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 = -0.024691358024691 + 0.68448969262366

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) \(x=-4\pm\frac{\sqrt{4^2-4*81*38}}{2*81}\)

which simplifies to:

(4) \(x=-4\pm\frac{\sqrt{16-12312}}{162}\)

(5) \(x=-4\pm\frac{\sqrt{-12296}}{162}\)

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) \(x=-4\pm\frac{110.88733020503i}{162}\)

This equation further simplifies to:

(7) \(x=-\frac{-4}{162}\pm0.68448969262366i\)

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

x = -0.024691358024691 + 0.68448969262366

and

x = -0.024691358024691 - 0.68448969262366

Both of these solutions are complex numbers.

These are the two solutions that will satisfy the equation

ax

\ In this equation, x is a variable which is not known, and a, b, and c are constants. The constants a and b, are referred to as coefficients. Interestingly, a cannot equal to zero in the equation ax

In contrast to solving a linear equation, solving a quadratic equation requires some more advanced mathematics.. However, you have this handy-dandy quadratic equation solver. All kidding aside, quadratic equations can be always solved using the quadratic formula, which is the same technique used by this quadratic equation solver. Try it, and it will explain each of the steps to you. The quadratic formula is:

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. Depending on the values of a, b, and c, these two roots may be the same, meaning there will only be one solution for x.

There are many uses for quadratic equations. Quadratic equations are needed to find 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 hope you find this quadratic equation calculator useful. We encourage you to plug in different values for a, b, and c. But, if you just want to use it to calculate the answers to your quadratic equations, that's cool too. Thank you for your interest in Quadratic-Equation-Calculator.com.

click here for a random example of a quadratic equation.