Why do i need to factor out the minus

√13 > 1

Hence, 1 - √13 is negative.

In the Real number system, we cannot take the square root of a negative quantity.

Squaring 1-√13 yields a positive radicand, so we may take its square root. But we lose sign information, when we square a negative quantity.

Remember: When we square a quantity and subsequently take the square root of that square, we must always check for an extraneous result (that is, a "false solution").

Here's the general way to think about taking the square root of an unknown square:

\(\displaystyle \sqrt{x^2} = |x| = ±x\)

Think of x as (1 - √13), and remember to check your final results. You want the expression for x that yields a positive value (the "principal square root").

By the way, the radical sign in your result is not correct.

?
 
First, the answer here is wrong! You wrote the radical incorrectly.

The original problem is asking for the positive number whose square is [imath](1-\sqrt{13})^2[/imath]. You might at first think the answer is obviously [imath]1-\sqrt{13}[/imath]. But there are two numbers that have the same square (one positive and one negative); for example, both 3 and -3 have their square equal to 9, and the square root of 9 is the positive one, 3. In your case, the other number is [imath]\sqrt{13}-1[/imath].

You need to determine which of the two is positive, and it turns out to be the latter.

A better way to show the work, in my mind, would be to use the formula [imath]\sqrt{x^2}=|x|[/imath]. Can you see why this would be true regardless of whether x is positive or negative? So you'd write [imath]\sqrt{(1-\sqrt{13})^2} = |1-\sqrt{13}| = \sqrt{13}-1[/imath].
 
Dr.Peterson said:
First, the answer here is wrong! You wrote the radical incorrectly.

The original problem is asking for the positive number whose square is [imath](1-\sqrt{13})^2[/imath]. You might at first think the answer is obviously [imath]1-\sqrt{13}[/imath]. But there are two numbers that have the same square (one positive and one negative); for example, both 3 and -3 have their square equal to 9, and the square root of 9 is the positive one, 3. In your case, the other number is [imath]\sqrt{13}-1[/imath].

You need to determine which of the two is positive, and it turns out to be the latter.

A better way to show the work, in my mind, would be to use the formula [imath]\sqrt{x^2}=|x|[/imath]. Can you see why this would be true regardless of whether x is positive or negative? So you'd write [imath]\sqrt{(1-\sqrt{13})^2} = |1-\sqrt{13}| = \sqrt{13}-1[/imath].
Click to expand...
Okay got it! Thanks
 
Otis said:
\(\displaystyle \sqrt{x^2} = |x| = ±x\)
Click to expand...


\(\displaystyle \sqrt{x^2} \ = \ |x|\)

\(\displaystyle |x| \ \ne \ \pm x \)


For example, \(\displaystyle \ |3| \ \ne \ \pm 3.\)

\(\displaystyle |3| \ \ just \ \ equals \ \ 3.\)


For another example, \(\displaystyle \ |-3| \ \ne \ \pm 3.\)

\(\displaystyle |-3| \ \ just \ \ equals \ \ 3.\)
 
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lookagain said:
\(\displaystyle |x| \ \ne \ \pm x \)
Click to expand...
Hi lookagain. In the phrase "plus or minus", how do you interpret the meaning of "or"?

I interpret ±x to mean: Either quantity x is negated OR it is not. So, for me, the ± symbol is an abbreviation of a conditional statement.

|x| = x
OR
|x| = -x

That conditional statement may be abbreviated as:

|x| = ±x

[imath]\;[/imath]
 
Otis said:
Hi lookagain. In the phrase "plus or minus", how do you interpret the meaning of "or"?

I interpret ±x to mean: Either quantity x is negated OR it is not. So, for me, the ± symbol is an abbreviation of a conditional statement.

|x| = x
OR
|x| = -x

That conditional statement may be abbreviated as:

|x| = ±x
Click to expand...

You have to interpret it as two different values, unless x = 0, just as the plus or minus symbol is used in the
Quadratic Formula.
 
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