6. (a) Write √80 in the form c√5, where c is a positive constant - Edexcel - A-Level Maths Pure - Question 8 - 2014 - Paper 1

Given:

• Length of the rectangle R = (1 + √5) cm
• Area of rectangle R = √80 cm²

We can express the area of a rectangle as:

ext{Area} = ext{Length} imes ext{Width}\ ext{Width} = rac{ ext{Area}}{ ext{Length}}\

Substituting the known values:

ext{Width} = rac{√80}{1 + √5}\

To simplify this expression, we will rationalize the denominator. Multiplying both the numerator and the denominator by (1 - √5):

ext{Width} = rac{√80(1 - √5)}{(1 + √5)(1 - √5)} = \ rac{√80(1 - √5)}{1 - 5} = rac{√80(1 - √5)}{-4}\

This leads to:

ext{Width} = -rac{√80(1 - √5)}{4}\

Substituting √80 = 4√5:

ext{Width} = -rac{4√5(1 - √5)}{4} = -√5(1 - √5)\ = -√5 + 5\

Thus, the width can be expressed as: $ext{Width} = 5 - √5$

This can be rewritten in the form p + q√5 as: $5 - 1√5 = 5 + (-1)√5$

This means p = 5 and q = -1.