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 that has 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. Further, a cannot equal to 0 in the equation ax2+bx+c=0. If a is 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 straightforward. Solving a quadratic equation requires more work. Fortunately, you have this handy-dandy quadratic equation calculator. All kidding aside, quadratic equations can be quickly 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:


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, both roots may have the same value, producing one solution for x.

You may be asking yourself, "Why is this stuff so important?" Quadratic equations are needed to find answers to many real-world problems. The geometry of a parablolic dish antenna 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 hope you find this quadratic equation calculator useful. 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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