Question 1168999: Factor,
a^2 - 14a + 49 - b^2
I know you can use difference of squares but i have no idea how to do it
Found 4 solutions by ikleyn, Theo, Alan3354, MathTherapy:
Answer by ikleyn(52832)   (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! the general formula for difference of squares is:
a^2 - b^2 = (a - b) * (a + b)
a is the square root of a^2 and b is the square root of b^2.
because of that, the general form for the equation of the difference of squares could be written as:
a^2 - b^2 = (sqrt(a^2) - sqrt(b^2)) * (sqrt(a^2) + sqrt(b^2))
since the square root of b^2 is equal to b, then it could be written as:
a^2 - b^2 = (sqrt(a^2) - b) * (sqrt(a^2) + b)
in this problem, b is left as is and a is replaced by sqrt(a^2 - 14a + 49), so the equation becomes:
sqrt(a^2 - 14a + 49))^2 - b^2 = (sqrt(a^2 - 14a + 49) - b) * (sqrt(a^2 - 14a + 49) + b)
what you need to do is find the square root of a^2 - 14a + 49.
fortunately, sqrt(a^2 - 14a + 49) is equal to a-7, because (a-7)^2 = a^2 - 14a + 49.
the equation of:
sqrt(a^2 - 14a + 49))^2 - b^2 = (sqrt(a^2 - 14a + 49) - b) * (sqrt(a^2 - 14a + 49) + b) becomes:
(a - 7)^2 - b^2 = (a - 7 + b) * (a - 7 - b)
to confirm this is accurate, you would perform the indicated operations to get:
(a - 7 + b) * (a - 7 - b) equals:
a * (a - 7 - b) - 7 * (a - 7 - b) + b * (a - 7 - b) which equals:
a^2 - 7a - ab -7a + 49 + 7b + ab - 7b - b^2.
group like terms together to get:
a^2 - 7a - 7a - ab + ab + 49 + 7b - 7b - b^2
combine like terms to get:
a^2 - 14a + 49 - b^2
that's the same as your original expression, so we're good.
your solution is:
a^2 - 14a + 49 - b^2 is equivalent to:
(a - 7 + b) * (a - 7 - b)
Answer by Alan3354(69443)  (Show Source):
Answer by MathTherapy(10555)  (Show Source):
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