Solve the equation

Question

[a-x b-x c
a-x c b-x
a b-x c-x]= 0

Answer 1

To solve this equation, we can first expand the determinant along the first row.

[a-x b-x c = (a-x)[(b-x)(c-x) - (c)(b-x)]
a-x c b-x - (c)[(a-x)(b-x) - (a)(b-x)]
a b-x c-x] + (b-x)[(a-x)(c) - (a)(c)]

= (a-x)[bc - bx - cx + x^2 - cb + cx] - c[ab - ax - bx + x^2 - ab + ax]
+ (b-x)[ac - axc - ac + acx - ac + ac]

= (a-x)[x^2 - cx - cb + x^2 - cb]
- c[x^2 - bx - ab + x^2 - ab]
+ (b-x)[acx - acx]

= (2x^2 - 2cx - 2cb) - (2x^2 - 2bx - 2ab)

= 2bx + 2ab - 2cx - 2cb

Setting this equal to zero and solving for x, we get:

2bx + 2ab = 2cx + 2cb
bx + ab = cx + cb
bx - cx = cb - ab
x(b - c) = b(c - a)
x = b(c - a) / (b - c)