Question

$prove space that space open vertical bar table row straight a straight b cell space space space space space ax plus by end cell row straight b straight c cell space space space space space bx plus cy end cell row cell ax plus by space end cell cell space space bx plus cy end cell 0 end table close vertical bar space equals left parenthesis straight b squared minus ac right parenthesis left parenthesis ax squared plus 2 space straight b space straight x space straight y space plus straight c space straight y cubed right parenthesis$

Solution

Let space space increment space equals space open vertical bar table row straight a straight b cell space space space space space ax plus by end cell row straight b straight c cell space space space space space bx plus cy end cell row cell ax plus by space end cell cell space space bx plus cy end cell 0 end table close vertical bar space space space space space space space space space space space space space equals space begin inline style 1 over xy end style space space space open vertical bar table row straight a straight b cell space space space space space ax plus by end cell row straight b straight c cell space space space space space bx plus cy end cell row cell ax plus by space end cell cell space space bx plus cy end cell 0 end table close vertical bar space space Multiplying space straight C subscript 1 space end subscript by space straight x. straight C subscript 2 space by space straight y space space space space space space space space space space space space equals space begin inline style 1 over xy end style space space space open vertical bar table row straight a straight b cell space space space space space ax plus by end cell row straight b straight c cell space space space space space bx plus cy end cell row cell ax plus by space end cell cell space space bx plus cy end cell 0 end table close vertical bar space by space straight C subscript 1 space end subscript plus. straight C subscript 2 minus straight C subscript 3 space space space space space space space space space space space space equals space begin inline style 1 over xy space space left parenthesis straight a space straight x end style squared space plus 2 space straight b space straight x space straight y space plus straight c space straight y squared right parenthesis space open vertical bar table row cell straight b space straight y end cell cell straight a space straight x plus straight b space straight y end cell row cell straight c space straight y end cell cell straight b space straight x plus straight c space straight y end cell end table close vertical bar space comma space expanding space by space first space column space space space space space space space space space space space space equals space begin inline style fraction numerator straight a space straight x squared plus 2 space straight b space straight x space straight y plus straight c space straight y squared over denominator straight x space straight y end fraction end style open square brackets table row cell by space left parenthesis straight b space straight x plus straight c space straight y right parenthesis minus cy left parenthesis straight a space straight x plus straight b space straight y right parenthesis end cell end table close square brackets space space space space space space space space space space space space equals space begin inline style fraction numerator straight a space straight x squared plus 2 space straight b space straight x space straight y plus straight c space straight y squared over denominator straight x space straight y end fraction space open square brackets table row cell straight b squared end cell end table space straight x space straight y space plus straight b space straight c space straight y squared space minus straight c space straight a space straight x space straight y space minus straight b space straight c space straight y squared close square brackets end style space space space space space space space space space space space space equals space begin inline style fraction numerator straight a space straight x squared plus 2 space straight b space straight x space straight y plus straight c space straight y squared over denominator straight x space straight y end fraction space space open square brackets table row cell straight b squared end cell end table xy minus space straight c space straight a space straight x space straight y close square brackets space equals space fraction numerator straight a space straight x squared space plus 2 space straight b space straight x space straight y plus straight c space straight y squared over denominator straight x space straight y end fraction space space open square brackets table row cell left parenthesis straight b squared minus ca right parenthesis space straight x space straight y end cell end table close square brackets space end style space space space space space space space space space space space space equals space space space left parenthesis straight b squared space minus ca right parenthesis space right parenthesis left parenthesis straight a space straight x squared space plus 2 space straight b space straight x space straight y plus straight c space straight y squared space right parenthesis space space space space space space space space space space space space space space space space space

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