Answer:
[tex]2a - 3b + 4c = 1[/tex]
Step-by-step explanation:
Given
[tex]a^2 + b^2 + c^2 = 2(a - b - c) - 3[/tex]
Required
Determine [tex]2a - 3b + 4c[/tex]
[tex]a^2 + b^2 + c^2 = 2(a - b - c) - 3[/tex]
Open bracket
[tex]a^2 + b^2 + c^2 = 2a - 2b - 2c - 3[/tex]
Equate the equation to 0
[tex]a^2 + b^2 + c^2 - 2a + 2b + 2c + 3 = 0[/tex]
Express 3 as 1 + 1 + 1
[tex]a^2 + b^2 + c^2 - 2a + 2b + 2c + 1 + 1 + 1 = 0[/tex]
Collect like terms
[tex]a^2 - 2a + 1 + b^2 + 2b + 1 + c^2 + 2c + 1 = 0[/tex]
Group each terms
[tex](a^2 - 2a + 1) + (b^2 + 2b + 1) + (c^2 + 2c + 1) = 0[/tex]
Factorize (starting with the first bracket)
[tex](a^2 - a -a + 1) + (b^2 + 2b + 1) + (c^2 + 2c + 1) = 0[/tex]
[tex](a(a - 1) -1(a - 1)) + (b^2 + 2b + 1) + (c^2 + 2c + 1) = 0[/tex]
[tex]((a - 1) (a - 1)) + (b^2 + 2b + 1) + (c^2 + 2c + 1) = 0[/tex]
[tex]((a - 1)^2) + (b^2 + 2b + 1) + (c^2 + 2c + 1) = 0[/tex]
[tex]((a - 1)^2) + (b^2 + b+b + 1) + (c^2 + 2c + 1) = 0[/tex]
[tex]((a - 1)^2) + (b(b + 1)+1(b + 1)) + (c^2 + 2c + 1) = 0[/tex]
[tex]((a - 1)^2) + ((b + 1)(b + 1)) + (c^2 + 2c + 1) = 0[/tex]
[tex]((a - 1)^2) + ((b + 1)^2) + (c^2 + 2c + 1) = 0[/tex]
[tex]((a - 1)^2) + ((b + 1)^2) + (c^2 + c+c + 1) = 0[/tex]
[tex]((a - 1)^2) + ((b + 1)^2) + (c(c + 1)+1(c + 1)) = 0[/tex]
[tex]((a - 1)^2) + ((b + 1)^2) + ((c + 1)(c + 1)) = 0[/tex]
[tex]((a - 1)^2) + ((b + 1)^2) + ((c + 1)^2) = 0[/tex]
Express 0 as 0 + 0 + 0
[tex](a - 1)^2 + (b + 1)^2 + (c + 1)^2 = 0 + 0+ 0[/tex]
By comparison
[tex](a - 1)^2 = 0[/tex]
[tex](b + 1)^2 = 0[/tex]
[tex](c + 1)^2 = 0[/tex]
Solving for [tex](a - 1)^2 = 0[/tex]
Take square root of both sides
[tex]a - 1 = 0[/tex]
Add 1 to both sides
[tex]a - 1 + 1 = 0 + 1[/tex]
[tex]a = 1[/tex]
Solving for [tex](b + 1)^2 = 0[/tex]
Take square root of both sides
[tex]b + 1 = 0[/tex]
Subtract 1 from both sides
[tex]b + 1 - 1 = 0 - 1[/tex]
[tex]b = -1[/tex]
Solving for [tex](c + 1)^2 = 0[/tex]
Take square root of both sides
[tex]c + 1 = 0[/tex]
Subtract 1 from both sides
[tex]c + 1 - 1 = 0 - 1[/tex]
[tex]c = -1[/tex]
Substitute the values of a, b and c in [tex]2a - 3b + 4c[/tex]
[tex]2a - 3b + 4c = 2(1) - 3(-1) + 4(-1)[/tex]
[tex]2a - 3b + 4c = 2 +3 -4[/tex]
[tex]2a - 3b + 4c = 1[/tex]
