Chapter 1.5, Problem 36E

To determine

To solve: The equation $2^x+2^{-x}=5$ and support the solution graphically.

Answer to Problem 36E

The solution of the equation $2^x+2^{-x}=5$ is approximately $\pm 2.26$.

Explanation of Solution

Given information: The equation is $2^x+2^{-x}=5$.

Calculation:

Simplify the equation $2^x+2^{-x}=5$ as follows.

  $\begin{array}{c}{2}^x+2^{-x}=5\\ 2^x-5+2^{-x}=0\end{array}$

Multiply both side of the equation by $2^x$.

  $\begin{array}{c}{2}^x\left(2^x-5+2^{-x}\right)=2^x\left(0\right)\\ {\left(2^x\right)}^2-5\left(2^x\right)+1=0\end{array}$

Solve the quadratic equation by the quadratic formula $e^x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.

  $\begin{array}{c}{2}^x=\frac{5\pm \sqrt{{\left(-5\right)}^2-4\left(1\right)\left(1\right)}}{2\left(1\right)}\\ =\frac{5\pm \sqrt{25-4}}{2}\\ =\frac{5\pm \sqrt{21}}{2}\end{array}$

Take $\log$ on both sides of the above equation as follows.

  $\begin{array}{c}\log 2^x=\log \left(\frac{5\pm \sqrt{21}}{2}\right)\\ x=\log \left(\frac{5\pm \sqrt{21}}{2}\right)\\ x\approx -2.26\text{or}2.26\end{array}$

To verify the solution graphically, plot the graph of the equation $2^x+2^{-x}=5$ . The intersection point of these two graphs will be the solution.

Follow the steps to solve the equation by calculator:

• First press “ON” button on graphical calculator and press $\boxed{\text{Y}=}$ key.
• Enter right hand side of the equation after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(}\boxed{2}\boxed{)}\boxed{^}\boxed{\text{X}}\boxed{+}\boxed{(}\boxed{2}\boxed{)}\boxed{^}\boxed{-}\boxed{\text{X}}$ . Now, press the key $\boxed{\text{ENTER}}$ .
• Enter right left side of the equation after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{5}$ .
• Press $\boxed{\text{WINDOW}}$ key and adjust the window at $\left[-4,4\right]$ by $\left[-4,8\right]$ .
• Now, press keystrokes given below to find the value of $t$ as:

  $\boxed{2\text{nd}}\to \boxed{\text{TRACE}}\to \boxed{5}\to \boxed{\text{ENTER}}\to \boxed{\text{ENTER}}\to \boxed{\text{ENTER}}$

  

From the graph it can be observed that the intersection point is $\left(\pm 2.26,5\right)$.

Therefore, the solution of the equation $2^x+2^{-x}=5$ is approximately $\pm 2.26$.