Solving Quadratic Equation with Surd

2011 Oct Nov Paper 23 Q4

Without using a calculator, find the positive root of the equation
$(5-2 \sqrt{2}) x^{2}-(4+2 \sqrt{2}) x-2=0,$
giving your answer in the form $a+b \sqrt{2}$ , where $a$ and $b$ are integers.

a = 5 - 2 \sqrt{2}

$a=5-2 \sqrt{2}$

b = - (4 + 2 \sqrt{2})

$b=-(4+2 \sqrt{2})$

c = - 2

$c=-2$

x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

$x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$

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

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

= \frac{4 + 2 \sqrt{2} + \sqrt{(16 + 16 \sqrt{2} + 8) + 40 - 16 \sqrt{2}}}{2 (5 - 2 \sqrt{2}}

$=\frac{4+2 \sqrt{2}+\sqrt{(16+16 \sqrt{2}+8)+40-16 \sqrt{2}}}{2(5-2 \sqrt{2}}$

= \frac{4 + 2 \sqrt{2} + \sqrt{64}}{2 (5 - 2 \sqrt{2})}

$=\frac{4+2 \sqrt{2}+\sqrt{64}}{2(5-2 \sqrt{2})}$

= \frac{12 + 2 \sqrt{2}}{2 (5 - 2 \sqrt{2})}

$=\frac{12+2 \sqrt{2}}{2(5-2 \sqrt{2})}$

= \frac{(6 + \sqrt{2}) (5 + 2 \sqrt{2})}{(5 - 2 \sqrt{2}) (5 + 2 \sqrt{2})}

$=\frac{(6+\sqrt{2})(5+2 \sqrt{2})}{(5-2 \sqrt{2})(5+2 \sqrt{2})}$

= \frac{30 + 15 \sqrt{2} + 5 \sqrt{2} + 4}{25 - 8}

$=\frac{30+15 \sqrt{2}+5 \sqrt{2}+4}{25-8}$

= 2 + \sqrt{2}

$=2+\sqrt{2}$

2014 May June Paper 23 Q5(ii)

Do not use a calculator in this question.
(ii) Solve the equation $(2 \sqrt{2}+3) x^{2}-(2 \sqrt{2}+4) x+2=0$ , giving your answer in the form $a+b \sqrt{2}$ where $a$ and $b$ are integers.

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Solving Quadratic Equation with Surd

2011 Oct Nov Paper 23 Q4

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2014 May June Paper 23 Q5(ii)

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