Solving 81x2+4x+38 using the Quadratic Formula

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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

What is a quadratic equation? Any equation that can be written in the form: ax2 + bx + c = 0. In this equation, x is an unknown, and a, b, and c are constants. A and b are called coefficients. Further, a cannot be equal to zero in the equation ax2+bx+c=0. If a equals 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.

Compared to solving a linear equation, solving a quadratic equation requires some more advanced mathematics.. Fortunately, there are a number of methods for solving quadratic equations. One of the most widely used is the quadratic formula. Here is the quadratic formula:


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. Under extraordinary circumstances, the two roots may have the same value, resulting in one solution for x.

Quadratic equations are important. Quadratic equations are needed to find answers to many real-world problems. For example, to compute the path of an accelerating object would require the use of s quadratic equation.

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.

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