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
Reset

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 an function that takes the form:
ax2 + bx + c = 0.
\ In this equation, x is a variable which is not known, and a, b, and c are constants. The constants a and b, are referred to as coefficients. Interestingly, a cannot equal to zero in the equation ax2+bx+c=0. If a=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.

In contrast to solving a linear equation, solving a quadratic equation requires some more advanced mathematics.. However, you have this handy-dandy quadratic equation solver. All kidding aside, quadratic equations can be always 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:


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. Depending on the values of a, b, and c, these two roots may be the same, meaning there will only be one solution for x.

There are many uses for quadratic equations. Quadratic equations are needed to find answers to many real-world problems. For example, to calculate how an object will rise and fall due to Earth's gravity would require the use of s quadratic equation.

Because equations can be rearranged without losing their meaning, sometimes you may see an equation that isn't written exactly this way, but it's still a quadratic equation. For example, you probably know that
ax2 + bx + c = 0 means exactly the same thing as 0 = c + bx + ax2. They're just written differently.
Here are some other examples of ways to write the quadratic equation. They all mean the same thing:
  (1)     \(ax^2+bx=d\), where d = -c
  (2)     \(x^2+bx-d=e\), where a=1 and d=e-c
  (3)     \(ax^2=ex+d\), where d=-c and e=-b
  (4)     \(\frac{x^2}{f}-d=ex\), where d=-c and e=-b and \(f=\frac{1}{a}\)
Look at each of the examples above. Do you understand why they are still quadratic equations, and how they can be rearranged to look like our familiar formula?


We hope you find this quadratic equation calculator useful. We encourage you to plug in different values for a, b, and c. But, if you just want to use it to calculate the answers to your quadratic equations, that's cool too. Thank you for your interest in Quadratic-Equation-Calculator.com.

click here for a random example of a quadratic equation.