A free quadratic equation calculator that shows and explains each step in solving your quadratic equation.

You entered:

There are no solutions in the real number domain.

There are two complex solutions: x = -0.83333333333333 + 5.5050481883046

where

(1)

For any quadratic equation

(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) \(x=-5\pm\frac{\sqrt{25-1116}}{6}\)

(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

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 + 5.5050481883046

and

x = -0.83333333333333 - 5.5050481883046

Both of these solutions are complex numbers.

These are the two solutions that will satisfy the 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

ax

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 hope the explanations showing how you can solve the equation yourself are educational and helpful. 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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