SOLUTION: Factor, a^2 - 14a + 49 - b^2 I know you can use difference of squares but i have no idea how to do it

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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(52834)   (Show Source): You can put this solution on YOUR website!

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): You can put this solution on YOUR website!
Factor,
a^2 - 14a + 49 - b^2
=====================
= (a-7)^2 - b^2
= (a-7+b)*(a-7-b)

Answer by MathTherapy(10555)   (Show Source): You can put this solution on YOUR website!

Factor,
a^2 - 14a + 49 - b^2
I know you can use difference of squares but i have no idea how to do it
You need to 1st group the expression as follows since you want to have a perfect square INTEGER/VARIABLE to the UTMOST right-side of the expression: 
Now, we use the "ac" method for the first grouped expression, which involves finding 2 INTEGERS that when MULTIPLIED gives us "+ 49", and when added, gives us "- 14."
These FACTORS are simply - 7 and - 7. We then get: , and finally:
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