6 (a) Write \( \sqrt{80} \) in the form \( c \sqrt{5} \), where \( c \) is a positive constant - Edexcel - A-Level Maths Pure - Question 7 - 2014 - Paper 1

Given the area of rectangle R is ( \sqrt{80} ) cm² and the length is ( (1 + \sqrt{5}) ) cm, we can find the width ( W ) using the formula for the area:

[ \text{Area} = \text{Length} \times \text{Width} \Rightarrow W = \frac{\text{Area}}{\text{Length}} ]

Substituting the known values, we get:

[ W = \frac{\sqrt{80}}{(1 + \sqrt{5})} ]

Next, simplify ( \sqrt{80} ):

[ W = \frac{4\sqrt{5}}{(1 + \sqrt{5})} ]

To simplify further, we rationalize the denominator by multiplying the numerator and denominator by ( (1 - \sqrt{5}) ):

[ W = \frac{4\sqrt{5}(1 - \sqrt{5})}{(1 + \sqrt{5})(1 - \sqrt{5})} ]

Calculating the denominator:

[ (1 + \sqrt{5})(1 - \sqrt{5}) = 1^2 - (\sqrt{5})^2 = 1 - 5 = -4 ]

Calculating the numerator:

[ 4\sqrt{5}(1 - \sqrt{5}) = 4\sqrt{5} - 20 ]

Thus, we combine them:

[ W = \frac{4\sqrt{5} - 20}{-4} = 5 - \sqrt{5} ]

Setting ( p = 5 ) and ( q = -1 ), we express the width in the required form.