SOLUTION: If a-b=-4 and a+c=4, find the value of {{{ 2a^2-b^2-c^2 }}}

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Question 1105538: If a-b=-4 and a+c=4, find the value of
Found 2 solutions by jim_thompson5910, ikleyn:
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!

Equation (1): a-b=-4
Equation (2): a+c=4

Solve equation (2) for 'a'
a+c=4
a+c-c=4-c
a = 4-c
Call the result equation (3)

Plug this equation (3) into equation (1) and solve for b
a-b = -4
4-c-b = -4
4-c-b-4 = -4-4
-b-c = -8
-1*(-b-c) = -1*(-8)
b+c = 8
b+c-c = 8-c
b = 8-c
Call the result equation (4)

-----------------------------------

To summarize so far, we found these equations
Equation (3): a = 4-c
Equation (4): b = 8-c

Plug them into the expression 2a^2-b^2-c^2 and simplify
2a^2-b^2-c^2 = 2(4-c)^2-(8-c)^2-c^2
2a^2-b^2-c^2 = 2(16-8c+c^2)-(64-16c+c^2)-c^2
2a^2-b^2-c^2 = 2(16)+2(-8c)+2(c^2)-1(64)-1(-16c)-1(c^2)-c^2
2a^2-b^2-c^2 = 32-16c+2c^2-64+16c-c^2-c^2
2a^2-b^2-c^2 = (2c^2-c^2-c^2)+(-16c+16c)+(32-64)
2a^2-b^2-c^2 = (0c^2)+(0c)+(-32)
2a^2-b^2-c^2 = -32 which is the final answer
Answer by ikleyn(52908)   (Show Source): You can put this solution on YOUR website!
.
1.  a - b = -4    (1)
    a + c =  4    (2)

               ==================>  subtract (1) from (2) to get

    c + b = 8.    (3)


2.   =  +  = 

    = (a-b)*(a+b) + (a-c)*(a+c) =  (replace here a-b  with -4  in accordance with (1) )
                                   (replace      a+c  with  4  in accordance with (2) )

    = -4*(a+b) +4*(a-c) = 4*(a -c - a - b) = = -4*(c + b)  =  (replace here c+b  with 8  in accordance with (3) )

    = -4*8 = -32.

Solved.


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