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Question 410486: Hi, Please help me solve this math problem...
Directions are :factor each expression as a difference of squares.
4(x+y)^2 -(2y-z)^2
my teacher told us to substitute, so i did : a= xty,b= 2y-z
4(a)^2-(b)^2
well i'm not sure i got the right answer and i'm still not 100% sure on what to do so if you could please try to explain it to me I would greatly appreciate it, Thank you XD
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! difference of squares:
a^2 - b^2 = (a-b) * (a+b)
equation is:
4*(x+y)^2 - (2y-z)^2
let a = 4*(x+y)^2 = 2^2*(x+y)^2 = (2*(x+y))^2 = (2x+2y)^2
let b = (2y-z)
equation becomes:
a^2 - b^2
which is the difference of squares which is equivalent to:
(2x+2y)^2 - (2y-z)^2
a^2 - b^2 = (a-b) * (a+b)
this is equivalent to:
(2x+2y)^2 - (2y-z)^2 = ((2x+2y) - (2y-z)) * ((2x+2y) + (2y-z))
because:
(2x+2y) = a
and:
(2y-z) = b
this should be the answer you are looking for:
(2x+2y)^2 - (2y-z)^2 = ((2x+2y) - (2y-z)) * ((2x+2y) + (2y-z))
you should now be able to solve as follows:
(2x+2y)^2 - (2y-z)^2 = ((2x+2y) - (2y-z)) * ((2x+2y) + (2y-z))
simplify expression on right of equal sign to get:
(2x+2y)^2 - (2y-z)^2 = (2x+2y -2y + z)) * (2x+2y + 2y-z)
simplify further to get:
(2x+2y)^2 - (2y-z)^2 = (2x+z)) * (2x+4y-z)
simplify further by multiplying expression on right of equal of sign to get:
(2x+2y)^2 - (2y-z)^2 = 4x^2 + 8xy + 4yz - z^2
this would be your answer after you've simplified as far as you can go:
(2x+2y)^2 - (2y-z)^2 = 4x^2 + 8xy + 4yz - z^2
you would confirm by substituting random numbers for x, y, and z.
I used x = 3, y = 4, z = 5 and I got:
187 = 187
this confirmed it for me.
I substituted in the original expression of:
4*(x+y)^2 - (2y-z)^2 to get 187
I then substituted in the final expression of:
4x^2 + 8xy + 4yz - z^2 to also get 187
the key was to get the original expression to look like a^2 - b^2
that was done up top when I did the following:
let a = 4*(x+y)^2 = 2^2*(x+y)^2 = (2*(x+y))^2 = (2x+2y)^2
let b = (2y-z)
the "a" part was the tricky part.
you needed to know that 4 = 2^2 and that 2^2 * (x+y)^2 = (2*(x+y)^2 = (2x+2y)^2
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