Question

Find the real solutions of the equation $x-5\sqrt{x}+6=0$.

Short answer

The given equation has two real solutions $x=4,9$.

Step-by-step solution

Step 1. Remove the radical of the given equation.

Rearrange the given equation by squaring it.

$$\begin{gathered} x-5\sqrt{x}+6=0....\left(\text{Given}\right) \\ x+6=5\sqrt{x} \\ {x}^2+36+12x=25x\mathrm{....}\left(\text{Squared}\right) \\ {x}^2-13x+36=0....\left(1\right) \end{gathered}$$

Step 2. Factoring (1).

Factorize the obtained equation (1),

$$\begin{gathered} x^2-9x-4x+36=0....\left(\text{From}\left(1\right)\right) \\ x\left(x-9\right)-4\left(x-9\right)=0 \\ \left(x-9\right)\left(x-4\right)=0 \\ x=9,4 \end{gathered}$$

Step 3. DESCRIPTION OF STEPS.

From the obtained values of solution the real solutions of the equation are$x=4,9$ and both of these solutions satisfy the given equation.