Manipulation of negative square root of a negative term/#

Suppose I have to solve for y:

$\displaystyle x\leq 1$

$\displaystyle (x - 1)^{2} = y$

So I know that (x - 1) will always be 0 or a negative, therefore I must take the negative square root of (x - 1):

$\displaystyle -\sqrt{(x - 1)^{2}} = -\sqrt{y}$

Am I to understand that this is the same as:

$\displaystyle -1 \cdot \sqrt{(x - 1)^{2}} = -1 \cdot \sqrt{y}$
$\displaystyle =$
$\displaystyle -1 \cdot (x - 1) = -1 \cdot \sqrt{y}$
$\displaystyle =$
$\displaystyle -x + 1 = -1 \cdot \sqrt{y}$

Or how does this work?

Quote:

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Originally Posted by daigo

Suppose I have to solve for y:
$\displaystyle x\leq 1$
$\displaystyle (x - 1)^{2} = y$
So I know that (x - 1) will always be 0 or a negative, therefore I must take the negative square root of (x - 1):
$\displaystyle -\sqrt{(x - 1)^{2}} = -\sqrt{y}$
Am I to understand that this is the same as:
$\displaystyle -1 \cdot \sqrt{(x - 1)^{2}} = -1 \cdot \sqrt{y}$
$\displaystyle =$
$\displaystyle -1 \cdot (x - 1) = -1 \cdot \sqrt{y}$
$\displaystyle =$
$\displaystyle -x + 1 = -1 \cdot \sqrt{y}$

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It is a bit awkward, and you missed a sign.
You you could have done this.
$\displaystyle (x - 1)^{2} = y \Rightarrow~|x-1|=\sqrt{y}$
But $\displaystyle x\le 1 \Rightarrow~1-x=\sqrt{y}$

Hello, daigo!

Quote:

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$\displaystyle \text{Solve for }y\!:\;\;\begin{Bmatrix}x\:\leq\: 1 & [1] \\ (x - 1)^2 \:=\: y & [2]\end{Bmatrix}$

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From [1]: .$\displaystyle x\,\le\,1 \quad\Rightarrow\quad x - 1\,\le\,0 \quad\Rightarrow\quad (x-1)^2\,\ge\,0$

Substitute into [2]: .$\displaystyle y \:=\:(x-1)^2 \:\ge\:0$

. . Therefore: .$\displaystyle y \:\ge\:0$

I don't really want to focus on solving the problem, I just need some insight on the negative square root operation.

I'm trying to understand the way it was done here:

$\displaystyle y = (x - 1)^{2}$

$\displaystyle -\sqrt{y} = -\sqrt{(x - 1)^{2}}$

$\displaystyle -\sqrt{y} = x - 1$

$\displaystyle -\sqrt{y} + 1 = x$

Is this correct or incorrect?

Incorrect.

Quote:

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Originally Posted by daigo

Suppose I have to solve for y:

$\displaystyle x\leq 1$

$\displaystyle (x - 1)^{2} = y$

So I know that (x - 1) will always be 0 or a negative, therefore I must take the negative square root of (x - 1):

$\displaystyle -\sqrt{(x - 1)^{2}} = -\sqrt{y}$

Am I to understand that this is the same as:

$\displaystyle -1 \cdot \sqrt{(x - 1)^{2}} = -1 \cdot \sqrt{y}$
$\displaystyle =$
$\displaystyle -1 \cdot (x - 1) = -1 \cdot \sqrt{y}$
$\displaystyle =$
$\displaystyle -x + 1 = -1 \cdot \sqrt{y}$

Or how does this work?

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In this problem, it does not matter what value of x is because the quantity $\displaystyle (x-1)^2$ will always be positive. So $\displaystyle y\geq0$.

Quote:

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Originally Posted by thevinh

In this problem, it does not matter what value of x is because the quantity $\displaystyle (x-1)^2$ will always be positive. So $\displaystyle y\geq0$.

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That may be a problem. But it is not the problem which is the OP refuses to see that:
$\displaystyle \sqrt{(x-1)^2}=|x-1|$.

That is the source of all this confusion.

Is there anything wrong with this.
(x-1)= + or -root y So rooty=(x-1) or (1-x) For example if x=-2 y=9 and root y =+3 or -3

Can I join in ? What's wrong with this ?

$\displaystyle x\leq1,$ so either $\displaystyle x-1<0$ or $\displaystyle x-1=0.$

If $\displaystyle x-1=0$ then $\displaystyle y=0,$ so concentrate on the first case.

If $\displaystyle (x-1)^{2}=y,$ then $\displaystyle x-1=\pm\sqrt{y},$ ($\displaystyle y$ is positive), but we know that $\displaystyle x-1<0$ so $\displaystyle x-1=-\sqrt{y}.$