Solving 1x2+-67x+-21 using the Quadratic Formula

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

For your equation of the form "ax2 + bx + c = 0," enter the values for a, b, and c:

= 0

You entered:

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

Here's how we found that solution:

You entered the following equation:
(1)           x2+-67x+-21=0.

For any quadratic equation ax2 + bx + c = 0, one can solve for x using the following equation, which is known as the quadratic formula:

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

Now, solving for x, we find two real solutions:
\(x=\frac{--67+67.623960250787}{2}\) = 67.311980125393,
\(x=\frac{--67-67.623960250787}{2}\) = -0.31198012539343,

Both of these solutions are real numbers.
These are the two solutions that will satisfy the quadratic equation x2+-67x+-21=0.


What is a quadratic equation? A quadratic equation is an equation that can be written in the form: ax2 + bx + c = 0. In this equation, x is unknown. A, b, and c are constants. The constants a and b are called coefficients. Furthermore, it is worth pointing out that a cannot be equal to zero in the equation ax2+bx+c=0. If a is 0, then ax2=0, and the equation becomes 0+bx+c=0, or bx+c=0. The equation bx+c=0 is a linear equation, and not a 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

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