Question 1131444: Solve The Equations and Check:
a)\sqrt(q+2)=7
b)\sqrt(5x-1)+3=0
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! a)sqrt(q+2)=7
square both sides to get q+2 = 49
subtract 2 from both sides to get q = 47
sqrt(47+2) = 7 becomes sqrt(49) = 7 becomes 7 = 7, confirming solution is good.
solution is q = 7.
b)sqrt(5x-1)+3=0
subtract 3 from both sides to get sqrt(5x-1) = -3
square both sides to get 5x-1 = 9
add 1 to both sides to get 5x = 10
divide both sides by 5 to get x = 2
sqrt(5*2-1) + 3 = 0 becomes sqrt(9) + 3 = 0 becomes 3 + 3 = 0 becomes 9 = 0 which is not true which means there is no solution.
you can argue that sqrt(9) is plus or minus 3, but that's not true because of a mathematical quirk that says that sqrt of a number is only the positive root and not the negativfe root.
for example:
if you are given x^2 = 9, then you can solve for x to get x = plus or minus 3.
but, if you are given that x = sqrt(9), then the only solution is x = 3.
x = -3 is not a solution.
the positive square root is called the principal square root.
here's an explanation from quora.
Every positive real number has two square roots; one is positive and the other is negative. In addition, a square root of a number is one of its two equal factors, for example, 2 is a square root of 4 because (2)(2) = 4, but ‒2 is also a square root of 4 because (‒2)(‒2) = 4. In general, a real number “a” is a square root of a nonnegative number “b” if a˛ = b. By definition, the symbol √ b is used to designate the nonnegative or principal square root of any nonnegative real number b, e.g., √4 = 2 is used to indicate the nonnegative or principal square root of 4, and ‒√4 = ‒2 is used to indicate the negative square root of 4. Although the principal square root of a positive number is only one of its two square roots, the designation “the square root” is often used to refer to the principal or nonnegative square root √ b. This is why √4 = 2, rather than ‒2.
while it's true that sqrt(9) is plus or minus 3, only the principal, or positive root is used, by convention.
if you were allowed to use the negative square root, then sqrt(10-1) + 3 = 0 could become sqrt(9) + 3 = 0 which could becomes -3 + 3 = 0 which would then be true.
this is very confusing, because while it looks like you can do that, but you can't, because of the convention.
at least that's what i understand.
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