Question

The number of the solutions of the equation $5^{2x-1}+5^{x+1}=250,$ is/are

• A. $0$
• B. $1$
• C. $2$
• D. $infinitelymany$

Answer 1

Answer: (B)

Given equation is $5^{2x-1}+5^{x+1}=250$
$\Rightarrow$ $5^{2x}{.5}^{-1}+5^x.5=250$
$\Rightarrow$ $\frac{{(5^x)}^2}{5}+5^x.5=250$ ?.(i) Let $5^x=t$ Then Eq. (i) becomes $\frac{t^2}{5}+t.5=250$
$\Rightarrow$ $t^2+25t=250\times 5$
$\Rightarrow$ $t^2+25t-1250=0$
$\therefore$ $t=\frac{-25\pm \sqrt{{(25)}^2-4\times 1\times (-1250)}}{2\times 1}$
$\Rightarrow$ $t=\frac{-25\pm \sqrt{625+5000}}{2}$
$\Rightarrow$ $t=\frac{-25\pm \sqrt{5625}}{2}$
$\Rightarrow$ $t=\frac{-25\pm 75}{2}$ On taking $+ve$ sign, we get $t=5^x=\frac{-25+75}{2}=\frac{50}{2}$
$\Rightarrow$ $5^x=25$
$\Rightarrow$ $5^x=5^2$
$\Rightarrow$ $x=2$ On taking $-ve$ sign, we get $t=5^x=\frac{-25-75}{2}$ $5^x=\frac{-100}{2}$ $5^x=-50$ Cannot express in terms of power of 5. Hence number of solution of given equation is 1.