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Answer :
(i) Given:\(a^2 + 8a + 16\)
Here, 4 + 4 = 8 and \(4 \times 4 = 16\)
\(a^2 + 8a +16\)
\(= a^2 + 4a + 4a + 4 \times 4\)
\(= (a^2 + 4a) + (4a + 16)\)
\(= a(a + 4) + 4(a + 4)\)
\(= (a + 4) (a + 4)\)
\(= (a + 4)^2\)
(ii) Given:\(p^2 – 10p + 25\)
Here, 5 + 5 = 10 and \(5 \times 5 = 25\)
\(=p^2 – 10p + 25\)
\(= p^2 – 5p – 5p + 5 \times 5\)
\(= (p^2 – 5p) + (-5p + 25)\)
\(= p(p – 5) – 5(p – 5)\)
\(= (p – 5) (p – 5)\)
\(= (p – 5)^2\)
(iii) Given:\(25m^2 + 30m + 9\)
Here, \(15 + 15 = 30\) and \(15 \times 15 = 25 \times 9 = 225\)
\(25m^2 + 30m + 9\)
\(= 25m^2 + 15m + 15m + 9\)
\(= (25m^2 + 15m) + (15m + 9)\)
\(= 5m(5m + 3) + 3(5m + 3)\)
\(= (5m + 3) (5m + 3)\)
\(= (5m + 3)^2\)
(iv) Given:\(49y^2 + 84yz + 36z^2\)
Here, \(42 + 42 = 84\) and \(42 \times 42 = 49 \times 36 = 1764\)
\(49y^2 + 84yz + 36z^2\)
\(= 49y^2 + 42yz + 42yz + 36z^2\)
\(= 7y(7y + 6z) +6z(7y + 6z)\)
\(= (7y + 6z) (7y + 6z)\)
\(= (7y + 6z)^2\)
(v) Given:\(4x^2 – 8x + 4\)
\(= 4(x^2 – 2x + 1) [Taking \;4 \;common]\)
\(= 4(x^2 – x – x + 1)\)
\(= 4[x(x – 1) -1(x – 1)]\)
\(= 4(x – 1)(x – 1)\)
\(= 4(x – 1)^2\)
(vi) Given: \(121b^2 – 88bc + 16c^2\)
Here, \(44 + 44 = 88\) and \(44 \times 44 = 121 \times 16 = 1936\)
\(121b^2 – 88bc + 16c^2\)
\(= 121b^2 – 44bc – 44bc + 16c^2\)
\(= 11b(11b – 4c) – 4c(11b – 4c)\)
\(= (11b – 4c) (11b – 4c)\)
\(= (11b – 4c)^2\)
(vii) Given:\((l + m)^2 – 4lm\)
Expanding the expression, \( (l + m)^2\), we have
\(l^2 + 2lm + m^2 – 4lm\)
\(= l^2 – 2lm + m^2\)
\(= l^2 – Im – lm + m^2\)
\(= l(l – m) – m(l – m)\)
\(= (l – m) (l – m)\)
\(= (l – m)^2\)
(viii) Given:\(a^4 + 2a^2b^2 + b^4\).So we have,
\(= a^4 + a^2b^2 + a^2b^2 + b^4\)
\(= a^2(a^2 + b^2) + b^2(a^2 + b^2)\)
\(= (a^2 + b^2)(a^2 + b^2)\)
\(= (a^2 + b^2)^2\)