find the distance & completing the square

[unregistered]

Hi all, I've been trying to figure out the best way to solve this:

Find x such that the distance between the points (3, -4), (x, 5) is 15

The book shows this:

√(x - 3)^2 + (5 - (-4))^2 = 15

(so they're getting rid of the square root on both sides I guess?)
(x - 3)^2 + 81 = 225

(then they move 81 to the other side, not sure why, are they trying to isolate x?)
(x - 3)^2 = 144

This is where I really get lost and I don't know how it got to this point:
x = 3 +12 = (-9, 15)

From what I know of the distance formula, I started to do it this way:

√(x - 3)^2 + (5 - (-4))^2 = 15
√(x - 3)^2 + (5 + 4)^2 = 15
√(x + 3)^2 + (9)^2 = 15
√(x + 3)^2 + 81= 15
√(x + 3)^2 + 9 = 15

This is what I can wrap my brain around. I looked up completing the square and it shows this formula for (x + 3)^2:

x2 + 2(3)x + 3^2

Thus

√x^2 + 6x + 9

Adding this with the rest of the equation I have:

√x^2 + 6x + 9 + 9 = 15

and if I were to try to combine and solve I can see that I'm not going to get the right answer. Is there a better way to solve this, like a rule by rule way so that each time I try to solve it I can just follow each rule to get the best way to solve. I guess I need real breakdown b/c I'm getting lost towards the end.

If anyone has the time I would be really really grateful,

Thank-you

 

[galactus]

Find x such that the distance between the points (3, -4), (x, 5) is 15

The book shows this:

√(x - 3)^2 + (5 - (-4))^2 = 15

(so they're getting rid of the square root on both sides I guess?)
(x - 3)^2 + 81 = 225

(then they move 81 to the other side, not sure why, are they trying to isolate x?).

yep

\(\displaystyle \underbrace{(x - 3)^{2}} = \underbrace{144}\)
......\(\displaystyle \nwarrow\)......\(\displaystyle \uparrow\)

Square root of both sides. x-3=12, x=15

Check \(\displaystyle \sqrt{(15-3)^{2}+(5+4)^{2}}\Rightarrow\sqrt{12^{2}+9^{2}}\Rightarrow\sqrt{225}=15\)

 

[unregistered]

Hey thanks galactus, I think I'm closer yet, but I can't seem to figure out:

√(x + 3)^2

Or how to break that down. I've read several things on completing the square but it never gets me to the result I need.

Thanks again for your help.

 

[galactus]

Hey thanks galactus, I think I'm closer yet, but I can't seem to figure out:

√(x + 3)^2

Or how to break that down. I've read several things on completing the square but it never gets me to the result I need.

Thanks again for your help.

My goodness, the square root and square cancel one another.

It's just x+3.

 

[unregistered]

yikes, sorry about that, thanks.