Solve the following: `|[1,x,x^2], [1,a, b^2], [1,b,c^2]|=0` , `a!=b`

Explanation

`|[1,x,x^2], [1,a, b^2], [1,b,c^2]|=0`


`R_{1} rightarrow R_{1}-R_{2}`

`R_{2} rightarrow R_{2}-R_{3}`


Now, `|[0 , x-a , x^{2}-b^{2} ],[ 0 , a-b , b^{2}-c^{2}], [1 , b, c^{2}]|=0`


i.e. `1(x-a)(b^{2}-c^{2})-(a-b)(x^{2}-b^{2})=0`


i.e.`x b^{2}-x c^{2}-a b^{2}+a c^{2}-a x^{2}+a b^{2}+b x^{2}`
`-b^{3}=0`


i.e. `x^{2}(b-a)+x(b^{2}-c^{2})-b^{2} a+c^{2} a+b^{2} a-b^{3}=0`


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