Question

State whether the following quadratic equations have two distinct real roots. Justify your answer.

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

Answer 1

The equation `(x - sqrt(2))^2 - 2(x + 1)` = 0 has two distinct and real roots.

Simplifying the above equation,

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

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

`x^2 - 3sqrt(2)x + (2 - sqrt(2))` = 0

D = b² – 4ac

= `(- 3sqrt(2))^2 – 4(1)(2 - sqrt(2))`

= `18 - 8 + 4sqrt(2) > 0`

Hence, the roots are real and distinct.