Find the roots of the following quadratic (if they exist) by the method of completing the square. x^2 - (√ 2 + 1) x + √ 2 = 0

Question

Find the roots of the following quadratic (if they exist) by the method of completing the square.

\(x^2-(\sqrt{2}+1)x+\sqrt{2}=0\)

Answer 1

We have to make the quadratic equation a perfect square if possible or sum of perfect square with a constant.

(a + b)² = a² + 2ab + b²

\(x^2-(\sqrt{2}+1)x+\sqrt{2}=0\)

⇒ x² – (√2 + 1)x + ((√2 + 1)/2)² - ((√2 + 1)/2)² + √2 = 0

⇒ (x - (√2 + 1)/2)² = (2 + 1 + 2√2)/4 - √2

⇒ (x – (√2 + 1)/2)² = (2 + 1 – 2√2)/4 = ((√2 – 1)/2)²

⇒ x - (√2 + 1)/² = (√2 – 1)/²

⇒ x = √2, 1