Solving a Complex Exponential Equation

00:00 5 marks

Consider the equation \(e^{2x} – 5e^{x} + 6 = 0\).

(a) Rearrange the equation by substituting \(u = e^x\) to express it as a quadratic equation in \(u\).

(b) Solve the quadratic equation and then find the corresponding values of \(x\) (give your answers to three significant figures).

Part a

4 minutes 2 marks

Question

Substitute \(u = e^x\) in \(e^{2x} – 5e^{x} + 6 = 0\) to rewrite the equation as a quadratic in \(u\).

Hint Hint

Hint

Recognize that \(e^{2x} = (e^x)^2\) and substitute accordingly.

Answer Answer
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Complete Answer

The equation becomes \(u^2 – 5u + 6 = 0\) with \(u = e^x\).

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Part b

6 minutes 3 marks

Question

Solve the quadratic equation \(u^2 – 5u + 6 = 0\) to find \(u\) and then determine \(x\) using \(x = \ln(u)\). Give your answers to three significant figures.

Hint Hint

Hint

Factor the quadratic equation and apply logarithms to find \(x\).

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Complete Answer

The solutions are \(x = \ln(2) \approx 0.693\) and \(x = \ln(3) \approx 1.099\).

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