(a) Solve : (2x^{2} + x – 6 < 0)
(b) Express (frac{5 – 2sqrt{10}}{3sqrt{5} + sqrt{2}}) in the form (msqrt{2} + nsqrt{5}) where m and n are rational numbers.
Explanation
(a) (2x^{2} + x – 6 < 0 implies 2x^{2} + 4x – 3x – 6 < 0)
(2x(x + 2) – 3(x + 2) < 0 implies (2x – 3)(x + 2) < 0)
If ((2x – 3)(x + 2) < 0), then either (2x – 3) < 0 or (x + 2) < 0.
((2x – 3) < 0 implies x < frac{3}{2}), then ((x + 2) > 0 implies x > -2)
((x + 2) < 0 implies x < -2), then ((2x – 3) > 0 implies x > frac{3}{2})
Check: Let x = -3,
((2x – 3)(x + 2) = (-9)(-1) = 9 > 0)
Let x = 1,
((2x – 3)(x + 2) = (-1)(3) = -3 < 0)
(therefore -2 < x < frac{3}{2}).
(b) (frac{5 – 2sqrt{10}}{3sqrt{5} + sqrt{2}})
= ((frac{5 – 2sqrt{10}}{3sqrt{5} + sqrt{2}})(frac{3sqrt{5} – sqrt{2}}{3sqrt{5} – sqrt{2}}))
= (frac{15sqrt{5} – 5sqrt{2} – 30sqrt{2} + 4sqrt{5}}{9(5) – 3sqrt{10} + 3sqrt{10} – 2})
= (frac{19sqrt{5} – 35sqrt{2}}{45 – 2})
= (frac{19sqrt{5} – 35sqrt{2}}{43})
= (frac{-35}{43} sqrt{2} + frac{19}{43} sqrt{5})
(therefore m = -frac{-35}{43} ; n = frac{19}{43}).