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

You entered:

There are two real solutions: x = -0.33066854136136, and x = -4.1238769131841.

(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) \(x=-49\pm\frac{\sqrt{49^2-4*11*15}}{2*11}\)

which simplifies to:

(4) \(x=-49\pm\frac{\sqrt{2401-660}}{22}\)

\(x=\frac{-49+41.72529209005}{22}\) = -0.33066854136136,

and

\(x=\frac{-49-41.72529209005}{22}\) = -4.1238769131841,

Both of these solutions are real numbers.

These are the two solutions that will satisfy the quadratic equation

Solving a linear equation is pretty basic. Solving a quadratic equation requires some more advanced mathematics. Fortunately, any quadratic equation can always 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. Rarely, both 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 compute 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 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, 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.

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