Kenneth S. answered • 09/19/17
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ANDREW's Comment IS WRONG. What is desired is the PRINCIPAL SQUARE ROOT, which can never be negative.
You're working with Real numbers here, not complex numbers.
In support of my assertion, consider the basic graphs of functions (which pass the vertical line test).
You should be familiar with the graph of y = √x. x is any non-negative real, and y is any non-negative real.
With these basics, in Calculus we can find the derivative of this and many other functions; recall that all of the theorems of Calculus I and II require FUNCTIONS conforming to this understanding.
Now in the subject matter of Complex numbers, we do encounter facts such as there are n roots of any complex number but that's beyond the scope of this arithmetic/Algebra question's context.
The solution to x2 = 4 is two solutions (2 roots, as Gauss's theorem on polynomials suggests); they are found by solving x2-4 = 0; (x-2)(x+2) = 0, so x = ±2. The shortcut technique can be used by saying that the equation x2 = plus or minus THE SQUARE ROOT OF 4.
Kenneth S.
Andrew, i think that you should review this one more time.
The context here is not complex numbers, so √36 = six, and only six.
If you take the fact √36 = 6 AND THEN SQUARE THIS EQUATION, you are introducing an extraneous root because that equation has two solutions: {6, -6}.
But the symbolism √36 refers only to the principal square root.
Consult a graph of y = √x. Nowhere on that graph does y ever equal -6.
On the other hand, y2 = x, a sideways opening-toward-the-right parabola, does contain (36,±6) but this is a different relation, NOT A FUNCTION.
When you solve your quadratic, you do get two roots but your quadratic
x2 - 36 = 0 IS NOT EQUIVALENT to the function y = √x.
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09/19/17
Andrew M.
Thank you Kenneth. Have a great day. :-)
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09/19/17
Andrew M.
The solution to x2 = 4 is two solutions (2 roots, as Gauss's theorem on polynomials suggests); they are found by solving x2-4 = 0; (x-2)(x+2) = 0, so x = ±2. The shortcut technique can be used by saying that the equation x2 = plus or minus THE SQUARE ROOT OF 4.
09/19/17