Solving Quadratic Equation with Surd

2011 Oct Nov Paper 23 Q4

Without using a calculator, find the positive root of the equation
(522)x2(4+22)x2=0,(522)x2(4+22)x2=0,
giving your answer in the form a+b2a+b2, where aa and bb are integers.

a=522a=522
b=(4+22)b=(4+22)
c=2c=2

x=b±b24ac2ax=b±b24ac2a
=[(4+22)]+[(4+2)]24(522)(2)2(522)=[(4+22)]+[(4+2)]24(522)(2)2(522)
=4+22+(16+162+8)+401622(522=4+22+(16+162+8)+401622(522
=4+22+642(522)=4+22+642(522)
=12+222(522)=12+222(522)
=(6+2)(5+22)(522)(5+22)=(6+2)(5+22)(522)(5+22)
=30+152+52+4258=30+152+52+4258
=2+2=2+2

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2014 May June Paper 23 Q5(ii)

Do not use a calculator in this question.
(ii) Solve the equation (22+3)x2(22+4)x+2=0(22+3)x2(22+4)x+2=0, giving your answer in the form a+b2a+b2 where aa and bb are integers.

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