Expert: Josh - 8/5/2008

Question
Question:

Why x is the anwser to the square root of x^2 only when x is positive?

I'm not sure if I understand the question completely or what its asking for?

Answer
Hi Amanda,

The statement "x is the answer to the square root of x^2" only tells half the story. In fact, +x and -x are BOTH solution to sqrt(x^2).
To see this, note that x*x equals x^2 while (-x)(-x) also equals x^2.
If you're a little confused, just plug in some numbers to check.

What is the square root (sqrt) of 9?
Ans: sqrt(9) equals +3 or -3.

EXAMPLE: The equation y=x^2-9 (which we can factorize as y=(x-3)(x+3)) represents a U-shape parabola. It is symmetrical about the y-axis and has solution at x=-3 and x=+3. Finding where the parabola intersects with the x-axis is equivalent to solving 0=x^2-9. It amounts to finding x=sqrt(9).

NOTE 1:
In the above question, both answers x=3 and x=-3 are legitimate. However, in physical problems, sometimes we cannot allow a negative number (i.e., -x) to be part of the solution of sqrt(x^2).

Consider the area of a square with dimension x. If the area is 9 square units, what is x? Here, an answer like x=-3 clearly doesn't make sense. Negative length does not exist in the real world and cannot be measured. So, the only acceptable value is x=3.

COMMENT: You may encounter "Complex Numbers" in a more advanced course. Until then, the equation y=sqrt(c) has no real solution if c is negative. So the lesson for now is to remember that +x and -x are both "possible" solutions to y=sqrt(x^2) depending on the context of the problem. You should check to see if they make sense.