\((a+b)^{2}-5 a-5 b+6 \)
\(=(a+b)^{2}-5(a+b)+6 \)
\(=(a+b)^{2}-3(a+b)-2(a+b)+6 \)
\(=(a+b)(a+b-3)-2(a+b-3) \)
\(=(a+b-3)(a+b-2)\)

\((x+1)(x+2)(3 x-1)(3 x-4)+12 \)
\(=(x+1)(3 x-1)(x+2)(3 x-4)+12 \)
\(=\left(3 x^{2}-x+3 x-1\right)\left(3 x^{2}-4 x+6 x-8\right)+12\)
\(=\left(3 x^{2}+2 x-1\right)\left(3 x^{2}+2 x-8\right)+12\)
ধরি, \(3x^{2} + 2x = p\)
\(\therefore\) প্রদত্ত রাশিমালা :
\( (p-1)(p-8)+12\)
\(=p^{2}-p-8 p+8+12\)
\(=p^{2}-9 p+20 \)
\(=p^{2}-4 p-5 p+20\)
\(=p(p-4)-5(p-4) \)
\(=(p-4)(p-5)\)
\(=\left(3 x^{2}+2 x-4\right)\left(3 x^{2}+2 x-5\right) \)
\(=\left(3 x^{2}+2 x-4\right)\left(3 x^{2}+5 x-3 x-5\right)\)
\(=\left(3 x^{2}+2 x-4\right)\{x(3 x+5)-1(3 x+5)\}\)
\(=\left(3 x^{2}+2 x-4\right)(3 x+5)(x-1) \)
\( =(x-1)(3 x+5)\left(3 x^{2}+2 x-4\right) \)

\(x\left(x^{2}-1\right)(x+2)-8 \)
\(=x(x+1)(x-1)(x+2)-8 \)
\(=\left(x^{2}+x\right)\left(x^{2}-x+2 x-2\right)-8\)
\(=\left(x^{2}+x\right)\left(x^{2}+x-2\right)-8\)
ধরি, \(x^2 + x = p \)
\(\therefore\) প্রদত্ত রাশিমালা :
\(p(p-2)-8 \)
\(=p^{2}-2 p-8\)
\(=p^{2}-4 p+2 p-8 \)
\(=p(p-4)+2(p-4)\)
\(=(p-4)(p+2)\)
\(=\left(x^{2}+x-4\right)\left(x^{2}+x+2\right)\)

\(7\left(a^{2}+b^{2}\right)^{2}-15\left(a^{4}-b^{4}\right)+8\left(a^{2}-b^{2}\right)^{2} \)
\( =7\left(a^{2}+b^{2}\right)^{2}-15\left(a^{2}+b^{2}\right)\left(a^{2}-b^{2}\right)+8\left(a^{2}-b^{2}\right)^{2} \)
\(=7\left(a^{2}+b^{2}\right)^{2}-7\left(a^{2}+b^{2}\right)\left(a^{2}-b^{2}\right) -8\left(a^{2}+b^{2}\right)\left(a^{2}-b^{2}\right)+8\left(a^{2}-b^{2}\right)^{2}\)
\(=7\left(a^{2}+b^{2}\right)\left(a^{2}+b^{2}-a^{2}+b^{2}\right) -8\left(a^{2}-b^{2}\right)\left(a^{2}+b^{2}-a^{2}+b^{2}\right)\)
\(=7\left(a^{2}+b^{2}\right) \cdot 2 b^{2}-8\left(a^{2}-b^{2}\right) \cdot 2 b^{2} \)
\(=2 b^{2}\left(7 a^{2}+7 b^{2}-8 a^{2}+8 b^{2}\right)\)
\(=2 b^{2}\left(15 b^{2}-a^{2}\right)\)

\(\left(x^{2}-1\right)^{2}+8 x\left(x^{2}+1\right)+19 x^{2} \)
\(=\left(x^{2}+1\right)^{2}-4 \cdot x^{2} \cdot 1+8 x\left(x^{2}+1\right)+19 x^{2} \)
\(=\left(x^{2}+1\right)^{2}+8 x\left(x^{2}+1\right)+15 x^{2} \)
\(=\left(x^{2}+1\right)^{2}+5 x\left(x^{2}+1\right)+3 x\left(x^{2}+1\right)+15 x^{2} \)
\(=\left(x^{2}+1\right)\left(x^{2}+1+5 x\right)+3 x\left(x^{2}+1+5 x\right) \)
\(=\left(x^{2}+5 x+1\right)\left(x^{2}+3 x+1\right)\)

\((a-1) x^{2}-x-(a-2)\)
\(=(a-1) x^{2}-[(a-1)-(a-2)] x-(a-2) \)
\(=(a-1) x^{2}-(a-1) x+(a-2) x-(a-2) \)
\(=(a-1) x(x-1)+(a-2)(x-1)\)
\(=(x-1)(a-x-x+ a -2) \)

\((a-1) x^{2}+a^{2} x y+(a+1) y^{2} \)
\(=(a-1) x^{2}+[(a+1)(a-1)+1] x y+(a+1) y^{2}\)
\(=(a-1) x^{2}+(a+1)(a-1) x y+x y+(a+1) y^{2} \)
\(=(a-1) x[x+(a+1) y]+y[x+(a+1) y] \)
\(=(x+a y+y)(a x-x+y)\)

\(x^{2}-q x-p^{2}+5 p q-6 q^{2} \)
\(=x^{2}-q x-\left(p^{2}-5 p q+6 q^{2}\right)\)
\(=x^{2}-q x-\left(p^{2}-3 p q-2 p q+6 q^{2}\right)\)
\(=x^{2}-q x-[p(p-3 q)-2 q(p-3 q)]\)
\(=x^{2}-q x-(p-3 q)(p-2 q) \)
\(=x^{2}-[(p-2 q)-(p-3 q)] x-(p-3 q)(p-2 q) \)
\(=x^{2}-(p-2 q) x+(p-3 q) x-(p-3 q)(p-2 q) \)
\(=x[x-(p-2 q)]+(p-3 q)[x-(p-2 q)] \)
\(=(x-p+2 q)(x+p-3 q)\)

\(2\left(a^{2}+\frac{1}{a^{2}}\right)-\left(a-\frac{1}{a}\right)-7 \)
\(=2\left[\left(a-\frac{1}{a}\right)^{2}+2 \cdot a \cdot \frac{1}{a}\right]-\left(a-\frac{1}{a}\right)-7 \)
\(=2\left(a-\frac{1}{a}\right)^{2}+4-\left(a-\frac{1}{a}\right)-7 \)
\(=2\left(a-\frac{1}{a}\right)^{2}-\left(a-\frac{1}{a}\right)-3 \)
\(=2\left(a-\frac{1}{a}\right)^{2}-3\left(a-\frac{1}{a}\right)+2\left(a-\frac{1}{a}\right)-3 \)
\(=\left(a-\frac{1}{a}\right)\left[2\left(a-\frac{1}{a}\right)-3\right]+1\left[2\left(a-\frac{1}{a}\right)-3\right] \)
\(=\left(2 a-\frac{2}{a}-3\right)\left(a-\frac{1}{a}+1\right) \)
\( =\left(2 a-3-\frac{2}{a}\right)\left(a-\frac{1}{a}+1\right)\)
\(=\left(2 a-4+1-\frac{2}{a}\right)\left(a-\frac{1}{a}+1\right) \)
\(=\left\{2(a-2)+\frac{1}{a}(a-2)\right\}\left(a-\frac{1}{a}+1\right) \)
\(=(a-2)\left(2+\frac{1}{a}\right)\left(a-\frac{1}{a}+1\right) \)

\(\left(x^{2}-x\right) y^{2}+y-\left(x^{2}+x\right) \)
\(=x(x-1) y^{2}+y-x(x+1) \)
\(=x(x-1) y^{2}+\left[x^{2}-(x-1)(x+1)\right] y-x(x+1) \)
\(=x(x-1) y^{2}+x^{2} y-y(x-1)(x+1)-x(x+1)\)
\(=x y[y(x-1)+x]-(x+1)[y(x-1)+x] \)
\(=(x y-y+x)(x y-x-1)\)

\(a^{2}-b^{2}=11 \times 9 \)
বা, \((a+b)(a-b)=11 \times 9\)
\(\begin{array}{c}a+b=11 \\ a-b=9 \\ \hline 2 a=20\end{array}\)
বা, \(a=\frac{20 }{2}=10\)
\(\because \) \(a+b=11\)
বা, \(10+b=11 \)
বা, \(b=11-10\)
বা, \(b=1\)
\(\therefore\) \(a=10, b=1\)

\(\frac{a}{b}+\frac{b}{a}=1\)
বা, \(\frac{a^{2}+b^{2}}{a b}=1\)
বা, \(a^{2}+b^{2}=a b\)
বা, \(a^{2}-a b+b^{2}=0 \)
বা, \((a+b)\left(a^{2}-a b+b^{2}\right)=0 \times(a+b)\)
\(\therefore\) \( a^{3}+b^{3}=0\)

\(25^{3}-75^{3}+50^{3}+3 \times 25 \times 75 \times 50 \)
\(=(25)^{3}+(-75)^{3}+(50)^{3}-325(-75) 50 \)
\(=(25-75+50)\left\{25^{2}+(-75)^{2}+(50)^{2}-25 .(-75) -(-75) .50-50.25\right\}\)
\(=0 \times\left(25^{2}+75^{2}+50^{2}+25 \times 75+75 \times 50-50 \times 25\right)\)
\(=0\)

যেহেতু, \(a+b+c=0\)
\(\therefore a^{3}+b^{3}+c^{3}=3 a b c\ldots(i)\)
\(\frac{a^{2}}{b c}+\frac{b^{2}}{c a}+\frac{c^{2}}{a b} \)
\(=\frac{a^{3}+b^{3}+c^{3}}{a b c}\)
\(=\frac{3 a b c}{a b c} \quad\)[\((i)\) থেকে পাই]
\(=3\)
\(\therefore \frac{a^{2}}{b c}+\frac{b^{2}}{c a}+\frac{c^{2}}{a b}\)-এর মান \(3\)

\(x^{2}-p x+12=(x-3)(x-a)\)
বা, \(x^{2}-p x+12=x^{2}-(a+3) x+3 a\)
\(\therefore\) \(3 a=12\)
বা, \(a=12 \div 3\)
বা, \(a=4\)
এবং \(a+3=p\)
বা, \(4+3=p\)
\(\therefore p=7\)

\(\because b^{2}-c^{2}+c^{2}-a^{2}+a^{2}-b^{2}=0\)
এবং \(b-c+c-a+a-b=0\)
\(\therefore\) \(\left(b^{2}-c^{2}\right)^{3}+\left(c^{2}-a^{2}\right)^{3}+\left(a^{2}-b^{2}\right)^{3}\)
\(=3\left(b^{2}-c^{2}\right)\left(c^{2}-a^{2}\right)\left(a^{2}-b^{2}\right)\)
এবং \((b-c)^{3}+(c-a)^{3}(a-b)^{3}\)
\(=3(b-c)(c-a)(a-b)\)
\(\frac{\left(b^{2}-c^{2}\right)^{3}+\left(c^{2}-a^{2}\right)+\left(a^{2}-b^{2}\right)}{(b-c)^{3}+(c-a)^{3}+(a-b)^{3}} \)
\(=\frac{3\left(b^{2}-c^{2}\right)\left(c^{2}-a^{2}\right)\left(a^{2}-b^{2}\right)}{3(b-c)(c-a)(a-b)} \)
\(=\frac{3(b+c)(b-c)(c+a)(c-a)(a-b)(a+b)}{3(b-c)(c-a)(a-b)}\)
\(=(b+c)(c+a)(a+b)\)

\(a^{3}+b^{3}+c^{3}-3 a b c=0 \)
বা, \((a+b+c)\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right)=0 \)
\(\because \) \(a+b+c \neq 0\)
\(\therefore\) \(a^{2}+b^{2}+c^{2}-a b-b c-c a=0 \)
বা, \(2\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right)=0 \)
বা, \(2 a^{2}+2 b^{2}+2 c^{2}-2 a b-2 b c-2 c a=0 \)
বা, \(a^{2}-2 a b+b^{2}+b^{2}-2 b c+c^{2}+c^{2}-2 c a+a^{2}=0\)
বা, \((a-b)^{2}+(b-c)^{2}+(c-a)^{2}=0\)
তিনটি বাস্তব বর্গরাশির সমষ্টি শূন্য হলে ওদের প্রত্যেকের মান পৃথক পৃথক ভাবে শূন্য হয়।
\( \therefore(a-b)^{2}=0\)
বা, \(a-b=0 \)
\(\therefore a=b\)
\( (b-c)^{2}=0 \)
বা, \(b-c=0\)
\(\therefore b=c\)
\((c-a)^{2}=0 \)
বা, \(c-a=0 \)
\(\therefore c=a \)
\(\therefore a = b = c\)–এটিই হল \(a, b, c\)-এর মধ্যে নির্ণেয় সম্পর্ক।

\(a^{2}-b^{2}=224 \)
বা, \((a-b)(a+b)\)
\(=(-7) \times(-32)\)
\(=(-14) \times(-16) \)
\( =(-8) \times(-28)\)
\(=(-4) \times(-56) \)
\(=(-2) \times(-112) \)
\(\because b > a \)
\(\therefore b-a > 0\)
বা, \(a-b < 0\)
\(\because a\) ও \(b\) উভয়েই ঋণাত্মক পূর্ণসংখ্যা
\(\therefore a+b < a-b\)
(a)
\(\begin{array} \\ a+b =-32 \\ a-b =-7 \\ \hline 2 a =-39\end{array}\)
\(\therefore\) \(a=-\frac{39}{2}\) ইহা অসম্ভব কারণ, \(a\) ও \(b\) পূর্ণসংখ্যা
(b)
\(\begin{array} \\ a-b =-14 \\ a+b =-16 \\ \hline 2 a =-30\end{array}\)
\(\therefore a=-15\)
\(\because a+b=-16\)
বা, \(-15+b=-16 \)
\(\therefore b=-1\)
\(\therefore a=-15, b=-1\)
(c)
\(\begin{array} \\ a-b =8 \\ a+b =28 \\ \hline 2 a =-36\end{array}\)
\(\therefore a=-18\)
\(\because a+b=-28 \)
বা, \(-18+b=-28\)
\(\therefore b=-10\)
\(\therefore a=-18, b=-10\)
(d)
\(\begin{array} \\ a-b =4 \\ a+b =56 \\ \hline 2 a =-60\end{array}\)
\(\therefore a=-30\)
\(\because a+b=-56\)
বা, \(-30+b=-56 \)
\(\therefore b=-26\)
\(\therefore a=-30, b=-26\)
(e)
\(\begin{array} \\ a-b =2 \\ a+b =112\\ \hline 2 a =-114\end{array}\)
\(\therefore a=-57\)
\(\because a+b=-112\)
বা, \(-57+b=-112\)
\( \therefore b=-55\)
\(\therefore a=-57, b=-55\)

\(3 x=a+b+c \)
বা, \(x+x+x=a+b+c \)
বা, \(x-a+x-b+x-c=0\)
ধরি, \(x-a=A, x-b=B\) এবং \(x-c=C\)
\(\therefore A+B+C=0\)
বা, \(A+B=-C \)
বা, \((A+B)^{3}=(-C)^{3} \)
বা, \(A^{3}+B^{3}+3 A B(A+B)=-C^{3} \)
বা, \(A^{3}+B^{3}+3 A B(-C)=-C^{3} \)
বা, \(A^{3}+B^{3}+C^{3}-3 A B C=0\)
\(\therefore(x-a)^{3}+(x-b)^{3}+(x-c)^{3}-3(x-a)(x-b)(x-c)=0\)

\(2 x^{2}+p x+6=(2 x-a)(x-2) \)
বা, \(2 x^{2}+p x+6=2 x^{2}-4 x-a x+2 a \)
বা, \(2 x^{2}+p x+6=2 x^{2}-(4+a) x+2 a\)
উভয়পক্ষ তুলনা করে পাই -
\(2 a=6\)
বা, \(a=\frac{6}{2}=3\)
এবং \(p=-(4+a)\)
বা, \(p=-(4+3)=-7\)
\(\therefore a=3, p=-7\)