## 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.

\(a=5-2 \sqrt{2}\)

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

\(c=-2\)

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

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

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

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

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

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

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

\(=2+\sqrt{2}\)

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

\(c=-2\)

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

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

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

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

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

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

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

\(=2+\sqrt{2}\)

## More Similar Questions

## 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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