Solving a hyperbolic equation with a restriction

Choose your method, connect the parts and explain your conclusions. These original questions include written reasoning and sketches as well as exact answer checks.

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Before you start

Hyperbolic functions. Read the relevant method, then decide which parts you can solve without hints.

Check a misconception first · Try a connected problem

Read an original example question

$\cosh x=\frac{17}{8}$ , with $x\geq0$ .

Part a

Find x exactly.

Part b

State $e^{x}$ for the permitted solution.

Part c

State how many real solutions exist without x ≥ 0, and explain.

Model solution and review criteria

Part a

Put u = $e^{x}$ > 0. Then $u^{2}$ − 17/4u + 1 = 0, with roots 4 and $\frac{1}{4}$ . Since x ≥ 0, u ≥ 1; hence x = ln 4.

Use the exponential definition.
Solve the quadratic and keep both positive candidates initially.
Apply x ≥ 0 to choose a branch.

Part b

The permitted exponential is 4.

Use the branch restriction.

Part c

There are two, x = ±ln 4, because cosh is even and both exponential roots are positive. A negative u would never represent $e^{x}$ .

State both real values.
Distinguish a negative x from a negative $e^{x}$ .

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