(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 \(p + q\sqrt{r}\), whe...
Question 1 Report
(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.
Answer Details
(a) To find \(f(5)\) from \( f(x+2) = 6x^2 + 5x - 8 \), set \( x + 2 = 5 \), i.e. \( x = 3 \):
\[ f(5) = 6(3)^2 + 5(3) - 8 = 54 + 15 - 8 = 61. \]
(b) Rationalise by multiplying numerator and denominator by \( 4\sqrt2 + 2\sqrt3 \):
Denominator: \( (4\sqrt2 - 2\sqrt3)(4\sqrt2 + 2\sqrt3) = (4\sqrt2)^2 - (2\sqrt3)^2 = 32 - 12 = 20. \)
Numerator: \( (7\sqrt2 + 3\sqrt3)(4\sqrt2 + 2\sqrt3) = 56 + 14\sqrt6 + 12\sqrt6 + 18 = 74 + 26\sqrt6. \)
\[ \frac{74 + 26\sqrt6}{20} = \frac{37}{10} + \frac{13}{10}\sqrt6. \]
So \( p = \dfrac{37}{10},\ q = \dfrac{13}{10},\ r = 6 \).
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