Expand polynomial with surds

A question is this type if and only if it asks to expand and simplify a polynomial expression containing surds into the form ax^2 + bx√c + d, such as f(x) = (x + √2)^2 + (3x - 5√8)^2.

2 questions · Moderate -1.0

1.02j Manipulate polynomials: expanding, factorising, division, factor theorem
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Edexcel P1 2022 January Q3
7 marks Moderate -0.8
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
  1. $$f ( x ) = ( x + \sqrt { 2 } ) ^ { 2 } + ( 3 x - 5 \sqrt { 8 } ) ^ { 2 }$$ Express \(\mathrm { f } ( x )\) in the form \(a x ^ { 2 } + b x \sqrt { 2 } + c\) where \(a , b\) and \(c\) are integers to be found.
  2. Solve the equation $$\sqrt { 3 } ( 4 y - 3 \sqrt { 3 } ) = 5 y + \sqrt { 3 }$$ giving your answer in the form \(p + q \sqrt { 3 }\) where \(p\) and \(q\) are simplified fractions to be found.
Edexcel C1 Q1
3 marks Easy -1.2
1. $$f ( x ) = ( \sqrt { x } + 3 ) ^ { 2 } + ( 1 - 3 \sqrt { x } ) ^ { 2 }$$ Show that \(\mathrm { f } ( x )\) can be written in the form \(a x + b\) where \(a\) and \(b\) are integers to be found.