91Ó°ÊÓ

The given equation is $2x − 7x^{\frac{1}{2}}= 15$.

Simplify $$\begin{gathered} 2x − 7x^{\frac{1}{2}}= 15 \\ 2x-15=7x^{\frac{1}{2}} \end{gathered}$$

Squaring both sides , we get:-

$$\begin{gathered} {(2x-15)}^2={\left(7x^{\frac{1}{2}}\right)}^2 \\ 225-60x+4x^2=49x \\ 4x^2-60x-49x+225=0 \\ 4x^2-109x+225=0 \end{gathered}$$

Using Quadratic formula :-

$$\begin{gathered} x_{1, 2}=\frac{-\left(-109\right)Â\pm \sqrt{{\left(-109\right)}^2-4· 4· 225}}{2· 4} \\ {x}_{1, 2}=\frac{-\left(-109\right)Â\pm  91}{2· 4} \\ {x}_1=\frac{-\left(-109\right)+91}{2· 4}, x_2=\frac{-\left(-109\right)-91}{2· 4} \\ x=25, x=\frac{9}{4} \end{gathered}$$

Substitute the value of $x=25$in equation $2x − 7x^{\frac{1}{2}}- 15=0$, we get:-

$$\begin{gathered} 2\left(25\right)− 7{\left(25\right)}^{\frac{1}{2}}- 15=50-35-15 \\ 50-35-15=0 \end{gathered}$$

Similarly , Substitute the value of $x=\frac{9}{4}$in the equation $2x − 7x^{\frac{1}{2}}- 15=0$, we get:-

role="math" localid="1645356098661" $$\begin{gathered} 2\left(\frac{9}{4}\right) − 7{\left(\frac{9}{4}\right)}^{\frac{1}{2}}-15=\frac{9}{2}-\frac{21}{2}-15 \\ \frac{9}{2}-\frac{21}{2}-15=\frac{-12}{2}-15 \\ \frac{-12}{2}-15=-6-15 \\ -6-15=-21 \\ -21≠0 \end{gathered}$$

Since, The condition for $x=25$ is verified. Therefore, The solution of the equation is $x=25$.