Help, please?

I'd love it if you showed me the steps. I want to use it for future reference.

Solve:
x^2 + 24^2 = (x + 8)^2

Question

Help, please?

I'd love it if you showed me the steps. I want to use it for future reference.

Solve:
x^2 + 24^2 = (x + 8)^2

whpalmer4 · Answer

$$x^2+24^2 = (x+8)^2$$I think the first step would be to expand the right hand side.  Can you multiply $$(x+8)^2 = (x+8)(x+8) =$$and report back with the answer?

anonymous · Answer

x^2 + 576 = x^2 + 16x + 64 (expanding terms)
Both sides have x^2 so they cancel
576 = 16x + 64 (cancelled out x^2's)
16x = 512 (subtract 576 from both sides)
x = 32 (divide by 16 to isolate x)

anonymous · Answer

Oh, my gosh. I feel so dumb now. I knew you had to factor out (x + 8)^2 but my friend confused me saying you had to do quadratic formula in this problem and that's where I got stuck because x^2 cancelled each other out. Thank you so much for helping me. (:

whpalmer4 · Answer

Yeah, no need for the quadratic formula here.  

A clever way you could do this would be to recognize that this has the same general form as the Pythagorean theorem: $$a^2+b^2=c^2$$

24 = 8*3 and 3 is the middle value of the Pythagorean triple $3,4,5$ so we could immediately guess that a solution might be $8*3, 8*4, 8*5$ which would allow us to have $x=8*4=32$ and $x+8 = 8*5= 40$ as the other two values...

$$24^2+32^2 = 40^2$$$$576+1024 = 1600$$

In case you have never thought of it in these terms, you can multiply a Pythagorean triple by any factor to scale it up:

$$a^2+b^2= c^2$$$$(na)^2+(nb)^2=(nc)^2$$$$n^2a^2+n^2b^2=n^2c^2$$$$n^2(a^2+b^2) = n^2*c^2$$and of course that's equivalent to $$a^2+b^2 = c^2$$

whpalmer4 · Answer

er, sorry, 3 is the first value of the Pythagorean triple, not the middle value.  

Anyhow, multiplying 3,4,5 by 8 spaces the values out to 24, 32, 40, and if x = 32, x+8 = 40, which is what we want.

whpalmer4 · Answer

" had to factor out (x + 8)^2 "

that's incorrect terminology, as you are not factoring, you are expanding (by multiplication).  An example of factoring out would be this:
$$x^3+2x^2+6x = x(x^2+2x+6)$$where you have "factored out" a factor of $x$.

Math is easier when you use the terminology the same way as everyone else :-)