# 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.

$$a=5-2 \sqrt{2}$$
$$b=-(4+2 \sqrt{2})$$
$$c=-2$$

$$x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$$
$$=\frac{-[-(4+2 \sqrt{2})]+\sqrt{[-(4+\sqrt{2})]^{2}-4(5-2 \sqrt{2})(-2)}}{2(5-2 \sqrt{2})}$$
$$=\frac{4+2 \sqrt{2}+\sqrt{(16+16 \sqrt{2}+8)+40-16 \sqrt{2}}}{2(5-2 \sqrt{2}}$$
$$=\frac{4+2 \sqrt{2}+\sqrt{64}}{2(5-2 \sqrt{2})}$$
$$=\frac{12+2 \sqrt{2}}{2(5-2 \sqrt{2})}$$
$$=\frac{(6+\sqrt{2})(5+2 \sqrt{2})}{(5-2 \sqrt{2})(5+2 \sqrt{2})}$$
$$=\frac{30+15 \sqrt{2}+5 \sqrt{2}+4}{25-8}$$
$$=2+\sqrt{2}$$

## 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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