Chapter 1.1, Problem 38E

Solution

To find: The solution of the equation and plot the graph for verification of solution.

The solution of the equation is $x=\frac{3}{2}-\frac{\sqrt{5}}{2}$ and the graph of the equation is shown below.

  

Given information:

The algebraic equation is $\sqrt{x}+x=1$ .

Concept used:

Use the formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

The equation is $\sqrt{x}+x=1$ .

  $\begin{array}{c}\sqrt{x}+x=1\\ {\left(\sqrt{x}\right)}^2={\left(1-x\right)}^2\\ x=1+x^2-2x\\ x^2-3x+1=0\end{array}$

Compare the equation $x^2-3x+1=0$ with $a{x}^2+bx+c=0$ .

The value of $a=1$ , $b=-3$ , $c=1$ .

  $\begin{array}{l}x=\frac{3\pm \sqrt{{\left(-3\right)}^2-4\times 1\times 1}}{2}\\ x=\frac{3\pm \sqrt{5}}{2}\\ x=\frac{3}{2}+\frac{\sqrt{5}}{2},\frac{3}{2}-\frac{\sqrt{5}}{2}\end{array}$

The value of $x=\frac{3}{2}+\frac{\sqrt{5}}{2}$ does not satisfy the equation.

So the value of $x=\frac{3}{2}-\frac{\sqrt{5}}{2}$ .

The graph of the equation verifying the solution is shown below.

  

Conclusion:

The solution of the equation both algebraically and graphically is $x=\frac{3}{2}-\frac{\sqrt{5}}{2}$ .