Theory question 101
Polynomials(a) Solve 2x2 + x − 6 < 0
(b) Express (5 − 2√10)/(3√5 + √2) in the form m√2 + n√5 where m and n are rational numbers.
Show the model answer
Model answer
(a) (2x − 3)(x + 2) < 0
Negative between the roots: −2 < x < 3/2
(b) Multiply by (3√5 − √2)/(3√5 − √2); denominator 45 − 2 = 43
Numerator: 15√5 − 5√2 − 6√50 + 2√20 = 19√5 − 35√2
= −(35/43)√2 + (19/43)√5, so m = −35/43, n = 19/43
Explanation
Rationalise with the conjugate 3√5 − √2 and collect √2 and √5 terms.