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

You entered:

There are two real solutions: x = 67.311980125393, and x = -0.31198012539343.

(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) \(x=--67\pm\frac{\sqrt{-67^2-4*1*-21}}{2*1}\)

which simplifies to:

(4) \(x=--67\pm\frac{\sqrt{4489--84}}{2}\)

\(x=\frac{--67+67.623960250787}{2}\) = 67.311980125393,

and

\(x=\frac{--67-67.623960250787}{2}\) = -0.31198012539343,

Both of these solutions are real numbers.

These are the two solutions that will satisfy the quadratic equation

Solving a linear equation is rather basic. Solving a quadratic equation requires some more advanced mathematics. However, you have this handy-dandy quadratic equation solver. Acutally, quadratic equations can be reliably 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:

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. Depending on the values of a, b, and c, the two roots may be equal, producing one solution for x.

Why do we care about qudratic equations? Quadratic equations are needed to compute answers in many real-world fields, including physics, pharmacokinetics and architecture.

The quadratic formula has been known for centuries. Brahmagupta, a mathematician from India, first described the quadratic formula as a means to calculate solutions to quadratic equations in the 7th Century AD.

We this quadratic equation solver is useful to you. 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.

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