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)}$^{2}$ = 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.

9th Grade Math

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