Question
Simplify: $(a + b + c)^2 + (a - b + c)^2$
✓
Answer
In the given problem, we have to simplify the expressions
Given $(a + b + c)^2 + (a - b + c)^2$ By using identity $(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx$
Hence the equation becomes $\big(\text{a}+\text{b}+\text{c}\big)+\big(\text{a}-\text{b}+\text{c}\big)$
$=\Big[\text{a}^2+\text{b}^2+\text{c}^2+2\text{ab}+2\text{bc}+2\text{ca}\Big]$
$+\Big[\text{a}^2+(-\text{b})^2+\text{c}^2+2\text{a}(-\text{b})+2(-\text{b})(\text{c})+2\text{ca}\Big]$
$=\text{a}^2+\text{b}^2+\text{c}^2+2\text{ab}+2\text{bc}+2\text{ca}$
$+\text{a}^2+\text{b}^2+\text{c}^2-2\text{ab}-2\text{bc}+2\text{ca}$
$=\text{a}^2+\text{a}^2+\text{b}^2+\text{b}^2+\text{c}^2+\text{c}^2$
$+2\text{ab}-2\text{ab}+2\text{bc}-2\text{bc}+2\text{ca}+2\text{ca}$
$=2\text{a}^2+2\text{b}^2+2\text{c}^2+4\text{ca}$
Talking $2$ as common factor we get $=2\big(\text{a}^2+\text{b}^2+\text{c}^2+2\text{ca}\big)$
Hence the simplified value of $\big(\text{a}+\text{b}+\text{c}\big)^2+\big(\text{a}-\text{b}+\text{c}\big)^2$ is $2\big(\text{a}^2+\text{b}^2+\text{c}^2+2\text{ca}\big).$
Given $(a + b + c)^2 + (a - b + c)^2$ By using identity $(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx$
Hence the equation becomes $\big(\text{a}+\text{b}+\text{c}\big)+\big(\text{a}-\text{b}+\text{c}\big)$
$=\Big[\text{a}^2+\text{b}^2+\text{c}^2+2\text{ab}+2\text{bc}+2\text{ca}\Big]$
$+\Big[\text{a}^2+(-\text{b})^2+\text{c}^2+2\text{a}(-\text{b})+2(-\text{b})(\text{c})+2\text{ca}\Big]$
$=\text{a}^2+\text{b}^2+\text{c}^2+2\text{ab}+2\text{bc}+2\text{ca}$
$+\text{a}^2+\text{b}^2+\text{c}^2-2\text{ab}-2\text{bc}+2\text{ca}$
$=\text{a}^2+\text{a}^2+\text{b}^2+\text{b}^2+\text{c}^2+\text{c}^2$
$+2\text{ab}-2\text{ab}+2\text{bc}-2\text{bc}+2\text{ca}+2\text{ca}$
$=2\text{a}^2+2\text{b}^2+2\text{c}^2+4\text{ca}$
Talking $2$ as common factor we get $=2\big(\text{a}^2+\text{b}^2+\text{c}^2+2\text{ca}\big)$
Hence the simplified value of $\big(\text{a}+\text{b}+\text{c}\big)^2+\big(\text{a}-\text{b}+\text{c}\big)^2$ is $2\big(\text{a}^2+\text{b}^2+\text{c}^2+2\text{ca}\big).$
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