Solving 3x2+5x+93 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:
3x2+5x+93=0.

There are no solutions in the real number domain.
There are two complex solutions: x = -0.83333333333333 + 5.5050481883046i, and x = -0.83333333333333 - 5.5050481883046i,
where i is the imaginary unit.

Here's how we found that solution:

You entered the following equation:
(1)           3x2+5x+93=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)           \(x=-5\pm\frac{\sqrt{5^2-4*3*93}}{2*3}\)

which simplifies to:
(4)           \(x=-5\pm\frac{\sqrt{25-1116}}{6}\)

Now, note that b2-4ac is a negative number. Specifically in our case, 25 - 1116 = -1091.
(5)           \(x=-5\pm\frac{\sqrt{-1091}}{6}\)

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)           \(x=-5\pm\frac{33.030289129827i}{6}\)

This equation further simplifies to:
(7)           \(x=-\frac{-5}{6}\pm5.5050481883046i\)

Solving for x, we find two solutions which are both complex numbers:
x = -0.83333333333333 + 5.5050481883046i
  and
x = -0.83333333333333 - 5.5050481883046i

Both of these solutions are complex numbers.
These are the two solutions that will satisfy the equation 3x2+5x+93=0.






Notes

What is a quadratic equation? Any equation that can be written in the form: ax2 + bx + c = 0, where x is a variable which is not known. A, b, and c are constants. The constants a and b, are referred to as coefficients. It should be pointed out that a cannot be 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 is a more complicated task. Fortunately, any quadratic equation can reliably be solved using the quadratic formula. The quadratic formula is:


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 equal each other, 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. The contour of a parablolic dish antenna is one example of an application of quadratic equations.

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?


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