If the roots of the equation a x 2 + 2 b x + c = 0  be alpha and beta and those of the equation A x 2 + 2 B x + C = 0  be alpha+delta and beta+delta prove that,
b 2 - a c a 2 = B 2 - A C A 2

Dear Student,

ax2+2bx+c=0 has roots α and βα+β=-2baba=-α+β2and αβ=ca                                    eq1Ax2+2Bx+C=0 has roots α+δ and β+δ α+δ+β+δ=-2BABA=-α+2δ+β2and α+δβ+δ=CA                                         eq2now To prove b2-aca2=B2-ACA2L.H.S.=b2-aca2=b2a2-ca=-α+β22-αβ     using eq1=α+β22-αβ  =α2+β2+2αβ-4αβ4=α2+β2-2αβ4=α-β24R.H.S.=B2-ACA2=B2A2-CA=-α+2δ+β22-α+δβ+δ=α+2δ+β22-α+δβ+δ=α2+4δ2+β2+4αδ+4δβ+2αβ4-αβ+αδ+δβ+δ2=α2+4δ2+β2+4αδ+4δβ+2αβ-4αβ+αδ+δβ+δ24=α2+4δ2+β2+4αδ+4δβ+2αβ-4αβ-4αδ-4δβ-4δ24=α2+β2+2αβ-4αβ4=α2+β2-2αβ4=α-β24L.H.S.=R.H.S.

Regards

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