Q. 8.122 - Intermediate Algebra

Question

Solve: $\sqrt{n+1}-n+1=0$

Step-by-Step Solution

Answer

The solution is, $n=3$.

Step 1 Given equation is,

$\sqrt{n+1}-n+1=0.$

The objective is, we need to find the solution for the given equation.

Also, we know that,${\left(a-b\right)}^2=a^2-2ab+b^2$.

Step 2 First isolate the radical.

$\sqrt{n+1}=n-1$.

Take square on both sides.

$$\begin{gathered} n+1={\left(n-1\right)}^2 \\ n+1=n^2-2n+1 \end{gathered}$$

Simplify it.

$$\begin{gathered} n^2-2n+1-n-1=0 \\ {n}^2-3n=0 \\ n\left(n-3\right)=0 \end{gathered}$$

Now use the zero product property.

$$\begin{gathered} n=0 \\ n-3=0 \\ n=3 \end{gathered}$$

Step 3 Let us check the answer.

Substitute $n=0$ into the equation $\sqrt{n+1}-n+1=0$.

 $$\begin{gathered} \sqrt{0+1}-0+1=0 \\ \sqrt{1}+1=0 \\ 1+1=0 \\ 2\ne 0 \end{gathered}$$

which is not true.

Substitute $n=3$ into the equation $\sqrt{n+1}-n+1=0$.

  $$\begin{gathered} \sqrt{3+1}-3+1=0 \\ \sqrt{4}-2=0 \\ 2-2=0 \\ 0=0 \end{gathered}$$

which is true.

So, the solution is,  $n=3$.