Question

Show that `x = - (bc)/(ad)` is a solution of the quadratic equation `ad^2((ax)/b + (2c)/d)x + bc^2 = 0`.

Answer 1

Given:

Quadratic: `ad^2((ax)/b + (2c)/d)x + bc^2 = 0`

Proposed solution: `x = - (bc)/(ad)`.

Step-wise calculation:

1. Compute `(ax)/b` with `x = -(bc)/(ad)`: 

`a xx x = a xx (-(bc)/(ad))` 

= `(-(bc)/d)`

So, `(ax)/b = (-(bc)/d)/b`

= `-c/d`

2. Sum inside parentheses:

`(ax)/b + (2c)/d = −c/d + (2c)/d`

= `c/d`

3. Multiply by ad² and x:

`ad^2 xx (c/d) xx x = (acd) xx x`

4. Substitute `x = -(bc)/(ad)`:

`(acd) xx (-(bc)/(ad)) = -bc^2`

5. Add bc² the constant term:

–bc² + bc² = 0

Substituting `x = -(bc)/(ad)` into the quadratic yields 0, so `x = -(bc)/(ad)` is indeed a solution.