2021 WB Math Suggestion | একচলবিশিষ্ট দ্বিঘাত সমীকরণ | কষে দেখি – 1.2

WB Mathematics Suggestion with Answers | Class 10th Math Suggestion With Solution | একচলবিশিষ্ট দ্বিঘাত সমীকরণ | কষে দেখি – 1.2

(x) $\frac{x}{x+1}+\frac{x+1}{x}=2\frac{1}{12}$

$\Rightarrow&space;\frac{x^{2}+\left&space;(&space;x+1&space;\right&space;)^{2}}{x\left&space;(&space;x+1&space;\right&space;)}=\frac{25}{12}$

$\Rightarrow&space;\frac{x^{2}+x^{2}+x+x+1}{x^{2}+x}=\frac{25}{12}$
$\Rightarrow&space;25x^{2}+25x=24x^{2}+24x+12$
$\Rightarrow&space;25x^{2}-24x^{2}+25x-24x-12=0$
$\Rightarrow&space;x^{2}+x-12=0$
$\Rightarrow&space;x^{2}+4x-3x-12=0$
$\Rightarrow&space;x\left&space;(&space;x+4&space;\right&space;)-3\left&space;(&space;x+4&space;\right&space;)=0$
$\Rightarrow&space;\left&space;(&space;x+4&space;\right&space;)\left&space;(&space;x-3&space;\right&space;)=0$

∴ হয়  $x+4=0$

$\therefore&space;x=-4$

অথবা,
$x-3=0$
$\therefore&space;x=3$

নির্ণেয় দ্বিঘাত সমীকরণের সমাধান, x= -4, 3 Ans.

(xiii) $\frac{x+1}{2}+\frac{2}{x+1}=\frac{x+1}{3}+\frac{3}{x+1}-\frac{5}{6},\quad&space;x\ne&space;-1$
$\Rightarrow&space;\frac{x+1}{2}-\frac{x+1}{3}+\frac{5}{6}=\frac{3}{x+1}-\frac{2}{x+1}$
$\Rightarrow&space;\frac{3\left&space;(&space;x+1&space;\right&space;)-2\left&space;(&space;x+1&space;\right&space;)+5}{6}=\frac{3-2}{x+1}$
$\Rightarrow&space;\frac{3x+3-2x-2+5}{6}=\frac{1}{x+1}$
$\Rightarrow&space;\frac{x+6}{6}=\frac{1}{x+1}$
$\Rightarrow&space;\left&space;(&space;x+6&space;\right&space;)\left&space;(&space;x+1&space;\right&space;)=6$
$\Rightarrow&space;x^{2}+x+6x+6-6=0$
$\Rightarrow&space;x^{2}+7x=0$
$\Rightarrow&space;x\left&space;(&space;x+7&space;\right&space;)=0$

$\therefore&space;x=0$
অথবা,
$x+7=0$
$\therefore&space;x=-7$

∴ নির্ণেয় দ্বিঘাত সমীকরণের সমাধান, x= 0, -7 Ans.

(xvi) $\frac{1}{a+b+x}=\frac{1}{a}+\frac{1}{b}+\frac{1}{x},\quad&space;x\ne&space;0,-\left(&space;a+b&space;\right)$
$\Rightarrow&space;\frac{1}{a+b+x}-\frac{1}{x}=\frac{1}{a}+\frac{1}{b}$
$\Rightarrow&space;\frac{x-a-b-x}{x\left&space;(&space;a+b+x&space;\right&space;)}=\frac{b+a}{ab}$
$\Rightarrow&space;\frac{-\left&space;(&space;a+b&space;\right&space;)}{ax+bx+x^{2}}=\frac{a+b}{ab}$
$\Rightarrow&space;\frac{-1}{ax+bx+x^{2}}=\frac{1}{ab}$
$\Rightarrow&space;ax+bx+x^{2}=-ab$
$\Rightarrow&space;x^{2}+ax+bx+ab=0$
$\Rightarrow&space;x\left&space;(&space;x+a&space;\right&space;)+b\left&space;(&space;x+a&space;\right&space;)=0$
$\Rightarrow&space;\left&space;(&space;x+a&space;\right&space;)\left&space;(&space;x+b&space;\right&space;)=0$

হয় $x+a=0$
$\therefore&space;x=-a$
অথবা,
$x+b=0$
$\therefore&space;x=-b$

∴ নির্ণেয় দ্বিঘাত সমীকরণের সমাধান, x= -a, -b Ans.

(xviii) $\frac{1}{x}-\frac{1}{x+b}=\frac{1}{a}-\frac{1}{a+b},\quad&space;x\ne&space;0,-b$

$\Rightarrow&space;\frac{x+b-x}{x\left&space;(&space;x+b&space;\right&space;)}=\frac{a+b-a}{a\left&space;(&space;a+b&space;\right&space;)}$
$\Rightarrow&space;\frac{b}{x\left&space;(&space;x+b&space;\right&space;)}=\frac{b}{a\left&space;(&space;a+b&space;\right&space;)}$
$\Rightarrow&space;\frac{1}{\left&space;(&space;x^{2}+bx&space;\right&space;)}=\frac{1}{\left&space;(&space;a^{2}+ab&space;\right&space;)}$
$\Rightarrow&space;x^{2}+bx=a^{2}+ab$
$\Rightarrow&space;x^{2}-a^{2}+bx-ab=0$
$\Rightarrow&space;\left&space;(&space;x+a&space;\right&space;)\left&space;(&space;x-a&space;\right&space;)+b\left&space;(&space;x-a&space;\right&space;)=0$
$\Rightarrow&space;\left&space;(&space;x-a&space;\right&space;)\left&space;(&space;x+a+b&space;\right&space;)=0$
হয় $x-a=0$
$\therefore&space;x=a$
অথবা,
$x+a+b=0$
$\therefore&space;x=-a-b=-\left&space;(&space;a+b&space;\right&space;)$
∴ নির্ণেয় দ্বিঘাত সমীকরণের সমাধান x= a, -a(a+b) Ans.

(xix) $\frac{1}{\left(&space;x-1&space;\right)\left(&space;x-2&space;\right)}+\frac{1}{\left(&space;x-2&space;\right)\left(&space;x-3&space;\right)}+\frac{1}{\left(&space;x-3&space;\right)\left(&space;x-4&space;\right)}=\frac{1}{6}$
$\Rightarrow&space;\frac{1}{x-2}-\frac{1}{x-1}+\frac{1}{x-3}-\frac{1}{x-2}+\frac{1}{x-4}-\frac{1}{x-3}=\frac{1}{6}$
$\Rightarrow&space;\frac{1}{x-4}-\frac{1}{x-1}=\frac{1}{6}$
$\Rightarrow&space;\frac{x-1-x-4}{\left&space;(&space;x-4&space;\right&space;)\left&space;(&space;x-1&space;\right&space;)}=\frac{1}{6}$
$\Rightarrow&space;\frac{3}{x^{2}-x-4x+4}=\frac{1}{6}$
$\Rightarrow&space;\frac{3}{x^{2}-5x+4}=\frac{1}{6}$
$\Rightarrow&space;x^{2}-5x+4=18$
$\Rightarrow&space;x^{2}-5x+4-18=0$
$\Rightarrow&space;x^{2}-5x-14=0$
$\Rightarrow&space;x^{2}-7x+2x-14=0$
$\Rightarrow&space;x\left&space;(&space;x-7&space;\right&space;)+2\left&space;(&space;x-7&space;\right&space;)=0$
$\Rightarrow&space;\left&space;(&space;x-7&space;\right&space;)\left&space;(&space;x+2&space;\right&space;)=0$
হয় $x-7=0$
$\therefore&space;x=7$
অথবা,
$x+2=0$
$\therefore&space;x=-2$
∴  নির্ণেয় দ্বিঘাত সমীকরণের সমাধান x= 7,-2 Ans.

(xx) $\frac{a}{x-a}+\frac{b}{x-b}=\frac{2c}{x-c},\quad&space;x\ne&space;a,b,c$
$\Rightarrow&space;\frac{a}{x-a}-\frac{b}{x-b}=\frac{c}{x-c}+\frac{c}{x-c}$
$\Rightarrow&space;\frac{a}{x-a}-\frac{c}{x-c}=\frac{c}{x-c}-\frac{b}{x-b}$
$\Rightarrow&space;\frac{a\left&space;(&space;x-c&space;\right&space;)-c\left&space;(&space;x-a&space;\right&space;)}{\left&space;(&space;x-a&space;\right&space;)\left&space;(&space;x-c&space;\right&space;)}=\frac{c\left&space;(&space;x-b&space;\right&space;)-b\left&space;(&space;x-c&space;\right&space;)}{\left&space;(&space;x-c&space;\right&space;)\left&space;(&space;x-b&space;\right&space;)}$
$\Rightarrow&space;\frac{ax-ac-cx+ac}{\left&space;(&space;x-a&space;\right&space;)\left&space;(&space;x-c&space;\right&space;)}=\frac{cx-bc-bx+bc}{\left&space;(&space;x-c&space;\right&space;)\left&space;(&space;x-b&space;\right&space;)}$
$\Rightarrow&space;\frac{ax-cx}{x-a}=\frac{cx-bx}{x-b}$
$\Rightarrow&space;\left&space;(&space;x-a&space;\right&space;)\left&space;(&space;cx-bx&space;\right&space;)=\left&space;(&space;ax-cx&space;\right&space;)\left&space;(&space;x-b&space;\right&space;)$
$\Rightarrow&space;cx^{2}-bx^{2}-acx+abx=ax^{2}-abx-cx^{2}-bcx$
$\Rightarrow&space;cx^{2}+cx^{2}-bx^{2}-ax^{2}-acx+abx+abx-bcx=0$
$\Rightarrow&space;2cx^{2}-bx^{2}-ax^{2}-acx+2abx-bcx=0$
$\Rightarrow&space;x\left&space;(&space;2cx-bx-ax-ac+2ab-bc&space;\right&space;)=0$
$\therefore&space;x=0$
অথবা,
$2cx-bx-ax-ac+2ab-bc=0$
$\Rightarrow&space;x\left&space;(&space;2c-b-a&space;\right&space;)=ac+bc-2ab$
$\therefore&space;x=\frac{ac+bc-2ab}{2c-b-a}$
$\therefore$ নির্ণেয় দ্বিঘাত সমীকরণের সমাধান  :
$x=a, \frac{ac+bc-2ab}{2c-b-a}$
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