm\frac{\sqrt{-140}}{132}\)

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)           \(x=--38\pm\frac{11.832159566199i}{132}\)

This equation further simplifies to:
(7)           \(x=-\frac{--38}{132}\pm0.089637572471206i\)

Solving for x, we find two solutions which are both complex numbers:
x = 0.28787878787879 + 0.089637572471206i
  and
x = 0.28787878787879 - 0.089637572471206i

Both of these solutions are complex numbers.
These are the two solutions that will satisfy the equation 66x2+-38x+6=0.






Notes

A quadratic equation is an function
ax2 + bx + c = 0,
where x is a variable which is not known, and a, b, and c are constants. The constants a and b are called coefficients. It is worth noting that a cannot be zero. If a is equal to 0, then ax2=0, and the equation becomes 0+bx+c=0, or bx+c=0. The equation bx+c=0 is a linear equation, and not a quadratic equation.

Solving a linear equation is pretty basic. Solving a quadratic equation requires some more advanced mathematics. However, you have this handy-dandy quadratic equation solver. Acutally, quadratic equations can be readily 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:


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, these two roots may have the same value, resulting in one solution for x.

There are many uses for quadratic equations. 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.

The term "quadratic" comes from the Latin word quadratum, which means "square." Why? Because what defines a quadratic equation is the inclusion of some variable squared. In our equation above, the term x2 (x squared) is what makes this equation quadratic.

We this quadratic equation calculator is useful to you. 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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