In the "solving by square-rooting" section of the "solving quadratics" lesson, we had the following problem and solution:
My steps were:
x^{2} – 4 = 0
x^{2} = 4
x = ± 2
Then my solution was:
x = ± 2
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...and I explained the form of the solution by saying:
Why the "±" ("plus-or-minus") sign? Because it might have been a positive 2 or a negative 2 that was squared to get the 4.
While a valid explanation, we can be more mathematically precise about the source of that "plus-minus". The explanation can go like either of the following:
Suppose we are given the equation "x^{2} = 4" and we are told to solve. When we "take the square root" of either side, we are doing a shortcut from regular solving. If we instead apply regular factoring methods to this quadratic, we would first move everything over to one side of the "equals" sign, factor, and solve:
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x^{2} = 4
x^{2} – 4 = 0
(x – 2)(x + 2) = 0
x = 2, –2
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Because we were factoring a difference of squares, we arrived at two solutions, being equal other than for their signs. By "taking the square root" of either side of the first computational line above, and slapping a "plus-minus" sign in front of the numerical side of the equation, we would have found the same two solution values, including their different signs, in one or two fewer steps. But the reasoning involved is more clear in the factored form.
In the above example, we were taking the square root of a perfect square (namely, 4), but the process works just as well for when the strictly numerical portion of this sort of an equation is not a perfect square. For instance:
x^{2} = 7
This can be converted into a difference of squares, as long as we allow the square root of seven to be one of the values being squared:
Then we can factor as usual:
This is the same result we would have gotten by "taking the square root" of either side of the original equation, and then slapping a "±" in front of the numerical side:
The other explanation for why there is a "±" on the one side of the equation is very much more technical, in a mathematical definition-based way:
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Suppose we are given the equation "x^{2} = 4" and we are told to solve. When we take the square root of either side, we get the following:
That is, technically-speaking, we don't have a "±" on the square root sign on the right. However—
The technical definition of "the square root of x squared" is "the absolute value of x ". That is:
Because of this highly technical consideration, the equation actually simplifies as:
x^{2} = 4
| x | = 2
But x could be positive or negative (though not zero, obviously). To solve this absolute-value equation, we must consider each of the two cases. If x is positive, then we can remove the absolute-value bars without changing anything:
If x > 0, then | x | = x, so | x | = x = 2
On the other hand, if x is negative, then we must change the sign on x when we remove the absolute-value bars, so we get:
If x < 0, then| x | = –x, so | x | = –x = 2
Solving this, we get that x = –2.
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That is, while we place the "±" sign on the side with the number, the "plus-minus" actually (technically) comes from the side with the variable, because the square root of the squared variable returns the absolute value of that variable. By "taking the square root" of either side and placing a "±" in front of the numerical value, we save ourselved the trouble of solving the absolute-value equation that was (technically) created by taking the square root.
Most students find it simplest just to remember that, whenever you square-root both sides of an equation, you have to remember to put a "±" on the side opposite the variable. You should use whatever works best for you.
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