Question

Find the roots of the following equation, if they exist, by applying the quadratic formula:

`2x^2 - 2sqrt(2)x + 1 = 0`

Answer 1

Given: `2x^2 - 2sqrt(2)x + 1 = 0`

Step-wise calculation:

1. Identify coefficients:

a = 2, b = `-2sqrt(2)`, c = 1

2. Discriminant:

D = b² – 4ac 

= `(-2sqrt(2))^2 - 4(2)(1)`

= 8 – 8

= 0

3. Since D = 0, the quadratic has real and equal roots.

4. Quadratic formula:

`x = (-b ± sqrt(D))/(2a)` 

= `(-(-2sqrt(2)) ± 0)/(4)` 

= `(2sqrt(2))/4` 

= `sqrt(2)/2`

The equation has a repeated root `x = sqrt(2)/2` (multiplicity 2).