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