If P=a2−2bc+b2, Q=−b2+bc−c2 and R=c2+cb−a2 then, find the value of P+Q+R.
Answers
Answer:
\begin{gathered}\underline{\textsf{Given , }} \\ \\ \sf \implies P \: = \: a^2 \: - \: 2bc \: + \: b^2 \quad...(1) \\ \\ \textsf{And,} \\ \\ \sf \implies Q \: = \: -b^2 \: + \: bc \: - \: c^2 \quad...(2) \\ \\ \textsf{And,} \\ \\ \sf \implies R \: = \: c^2 \: + \: cb \: + \: a^2 \quad...(3) \end{gathered}
Given ,
⟹P=a
2
−2bc+b
2
...(1)
And,
⟹Q=−b
2
+bc−c
2
...(2)
And,
⟹R=c
2
+cb+a
2
...(3)
\begin{gathered}\underline{\textsf{Add all the equations , }} \\ \\ \sf \implies P \: + \: Q \: + \: R \: = \: a^2 \: - \: 2bc \: + \: \cancel{ b^2 }\: - \: \cancel{b^2} \: + \: bc \\ \sf \qquad \qquad \qquad \quad \: \: \: \: \: \: \: - \: \cancel{c^2} \: + \: \cancel{c^2} \: + \: bc \: + \: a^2 \\ \\ \sf \implies P \: + \: Q \: + \: R \: = \: 2 {a}^{2} \: + \: \cancel{2bc} \: - \: \cancel{2bc} \\ \\ \sf \: \: \therefore \: \: P \: + \: Q \: + \: R \: = \: 2 {a}^{2} \end{gathered}
Add all the equations ,
⟹P+Q+R=a
2
−2bc+
b
2
−
b
2
+bc
−
c
2
+
c
2
+bc+a
2
⟹P+Q+R=2a
2
+
2bc
−
2bc
∴P+Q+R=2a
2