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

A quadratic equation is an function that can be written as: ax2 + bx + c = 0. In this equation, x is an unknown, and a, b, and c are constants. The constants a and b, are referred to as coefficients. Furthermore, it should be pointed out that a cannot be equal to 0.

Solving a linear equation is simple. Solving a quadratic equation is less straightforward. 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:


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 be equal, producing one solution for x.

Who cares? Why do we care about qudratic equations? Quadratic equations are needed to calculate answers in many real-world fields, including physics, pharmacokinetics and business.

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

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