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Do a prime factorization of the two numbers. Use condensed (power) notation for each prime factor.
2 is a factor in N1 and N2, the smallest power of 2 is 4, thus 2^4 is common to N1, and N1. You could write 2^5=2*2^4
11 is a factor in N1, and N2, the smallest exponent of 11 is 1, thus 11 is a common factor of N1 and N2.
Highest common factor of N1 and N2 is (2^4)*11
The current area of the pond is 18m by 12 m, so, if A is the current area, A=18*12
Twice the area would be 2A and it is to be accomplished by adding the same amount, say x, to both dimensions. In symbols, that is:
2A = (18+x) * (12+x)
But we know A is 18*12, so
2(18*12) = (18+x) * (12+x) ; doing the arithmetic
432 = (18+x) * (12+x) ; doing the multiplication using FOIL (first, outside, inside, last)
432 = 18*12 + 18x +12x + x*x ; simplying
432 = 216 + 30x + x*x ; or
0 = -216 + 30x + x*x : or
x*x + 30x -216 = 0 ; using the quadratic equation (or factoring)
(x-6) (x+36) = 0
This has two solutions, 6 and (-36). Since a negative number doesn't make sense in this case, the answer is to add 6m to each side.
Check (18+6) (12+6) = ?
24 * 18 = ?
432 , which is indeed twice the original area of 216 ( = 18*12)