(a) If (f(x + 2) = 6x^{2} + 5x – 8), find (f(5)).
(b) Express (frac{7sqrt{2} + 3sqrt{3}}{4sqrt{2} – 2sqrt{3}}) in the form (p + qsqrt{r}), where p, q and r are rational numbers.
Explanation
(a) (f(x + 2) = 6x^{2} + 5x – 8)
(f(5) implies x + 2 = 5)
(therefore x = 3)
(f(5) = f(3 + 2) = 6(3^{2}) + 5(3) – 8)
= (54 + 15 – 8)
= (61)
(b) (frac{7sqrt{2} + 3sqrt{3}}{4sqrt{2} – 2sqrt{3}})
Rationalizing, we multiply the numerator and denominator by (4sqrt{2} + 2sqrt{3})
((frac{7sqrt{2} + 3sqrt{3}}{4sqrt{2} – 3sqrt{3}})(frac{4sqrt{2} + 2sqrt{3}}{4sqrt{2} + 2sqrt{3}}))
= (frac{28(2) + 14sqrt{6} + 12sqrt{6} + 6(3)}{16(2) – 4(3)})
= (frac{74 + 26sqrt{6}}{20})
= (3.7 + 1.3sqrt{6})