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Expansion of (a ± b ± c)2
We will discuss here about the expansion of (a ± b ± c)2.
(a + b + c)2 = {a + (b + c)}2 = a2 + 2a(b + c) + (b + c)2
= a2 + 2ab + 2ac + b2 + 2bc + c2
= a2 + b2 + c2 + 2(ab + bc + ca)
= sum of squares of a, b, c + 2(sum of the products of a, b, c taking two at a time}.
Therefore, (a – b + c)2 = a2 + b2 + c2 + 2(ac – ab – bc)
Similarly for (a – b – c)2, etc.
Corollaries:
(i) a2 + b2 + c2 = (a + b + c)2 – 2(ab + bc + ca)
(ii) ab + bc + ca = 12{(a + b + c)2 – (a2 + b2 + c2)}
Solved Examples on Expansion of (a ± b ± c)2
1. Expand (2x + y +3z)^2
Solution:
(2x + y +3z)2
= (2x)2 + y2 + (3z)2 + 2{2x ∙ y + y ∙ 3z + 3z ∙ 2x}
= 4x2 + y2 + 9z2 + 4xy + 6yz + 12zx.
2. Expand (a - b - c)2
Solution:
(a - b - c)2
= a2 + (-b)2 + (-c)2 + 2{a ∙ (-b) + (-b) ∙ (-c) + (-c) ∙ a}
= a2 + b2 + c2 - 2ab + 2bc - 2ca.
3. Expand (m - 12x + m2)2
Solution:
(m - 12x + m2)2
m2 + (-12m)2 + (m2)2 + 2{m ∙ (-12m) + (-12m) ∙ m2 + m2 ∙ m}
= m2 + 14m2+ m4 + 2{-12 - 12m + m3}
= m2 + 14m2+ m4 - 1 - m + 2m3.
4. If p + q + r = 8 and pq + qr + rp = 18, find the value of p2 + q2 + r2.
Solution:
We know that p2 + q2 + r2 = (p + q + r)2 - 2(pq + qr + rp).
Therefore, p2 + q2 + r2
= 82 - 2 × 18
= 64 – 36
= 28.
5. If x – y – z = 5 and x2 + y2 + z2 = 29, find the value of xy – yz – zx.
Solution:
We know that ab + bc + ca = 12[(a + b + c)2 – (a2 + b2 + c2)].
Therefore, xy + y(-z) + (-z)x = 12[(x + y - z)2 – (x2 + y2 + (-z)2)]
Or, xy – yz – zx = 12[52 – (x2 + y2 + z2)]
= 12[25 – 29]
= 12(-4)
= -2.
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