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Addition and Subtraction of Algebraic Expression

a What should...

Question

(a) What should be added to $2 x^{2} + 6 x - 5$ to get $3 x^{2} - 2 x + 6$ ?
(b) What must be subtracted from $a^{2} + b^{2} + c^{2} - 3 a b c$ to get $2 a^{2} - b^{2} - 3 c^{2} + a b c$

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Solution

(I). In order to get $(3 x^{2} - 2 x + 6) f r o m (2 x^{2} + 6 x - 5)$ we need to:
$(3 x^{2} - 2 x + 6) - (2 x^{2} + 6 x - 5)$
$= 3 x^{2} - 2 x + 6 - 2 x^{2} - 6 x + 5$
$= x^{2} - 4 x + 11$

(ii) In order to get $(2 a^{2} - b^{2} - 3 c^{2} + a b c) f r o m (a^{2} + b^{2} + c^{2} - 3 a b c)$ we need to:
$(a^{2} + b^{2} + c^{2} - 3 a b c) - (2 a^{2} - b^{2} - 3 c^{2} + a b c)$
$= a^{2} + b^{2} + c^{2} - 3 a b c - 2 a^{2} + b^{2} + 3 c^{2} - a b c$

$= a^{2} + 2 b^{2} + 4 c^{2} - 4 a b c$

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