Practice the questions given in the worksheet on expanding of (a ± b ± c)\(^{2}\) and its corollaries.
1. Expand the squares of the following trinomials.
(i) a + 2b + 3c
(ii) 2x + 3y + 4z
(iii) x + 2y – 3z
(iv) 3a – 4b – c
(v) 1 – x - \(\frac{1}{x}\)
(vi) 1 – a – a\(^{2}\)
2. Simplify:
(i) (x + y + z)\(^{2}\) + (x – y + z)\(^{2}\)
(ii) (a – 2b – 3c)\(^{2}\) + (2a + 3b – c)\(^{2}\) + (3a – b +2c)\(^{2}\)
3. (i) If a + b + c = 6 and ab + bc + ca = 11, find a\(^{2}\) + b\(^{2}\) + c\(^{2}\).
(ii) a + b – c = 7 and ab – bc – ca = 14, find a\(^{2}\) + b\(^{2}\) + c\(^{2}\).
[Hint: a\(^{2}\) + b\(^{2}\) + c\(^{2}\) = a\(^{2}\) + b\(^{2}\) + (-c)\(^{2}\)
= (a + b – c)\(^{2}\) – 2{ab + b(-c) + (-c)a}
= (a + b –c)\(^{2}\) – 2(ab – bc – ca)
= 7\(^{2}\) – 2 × 14]
(iii) If a\(^{2}\) + b\(^{2}\) + c\(^{2}\) = 50 and ab + bc + ca = 47, find a + b + c.
4. If x + y + z = 12 and x\(^{2}\) + y\(^{2}\) + z\(^{2}\) = 44, find xy + yz + zx.
Answers for the worksheet on on expanding of (a ± b ± c)\(^{2}\) and its corollaries are given below.
Answers
1. (i) a\(^{2}\) + 4b\(^{2}\) + 9c\(^{2}\) + 4ab + 12bc + 6ca
(ii) 4x\(^{2}\) + 9y\(^{2}\) + 16z\(^{2}\) + 12xy + 24yz + 16zx
(iii) x\(^{2}\) + 4y\(^{2}\) + 9z\(^{2}\) + 4xy – 12yz – 6zx
(iv) 9a\(^{2}\) + 16b\(^{2}\) + c\(^{2}\) – 24ab + 8bc – 6ca
(v) x\(^{2}\) + \(\frac{1}{x^{2}}\) – 2x + \(\frac{2}{x}\) - 1
(vi) 1 – 2a – a\(^{2}\)+ 2a\(^{3}\) + a\(^{4}\)
2. (i) 2x\(^{2}\) + 2y\(^{2}\) + 2z\(^{2}\) – 4yz
(ii) 14a\(^{2}\) + 14b\(^{2}\)+ 14c\(^{2}\) +2ab + 2bc + 2ca
3. (i) 14
(ii) 21
(iii) ±12
4. 50.
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