Irrational-exponential equation

Statement

Solve, or discuss the existence of the solutions of, the equation: $\sqrt{x+3} = 2\cdot 3^x$

Solution

Recognising the strategy. On the left there is an irrational function (which grows slowly), on the right an exponential function (which grows very rapidly): there is no common base nor a substitution that reduces it to an algebraic equation. It is therefore not an equation solvable “in closed form”: we tackle it with a qualitative/graphical study of the two sides. Existence condition: $x+3\ge 0$, that is $x\ge -3$. Set $h(x) = \sqrt{x+3} - 2\cdot 3^x$. The root grows slowly, whereas $2\cdot 3^x$ grows rapidly and for large $x$ dominates; for $x$ near $-3$, on the other hand, the root is small but positive and the exponential is even smaller. Hence $h$ goes from positive to negative just once: there is a single solution. Evaluating at a few points: $h(-1)=\sqrt{2}-\tfrac{2}{3}>0$ and $h(0)=\sqrt{3}-2<0$. The solution therefore lies between $-1$ and $0$ (numerically $x\approx -0{,}15$), and it can be approximated graphically or numerically.

Links

Topics: Unifying methods
Concepts: Exponential equation · Irrational equation · Graphical method
Functions: Exponential function
Skills: Interpreting a graph · Solving equations
Exercise type: Reading a graph · Solving an equation