WAEC 2016 · Paper 2 · Q3

(a)
If $f(x + 2) = 6x^2 + 5x - 8$ , find $f(5)$ .
(b)
Express $\dfrac{7\sqrt2 + 3\sqrt3}{4\sqrt2 - 2\sqrt3}$ in the form $p + q\sqrt r$ , where $p$ , $q$ and $r$ are rational numbers.

Worked solution (try it first)

(a)

Find the $x$ that makes the bracket 5: $x + 2 = 5$ , so $x = 3$ .
Put $x = 3$ into the right-hand side: $f(5) = 6(3)^2 + 5(3) - 8$
$= 54 + 15 - 8$
$= 61$ .

(b)

Multiply the top and the bottom by the conjugate of the bottom, $4\sqrt2 + 2\sqrt3$ .
The bottom: $(4\sqrt2)^2 - (2\sqrt3)^2 = 32 - 12 = 20$ .
The top: $(7\sqrt2 + 3\sqrt3)(4\sqrt2 + 2\sqrt3) = 56 + 14\sqrt6 + 12\sqrt6 + 18$
$= 74 + 26\sqrt6$ .
Divide each term by 20: $\dfrac{74}{20} + \dfrac{26}{20}\sqrt6 = \dfrac{37}{10} + \dfrac{13}{10}\sqrt6$ .
So $p = \frac{37}{10}$ , $q = \frac{13}{10}$ , $r = 6$ .

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