\((a+b)(a+b)=(a+b)^{2}-(i i)\)

\((x+7)^{2}=x^{2}+14 x+k\)

or, \(x^{2}+14 x+49=x^{2}+14 x+k\)

or, \(x^{2}+14 x+k=x^{2}+14 x+49\)

\(\therefore k=49\)

\(a^{2}+b^{2} \pm 2 a b=(a \pm b)^{2}\)
(iii) 2ab বা -2ab (যোগ করতে হবে) ।

\((a+b)^{2}\)
\(=(a)^{2}+6 a+9\)
\(=(a)^{2}+2 \cdot 3 \cdot a+(3)^{2}\)
\(=(a+3)^{2}\)
b-এর মান হবে 3---(iii)

\(x^{2}+\frac{1}{4}x\)
\(=(x)^{2}+2 \cdot \frac{1}{8} \cdot x +\left(\frac{1}{8}\right)^{2}\)
\(=\left(x+\frac{1}{8}\right)^{2}\)
\(\therefore\) \(\frac{1}{64}\) -(i) যোগ করলে যোগফল পূর্ণ বর্গ হবে।

\(c^{2}+k \cdot c+\frac{1}{9}\)
\(=c^{2}+2 \cdot c \cdot \frac{1}{3}+\left(\frac{1}{3}\right)^{2}\)
\(=\left(c+\frac{1}{3}\right)^{2}\)
\(\therefore\) k-এর মান \(\pm \frac{2}{3}\) হলে, \(c^{2}+k \cdot c+\frac{1}{9}\) পূর্ণবর্গ হবে।

\(9 p^{2}+\frac{1}{9 p^{2}}\)
\(=(3 p)^{2}+\left(\frac{1}{3 p}\right)^{2} \pm 2 \cdot 3 p \cdot \frac{1}{3 p}\)
\(\therefore\) ±2 যোগ করলে যোগফল পূর্ণবর্গ হবে।

\((x-y)^{2}\)
\(=(2)^{2}-2 \cdot 2 \cdot y+(y)^{2}\)
\(=(2-y)^{2}\)
\(\therefore\) \(x\) মান 2

\((c-3)^{2}=c^{2}+k c+9\)
বা, \(c^{2}-2 .\) c. \(3+3^{2}=c^{2}+k c+9\)
বা, \(c^{2}\) – 6c + 9 = \(c^{2}\) + kc + 9
বা,\(k=\frac{-6 c}{c}=-6\)
\(\therefore\) k-এর নির্ণেয় মান = - 6

\((2 q-3 z)^{2}-2(2 q-3 z)(q-3 z)+(q-3 z)^{2}\)
ধরি, \((2 q-3 z)=a\) এবং \((q-3 z)=b\)
\(\therefore\) প্রদত্ত রাশিমালা = \((a)^{2}-2 a b+(b)^{2}=(a-b)^{2}\)
এখন, a এবং b এর মান বসিয়ে পাই, \((2 q-3 z-q + 3 z)^{2}=(q)^{2}=q^{2}\)

\((3 p+2 q-4 r)^{2}+2(3 p+2 q-4 r)(4 r-2 p-q)\)
\( \quad \quad \quad \quad +(4 r-2 p-q)^{2}\)
ধরি, \((3 p+2 q-4 r)=a\)
এবং \((4 r-2 p-q)=b\)
\(\therefore\) প্রদত্তরাশিমালা = \(a^{2}+2 a b+b^{2}=(a+b)^{2}\)
\(\therefore\) a এবং b-এর মান বসিয়ে পাই,
\((3 p+2 q-4 r+4 r-2 p-q)^{2}=(p+q)^{2}\)

\(=(4 a)^{2}-2.4 a \cdot 5 c+(5 c)^{2}\)
\(=(4 a-5 c)^{2}\)

\(=(2 p)^{2}-2.2 p \cdot \frac{1}{2}+\left(\frac{1}{2}\right)^{2}\)
\(=\left(2 p-\frac{1}{2}\right)^{2}\)

\(=(1)^{2}+2 \cdot 1 \cdot \frac{2}{a}+\left(\frac{2}{a}\right)^{2}\)
\(=\left(1+\frac{2}{a}\right)^{2}\)

\(=(3 a)^{2}+2 \cdot 3 a \cdot 4 b+(4 b)^{2}\)
\(=(3 a+4 b)^{2}\)

\(=(8 a)^{2}+2\cdot 8 a \cdot 1+(1)^{2}\)
\(=(8 a+1)^{2}\)
\(a=1\) বসিয়ে পাই,
\((8 \cdot 1+1)^{2}\)
\(=(8+1)^{2}=(9)^{2}=81\)

\(=(5 a)^{2}-2\cdot 5 a \cdot 3 b+(3 b)^{2}\)
\(=(5 a-3 b)^{2}\)
\(a=3, b=2\) বসিয়ে পাই,
\(=(5\cdot 3-3\cdot 2)^{2}\)
\(=(15-6)^{2}=(9)^{2}=81\)

\(=(8)^{2}-2\cdot 8 \cdot \frac{1}{\mathrm{p}}+\left(\frac{1}{\mathrm{p}}\right)^{2}=\left(8-\frac{1}{\mathrm{p}}\right)^{2}\)
\(p=-1\) = বসিয়ে পাই \(=\left(8-\frac{1}{-1}\right)^{2}\)
\(=(8+1)^{2}=(9)^{2}=81\)

\(\mathrm{p}^{2} \mathrm{q}^{2}+10 \mathrm{pqr}+25 \mathrm{r}^{2}\)
\(=(\mathrm{pq})^{2}+2 \cdot \mathrm{pq} \cdot 5 \mathrm{r}+(5 \mathrm{r})^{2}=(\mathrm{pq}-5 \mathrm{r})^{2}\)
\(p=2, q=-1\) এবং \(r=3\) বসিয়ে পাই
\(\{(2 \cdot -1)+5\cdot 3\}^{2}=(-2+15)^{2}=(13)^{2}=169\)

st \(=\left(\frac{s+t}{2}\right)^{2}-\left(\frac{s-t}{2}\right)^{2}\)
\(=\left(\frac{12}{2}\right)^{2}-\left(\frac{8}{2}\right)^{2}\)
\(=6^{2}-4^{2}\)
\(=36-16\)
\(=20\)
\(s^{2}+t^{2}\)
\(=(s+t)^{2}-2 s t\)
\(=(12)^{2}-2\cdot 20\)
\(=144-40\)
\(=104\)

\(8 x y\left(x^{2}+y^{2}\right)\) এর মান লিখি যখন (\(x\) + y) = 5 এবং (\(x\) – y) =1
\(8 x y\left(x^{2}-y^{2}\right)=4 x y \cdot 2\left(x^{2}+y^{2}\right)\)
\(\left.=\left\{(x+y)^{2}-(x-y)^{2}\right)\right\}\left\{(x+y)^{2}+(x-y)^{2}\right\}\)
\(x+y=5\) এবং \(x-y=1\) বসিয়ে পাই।
\(\therefore\) প্রদত্ত রাশি হবে।
\(\left\{(5)^{2}-(1)^{2}\right\}\left\{(5)^{2}+(1)^{2}\right\}\)
\(=\{(25-1)(25+1)\}\)
\(=24 \times 26\)
\(=624\)

\(\left(\frac{x^{2}+y^{2}}{2 x y}\right)\) - এর মান লিখি যখন (\(x\) + y) = 9 এবং (\(x\) – y) = 5
\(\left(\frac{x^{2}+y^{2}}{2 x y}\right)\)
\(=\frac{(x+y)^{2}-2\left\{\left(\frac{x+y}{2}\right)^{2}-\left(\frac{x-y}{2}\right)^{2}\right\}}{2\left\{\left(\frac{x+y}{2}\right)^{2}-\left(\frac{x-y}{2}\right)^{2}\right\}}\)
\(=\frac{(9)^{2}-2\left\{\left(\frac{9}{2}\right)^{2}-\left(\frac{5}{2}\right)^{2}\right\}}{2\left\{\left(\frac{9}{2}\right)^{2}-\left(\frac{5}{2}\right)^{2}\right\}}\)
\(=\frac{81-2\left\{\frac{81}{4}-\frac{25}{4}\right\}}{2\left\{\frac{81}{4}-\frac{25}{4}\right\}}\)
\(=\frac{81-2\left\{\frac{81-25}{4}\right\}}{2\left\{\frac{81-25}{4}\right\}}\)
\(=\frac{81-2 \times \frac{56}{4}}{2 \times \frac{56}{4}}\)
\(=\frac{81-28}{28}\)
\(=\frac{53}{28}\)

\(36=9 \times 4=\left(\frac{4+9}{2}\right)^{2}-\left(\frac{4-9}{2}\right)^{2}\)

\(44=11 \times 4=\left(\frac{11+4}{2}\right)^{2}-\left(\frac{11-4}{2}\right)^{2}\)

\(8 x^{2}+50 y^{2}\)
\(=2\left(4 x^{2}+25 y^{2}\right)\)
\(=2\{(2 x)^{2}+(5 y)^{2}\}\)
\(=(2 x+5 y)^{2}+(2 x-5 y)^{2}\)

\(x=1 \cdot x\)
\(=\left(\frac{1+x}{2}\right)^{2}-\left(\frac{1-x}{2}\right)^{2}\)