Don't find a
Given that
a
2
+
2
a
c
+
2
b
c
−
b
2
=
1
0
9
2
a
+
2
c
−
b
=
7
Find
a
+
b
The answer is 156.
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3 solutions
a^2+2ac+2bc-b^2=(a^2+2ac+c^2)+(ac-ab+c^2-bc)-(ac+c^2-ab-bc)-(c^2-2bc+b^2)=(a+c)^2+(a+c)(c-b)-(a+c)(c-b)-(c-b)^2=((a+c)-(c-b))((a+c)+(c-b))=(a+b)(a+2c-b) Therefore, a+b=1092/7=156
Much simpler:
a 2 − b 2 + 2 a c + 2 b c = ( a + b ) ( a − b ) + 2 c ( a + b )
= ( a + b ) ( a − b + 2 c ) = ( a + b ) ( ( 7 − 2 c ) + 2 c )
= 7 ( a + b ) = 1 0 9 2 ⟹ a + b = 1 5 6
By factorising the first expression: a^2 + 2ac + 2bc - b^2 = 1092 (a + b)(a - b) + 2c( a + b ) = 1092 Hence, (a + b) ( a - b + 2c) = 1092 Notice how a - b + 2c is just the same as the second expression which is known to have the value 7. Now we can simply divide both sides by 7 to obtain the result a + b = 156
a 2 + 2 a c + c 2 − ( b 2 − 2 b c + c 2 ) = 1 0 9 2
= ( a + c ) 2 − ( b − c ) 2 = 1 0 9 2
= ( a + b ) ( a + 2 c − b ) = 1 0 9 2
thus
a + b = 1 5 6