Exponential equation with substitution

Statement

Solve the equation: $2^{x+1} + 2^{-x+1} = 5$

Solution

Recognising the strategy. Both $2^x$ and $2^{-x} = 1/2^x$ appear: the power $2^x$ recurs, so it is convenient to use the substitution $t = 2^x$ (with the constraint $t>0$), which reduces the equation to a quadratic one. We rewrite using $2^{x+1} = 2\cdot 2^x$ and $2^{-x+1} = 2\cdot 2^{-x}$: $2\cdot 2^x + 2\cdot 2^{-x} = 5$ Set $t = 2^x$, with $t>0$: $2t + \frac{2}{t} = 5 \;\Longrightarrow\; 2t^2 - 5t + 2 = 0 \;\Longrightarrow\; t = 2 \ \text{oppure}\ t = \tfrac{1}{2}$ Both values satisfy the constraint $t>0$. Returning to $x$: $2^x = 2 \Rightarrow x = 1; \qquad 2^x = \tfrac{1}{2} \Rightarrow x = -1$ $\boxed{x = 1 \quad\text{oppure}\quad x = -1}$

Links

Topics: Unifying methods
Concepts: Quadratic equation · Exponential equation · Substitution
Functions: Exponential function
Skills: Calculating · Solving equations
Exercise type: Solving an equation