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autodidude
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When you have something like x^2 > 2/3 and you root it, why does the left hand side become the plus or minus and not the right side? I only get the correct answer if I do it on the left hand side.
It doesn't. If [itex]x^2> 2/3[/itex] then either [itex]x<-\sqrt{2/3}[/itex] or [itex]x> \sqrt{2/3}[/itex].autodidude said:When you have something like x^2 > 2/3 and you root it, why does the left hand side become the plus or minus and not the right side? I only get the correct answer if I do it on the left hand side.
Mark44 said:Because, by definition, the square root of a nonnegative real number is nonnegative.
For example, many people erroneously believe that √4 = ±2. Although 4 does have two square roots, the principal square root of 4 is 2.
Granted, 4 has two square roots, and I mentioned this earlier in the thread. However, the notation ##\sqrt{a}## indicates the principal square root.DonAntonio said:Well, it is NOT erroneous to "believe" that [itex]\sqrt{4}=\pm 2[/itex] since, as it happens, both values on the RHS when
squared equal 4 and this is the primary definition of "square root.
DonAntonio said:It is DEFINED that [itex]\sqrt{4}=2[/itex] mostly, I think, to make [itex]\sqrt{x}[/itex] a function, which otherwise it wouldn't be. If one want to mess with the
negative root is thus customary to take [itex]-\sqrt{4}=-2[/itex] and everybody happy.
DonAntonio
Yes, of course. The above should be cos(x) = 1/2 OR -cos(x) = 1/2.autodidude said:Ooh, one more thing, does this apply to trig equations as well?
e.g. cos^2(x) = 1/4 becomes cos(x)=1/2 and -cos(x)=1/2
autodidude said:(from ±√(cos^2(x)) = √(1/2)) rather than cos(x)=±1/2 (which evaluates to the same answer, just being pedantic)
Square rooting an equation or inequality involves finding the number that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5, because 5 x 5 = 25.
Square rooting an equation or inequality is a way to solve for the unknown variable. By taking the square root of both sides of the equation or inequality, we can isolate the variable and find its value.
The +/- sign is always placed on the side of the equation/inequality that has the variable. For example, if the equation is x^2 = 25, when we square root both sides, it becomes x = +/- 5.
Yes, we can square root both sides of an equation/inequality at the same time. This is known as the square root property and is a useful method for solving equations and inequalities involving squares.
Yes, there are some restrictions when square rooting an equation/inequality. We cannot take the square root of a negative number, as it will result in an imaginary number. Also, when dealing with inequalities, we must consider the possibility of extraneous solutions, which may be introduced when squaring both sides.