 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-1/9/solution

Isolate the square root:

2 x^{2} + 17 = 3 - x

. Its right side must be nonnegative, so retain the condition

x \leq 3

before squaring. Squaring then gives

2 x^{2} + 17 = (3 - x)^{2}, x^{2} + 6 x + 8 = (x + 2) (x + 4) = 0.

Both candidates pass the original equation:

25 - 2 = 3

and

49 - 4 = 3

. Thus

x = - 2, x = - 4.

The substitution check excludes the possibility of extraneous roots introduced by squaring.

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