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If (7 - 4√3) raised to the power (x² - 4x + 3), together with (7 + 4√3) raised to the same power, adds up to 14, then the possible value(s) of x are

1. 2, 2 ± √2
2. 2 ± √3, 3
3. 3 ± √2, 2
4. None of these

Correct answer: 2, 2 ± √2

Solution

With k = x^2-4x+3, putting a = (7+4sqrt3)^k gives a + 1/a = 14, so a = 7 +/- 4sqrt3, i.e. k = 1 or k = -1. k=-1 gives x^2-4x+4=0 -> x=2; k=1 gives x^2-4x+2=0 -> x = 2 +/- sqrt2. Solutions: 2, 2 +/- sqrt2.

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