Question

Solve:$\sqrt{y-5}+5=y$.

Short answer

The solution is$y=5,6$.

Step-by-step solution

Step 1. Given Information   

We are given the quadratic equation $\sqrt{y-5}+5=y$and we need to solve it.

Step 2. Finding the solution   

Isolating the radical on one side of the equation:

$$\begin{gathered} \sqrt{y-5}+5=y \\ \Rightarrow \sqrt{y-5}+5-5=y-5 \\ \Rightarrow \sqrt{y-5}=y-5 \end{gathered}$$

Squaring both sides of the equation, we get:

localid="1647942171594" $$\begin{gathered} {\left(\sqrt{y-5}\right)}^2={\left(y-5\right)}^2 \\ \Rightarrow y-5=y^2-10y+25 \\ \Rightarrow y^2-10y+25-y+5=0 \\ \Rightarrow y^2-11y+30=0 \\ \Rightarrow y^2-5y-6y+30=0 \\ \Rightarrow y\left(y-5\right)-6\left(y-5\right)=0 \\ \Rightarrow \left(y-5\right)\left(y-6\right)=0 \end{gathered}$$

Using the zero product property, we get:

$$\begin{gathered} \left(y-5\right)=0 \\ \Rightarrow y=5 \\ \left(y-6\right)=0 \\ \Rightarrow y=6 \end{gathered}$$

Step 3. Checking the answer  

$$\begin{gathered} y=5, \\ \sqrt{y-5}+5=y \\ \Rightarrow \sqrt{5-5}+5=5 \\ \Rightarrow 0+5=5 \\ \Rightarrow 5=5 \\ y=6, \\ \sqrt{y-5}+5=y \\ \Rightarrow \sqrt{6-5}+5=6 \\ \Rightarrow \sqrt{1}+5=6 \\ \Rightarrow 1+5=6 \\ \Rightarrow 6=6 \end{gathered}$$