Question

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

`sqrt(2)x^2 + 7x + 5sqrt(2) = 0`

Answer 1

Given: `sqrt(2)x^2 + 7x + 5sqrt(2) = 0`

Step-wise calculation:

1. Compare with ax² + bx + c = 0: 

a = `sqrt(2)`, b = 7, c = `5sqrt(2)`

2. Discriminant:

D = b² – 4ac 

= `7^2 - 4(sqrt(2))(5sqrt(2))` 

= 49 – 4 × 5 × 2 

= 49 – 40

= 9

3. `sqrt(D) = 3`. By the quadratic formula `x = (-b ± sqrt(D))/(2a)`:

`x = (-7 ± 3)/(2sqrt(2))`

4. Compute each root:

`x_1 = (-7 + 3)/(2sqrt(2))` 

= `(-4)/(2sqrt(2))` 

= `(-2)/sqrt(2)`

= `-sqrt(2)`

`x_2 = (-7 - 3)/(2sqrt(2))` 

= `(-10)/(2sqrt(2))` 

= `(-5)/sqrt(2)`

= `-(5sqrt(2))/2`

The equation has two distinct real roots: `x = -sqrt(2)` and `x = -(5sqrt(2))/2`.