We will discuss here about the expansion of (a ± b ± c)\(^{2}\).
(a + b + c)\(^{2}\) = {a + (b + c)}\(^{2}\) = a\(^{2}\) + 2a(b + c) + (b + c)\(^{2}\)
= a\(^{2}\) + 2ab + 2ac + b\(^{2}\) + 2bc + c\(^{2}\)
= a\(^{2}\) + b\(^{2}\) + c\(^{2}\) + 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)\(^{2}\) = a\(^{2}\) + b\(^{2}\) + c\(^{2}\) + 2(ac – ab – bc)
Similarly for (a – b – c)\(^{2}\), etc.
Corollaries:
(i) a\(^{2}\) + b\(^{2}\) + c\(^{2}\) = (a + b + c)\(^{2}\) – 2(ab + bc + ca)
(ii) ab + bc + ca = \(\frac{1}{2}\){(a + b + c)\(^{2}\) – (a\(^{2}\) + b\(^{2}\) + c\(^{2}\))}
Solved Examples on Expansion of (a ± b ± c)\(^{2}\)
1. Expand (2x + y +3z)^2
Solution:
(2x + y +3z)\(^{2}\)
= (2x)\(^{2}\) + y\(^{2}\) + (3z)\(^{2}\) + 2{2x ∙ y + y ∙ 3z + 3z ∙ 2x}
= 4x\(^{2}\) + y\(^{2}\) + 9z\(^{2}\) + 4xy + 6yz + 12zx.
2. Expand (a - b - c)\(^{2}\)
Solution:
(a - b - c)\(^{2}\)
= a\(^{2}\) + (-b)\(^{2}\) + (-c)\(^{2}\) + 2{a ∙ (-b) + (-b) ∙ (-c) + (-c) ∙ a}
= a\(^{2}\) + b\(^{2}\) + c\(^{2}\) - 2ab + 2bc - 2ca.
3. Expand (m - \(\frac{1}{2x}\) + m\(^{2}\))\(^{2}\)
Solution:
(m - \(\frac{1}{2x}\) + m\(^{2}\))\(^{2}\)
m\(^{2}\) + (-\(\frac{1}{2m}\))\(^{2}\) + (m\(^{2}\))\(^{2}\) + 2{m ∙ (-\(\frac{1}{2m}\)) + (-\(\frac{1}{2m}\)) ∙ m\(^{2}\) + m\(^{2}\) ∙ m}
= m\(^{2}\) + \(\frac{1}{4m^{2}}\)+ m\(^{4}\) + 2{-\(\frac{1}{2}\) - \(\frac{1}{2}\)m + m\(^{3}\)}
= m\(^{2}\) + \(\frac{1}{4m^{2}}\)+ m\(^{4}\) - 1 - m + 2m\(^{3}\).
4. If p + q + r = 8 and pq + qr + rp = 18, find the value of p\(^{2}\) + q\(^{2}\) + r\(^{2}\).
Solution:
We know that p\(^{2}\) + q\(^{2}\) + r\(^{2}\) = (p + q + r)\(^{2}\) - 2(pq + qr + rp).
Therefore, p\(^{2}\) + q\(^{2}\) + r\(^{2}\)
= 8\(^{2}\) - 2 × 18
= 64 – 36
= 28.
5. If x – y – z = 5 and x\(^{2}\) + y\(^{2}\) + z\(^{2}\) = 29, find the value of xy – yz – zx.
Solution:
We know that ab + bc + ca = \(\frac{1}{2}\)[(a + b + c)\(^{2}\) – (a\(^{2}\) + b\(^{2}\) + c\(^{2}\))].
Therefore, xy + y(-z) + (-z)x = \(\frac{1}{2}\)[(x + y - z)\(^{2}\) – (x\(^{2}\) + y\(^{2}\) + (-z)\(^{2}\))]
Or, xy – yz – zx = \(\frac{1}{2}\)[5\(^{2}\) – (x\(^{2}\) + y\(^{2}\) + z\(^{2}\))]
= \(\frac{1}{2}\)[25 – 29]
= \(\frac{1}{2}\)(-4)
= -2.
From Expansion of (a ± b ± c)^2 to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Apr 20, 24 05:39 PM
Apr 20, 24 05:29 PM
Apr 19, 24 04:01 PM
Apr 19, 24 01:50 PM
Apr 19, 24 01:22 PM