here we have to find the quadratic equation whose one root is 1+ √2 and the sum of its roots is 2.
let the other root of the equation be x.
➡ 1 + √2 + x = 2
➡ x = 2 - 1 - √2
➡ x = 1 - √2
another root of the quadratic equation is 1 - √2
therefore product of the roots = (1 + √2)(1 - √2)
using identity (a + b)(a - b) = a² - b²
= (1)² - (√2)²
= 1 - 2
= -1
now we know that,
sum of roots = -b/a
product of roots = c/a
2 = -b/a
➡ b/a = -2
-1 = c/a
therefore a = 1, b = -2 and c = -1
standard form of quadratic equation = ax² + bx + c
hence, the quadratic equation is =
x² - 2x - c