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 = 0.28787878787879 + 0.089637572471206

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) \(x=--38\pm\frac{\sqrt{-38^2-4*66*6}}{2*66}\)

which simplifies to:

(4) \(x=--38\pm\frac{\sqrt{1444-1584}}{132}\)

(5) \(x=--38\pm\frac{\sqrt{-140}}{132}\)

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) \(x=--38\pm\frac{11.832159566199i}{132}\)

This equation further simplifies to:

(7) \(x=-\frac{--38}{132}\pm0.089637572471206i\)

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

x = 0.28787878787879 + 0.089637572471206

and

x = 0.28787878787879 - 0.089637572471206

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. However, any quadratic equation can readily be solved using the quadratic formula. This is the quadratic formula:

Solving a quadratic equation will always result in 2 solutions for x. These solutions are 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.

You may be asking yourself, "Why is this stuff so important?" Quadratic equations are needed to compute answers to many real-world problems. For example, to compute whether a braking car can stop fast enough to avoid hitting something 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 hope you find this quadratic equation solver useful. We encourage you to plug in different values for a, b, and c. But we totally understand if you just want to use it to find the answers you're looking for. Thank you for using Quadratic-Equation-Calculator.com.

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