Question

Prove that, if the sum of the squares of the roots of the equation aײ+ b×+ c =0 is 1, then b²=2ac+a²

Answer 1

Let the roots of the equation ax² + bx + c = 0 be α, β

Sum of roots = α + β = - b / a

Product of roots = αβ = c / a

Given :

Sum of squares of roots = α² + β² = 1

Using algebraic identity ( α + β )² = α² + β² + 2αβ

⇒ ( - b / a )² = 2( c / a ) + 1

⇒ b² / a² = 2c / a + 1

Multiplying every term with a²

⇒ a² × ( b² / a² ) = a² × ( 2c / a + 1 )

⇒ b² = 2ac + a²

Hence proved.