Question

Using the properties of determinants, solve the following for x:

`|[x+2,x+6,x-1],[x+6,x-1,x+2],[x-1,x+2,x+6]|=0`

Answer 1

`Let Delta=|[x+2,x+6,x-1],[x+6,x-1,x+2],[x-1,x+2,x+6]|`

Applying `C_2->C_2-C_1 and C_3->C_3-C_1`

`Delta=|[x+2,4,-3],[x+6,-7,-4],[x-1,3,7]|`

Applying `R_2->R_2-R_1 and R_3->R_3-R_1`

`Delta=|[x+2,4,-3],[4,-11,-1],[-3,-1,10]|`

Applying ` R_2->R_2+R_3`

`Delta=|[x+2,4,-3],[1,-12,9],[-3,-1,10]|`

Applying ` R_3->R_3+(3)R_2`

`Delta=|[x+2,4,-3],[1,-12,9],[0,-37,37]|`

Expanding along C₁

`Delta=(x+2)|[-12,9],[-37,37]|-1|[4,-3],[-37,37]|`

`Delta=(x+2)(-444+333)-1(148-111)`

`Delta=(x+2)(-111)-1(37)`

`Delta=0=-111x-259`

`x=-259/111=-7/3`