Solving 81x2+4x+38 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:
81x2+4x+38=0.

There are no solutions in the real number domain.
There are two complex solutions: x = -0.024691358024691 + 0.68448969262366i, and x = -0.024691358024691 - 0.68448969262366i,
where i is the imaginary unit.

Here's how we found that solution:

You entered the following equation:
(1)           81x2+4x+38=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)           \(x=-4\pm\frac{\sqrt{4^2-4*81*38}}{2*81}\)

which simplifies to:
(4)           \(x=-4\pm\frac{\sqrt{16-12312}}{162}\)

Now, note that b2-4ac is a negative number. Specifically in our case, 16 - 12312 = -12296.
(5)           \(x=-4\pm\frac{\sqrt{-12296}}{162}\)

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)           \(x=-4\pm\frac{110.88733020503i}{162}\)

This equation further simplifies to:
(7)           \(x=-\frac{-4}{162}\pm0.68448969262366i\)

Solving for x, we find two solutions which are both complex numbers:
x = -0.024691358024691 + 0.68448969262366i
  and
x = -0.024691358024691 - 0.68448969262366i

Both of these solutions are complex numbers.
These are the two solutions that will satisfy the equation 81x2+4x+38=0.






Notes

A quadratic equation is any function that can be written in the form:
ax2 + bx + c = 0.
\ In this equation, a, b, and c are constants. X is unknown. A and b are called coefficients. Interestingly, a cannot be zero in the equation ax2+bx+c=0.

Calculating a solution to a quadratic equation can seem challenging. Fortunately, you have this handy-dandy quadratic equation solver. All kidding aside, quadratic equations can be reliably 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:


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

So what? Why do we care about qudratic equations? Quadratic equations are needed to compute answers to many real-world problems. For example, to calculate the path of an accelerating object would require the use of s quadratic equation.

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

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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