If x = (2 − √3)-1, then the value of x3 − 2x2 − 7x + 5 is

Concept

If a nonzero number has a simple reciprocal form, rationalising it can expose a quadratic identity. That identity lets higher powers be reduced without expanding radicals.

Application

1. Rationalise the denominator: x = 1/(2 − √3) = (2 + √3)/(4 − 3) = 2 + √3.

2. Therefore 1/x = 2 − √3, so x + 1/x = 4. Multiplying by x gives x2 − 4x + 1 = 0, hence x2 = 4x − 1.

3. Multiplying x2 = 4x − 1 by x gives x3 = 4x2 − x = 4(4x − 1) − x = 15x − 4.

4. Substitute these identities: x3 − 2x2 − 7x + 5 = (15x − 4) − 2(4x − 1) − 7x + 5 = 3.

Cross-check

Using x ≈ 3.732, direct substitution in the original expression gives approximately 3, matching the exact reduction. Therefore, the value is 3.