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 = 0.3 + 0.60377599699347

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) \(x=--33\pm\frac{\sqrt{-33^2-4*55*25}}{2*55}\)

which simplifies to:

(4) \(x=--33\pm\frac{\sqrt{1089-5500}}{110}\)

(5) \(x=--33\pm\frac{\sqrt{-4411}}{110}\)

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) \(x=--33\pm\frac{66.415359669281i}{110}\)

This equation further simplifies to:

(7) \(x=-\frac{--33}{110}\pm0.60377599699347i\)

Solving for x, we find two solutions which are both complex numbers:

x = 0.3 + 0.60377599699347

and

x = 0.3 - 0.60377599699347

Both of these solutions are complex numbers.

These are the two solutions that will satisfy the equation

Compared to solving a linear equation, solving a quadratic equation requires a few more steps. However, any quadratic equation can quickly be solved using the quadratic formula. This is the quadratic formula:

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. Under extraordinary circumstances, these two roots may equal each other, meaning there will only be one solution for x.

Quadratic equations are more than just mathematical chores we have to endure. Quadratic equations are needed to calculate 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

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 this quadratic equation solver is useful to you. We encourage you to try it with different values, and to read the explanation for how to reach your answer. But we totally understand if you just want to use it to find the answers you're looking for. Thank you for your interest in Quadratic-Equation-Calculator.com.

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