Question

find the roots of given equation by the method of completing the square x^2–(√2+1)x+√2=0 ​

Answer 1

${x}^{2} - ( \sqrt{2} + 1)x + \sqrt{2} = 0 \\ \\ {x}^{2} - 2 \times ( \frac{ \sqrt{2} + 1}{2} )x = - \sqrt{2} \\ \\ {x}^{2} - ( \sqrt{2} + 1)x + \frac{3 + 2 \sqrt{2} }{4} - ( \frac{3 + 2 \sqrt{2} }{4} ) = - \sqrt{2} \\ \\ {(x - ( \frac{ \sqrt{2} + 1}{2} ))}^{2} = \frac{3 + 2 \sqrt{2} }{4} - \sqrt{2} \\ \\ {(x - ( \frac{ \sqrt{2} + 1}{2} ))}^{2} = \frac{3 - 2 \sqrt{2} }{4} \\ \\ {(x - ( \frac{ \sqrt{2} + 1}{2} ))} = | \frac{ \sqrt{3 - 2 \sqrt{2} } }{2} | \\ \\ x = \frac{ \sqrt{2} + 1 + \sqrt{3 - 2 \sqrt{2} } }{2} \\ \\ x = \frac{ \sqrt{2} + 1 - \sqrt{3 - 2 \sqrt{2} } }{2}$

hope it helps.