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Class-9»Maths

If a, b, c are real and a² + b² + c² = 2(a-b-c) -3 , then find the value of 2a - 3b + 4c. Also show the method please.

$a^{2} + b^{2} + c^{2} = 2 (a - b - c) - 3 \Rightarrow a^{2} + b^{2} + c^{2} - 2 (a - b - c) + 3 = 0 \Rightarrow a^{2} + b^{2} + c^{2} - 2 a + 2 b + 2 c + 3 = 0 \Rightarrow (a^{2} - 2 a + 1) + (b^{2} + 2 b + 1) + (c^{2} + 2 c + 1) = 0 \Rightarrow {(a - 1)}^{2} + {(b + 1)}^{2} + {(c + 1)}^{2} = 0 T h i s i s p o s s i b l e o n l y w h e n {(a - 1)}^{2} = 0, {(b + 1)}^{2} = 0, a n d {(c + 1)}^{2} = 0 \Rightarrow a = 1, b = - 1, a n d c = - 1 T h u s, 2 a - 3 b + 4 c = 2 \times 1 - 3 \times (- 1) + 4 \times (- 1) = 2 + 3 - 4 = 1 H e n c e, 2 a - 3 b + 4 c = 1$

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