Thursday, August 22, 2013

Does √(3*x + 1) = -2 has a solution?

Solve for x:

√(3*x + 1) = -2

One could approach then problem algebraically:

(1) Square both sides.
(3*x + 1) = (-2)^2 = 4

(2) Subtract 1 from both sides and then divide by 3.
3*x + 1 = 4
3*x = 3
x = 1

That could be perfectly reasonable. After all, 3*1 + 1 = 4 and 4 = -2 * -2 = 2 * 2.

However, use any calculator or mathematical software with solving capability, attempting to solve √(3*x + 1) = -2 will yield "No Solution".

Why is that?

In practice, when we take square roots of numbers, we take the positive root, which is practical for majority of applications. The positive root is referred to the principal square root. That is what you get when you use the square root function on your calculator or mathematics software.

Below are two graphs, used by the TI-nSpire App:
1. The principal square root function y = √x, plotted in red.
2. Solutions to the equation x^2 = y, allowing both roots. This is shown by the parametric equation, x = t^2, y = t, plotted in blue.

So keep in mind, when dealing with roots with a calculator or software, you are dealing with the principal root. (If possible, real, then positive).

Source: Dr. Math - Square Root Function

Until next time, have a great weekend!


This blog is property of Edward Shore. 2013

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