Question 1111814: If a+b=x and a-b=y, find the value of
a) a^2 - b^2
b) ab
Found 2 solutions by duckness73, ikleyn:
Answer by duckness73(47)   (Show Source):
You can put this solution on YOUR website! We know a + b = x and a - b = y
a) a^2 - b^2 can be factored into (a + b)(a - b). Using the substitution from above, then a^2 - b^2 = (a + b)(a - b) = xy
b) To determine the product of a and b, that is, ab, we need to express a or b in terms of x and y alone. Let's start with a:
Since a + b = x and
a - b = y let's add the equations together:
(a + b) + (a - b) = x + y
Combining like terms, we have
2a = x + y
Dividing both sides by 2, we have
a = (x + y)/2
Now, let's look at expressing b in terms of x and y alone. Looking at the first equation that was given to us (a + b = x) we can solve for b:
a + b = x
Subtracting a from both sides:
b = x - a
But, we know from the first part of this problem that a = (x + y)/2. So, let's make that substitution in the above equation:
b = x - [(x + y)/2]
Now, it becomes an exercise in combining like terms:
b = x - (x/2) - (y/2)
b = (x/2) - (y/2)
b = (x - y)/2
Now we have both a and b expressed in terms of x and y alone, so the product ab becomes:
ab = [(x + y)/2][(x - y)/2]
= (x + y)(x - y)/4
= (x^2 - y^2)/4
Answer by ikleyn(52781)  (Show Source):
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