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. $infinitely \,many$

Answer 1

Answer: (B)

$1$

Answer 2

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 E (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.