In quadratic equation ax^2 + bx + c = 0 , Identities the sum and product of Roots? -Maths 9th

In quadratic equation ax^2 + bx + c = 0 , Identities the sum and product of Roots? -Maths 9th
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If the two roots of the quadratic equation ax2 + bx + c = 0 obtained by the quadratic formula be denoted by a and b, then we have α = \(\frac{-b+\sqrt{b^2-4ac}}{2a},\) β = \(\frac{-b-\sqrt{b^2-4ac}}{2a}\) ∴ Sum of roots = α + β = \(\frac{-b+\sqrt{b^2-4ac}}{2a}\) + \(\frac{-b-\sqrt{b^2-4ac}}{2a}\) = \(\frac{-2b}{2a}=\frac{-b}{a}\) Product of roots = αβ = \(\bigg(\frac{-b+\sqrt{b^2-4ac}}{2a}\bigg)\) x \(\bigg(\frac{-b-\sqrt{b^2-4ac}}{2a}\bigg)\) = \(\frac{(-b)^2-(\sqrt{b^2-4ac})^2}{4a^2}\) = \(\frac{b^2-(b^2-4ac)}{4a^2}\) = \(\frac{4ac}{4a^2}\) = \(\frac{c}{a}.\) Thus,  Sum of roots = \(\frac{\text{-Coeff. of x}}{Coeff.\,of\,x^2}\); Product of roots = \(\frac{\text{Constant term}}{Coeff.\,of\,x^2}\) Thus, if α, β be the roots of the equation 6x2 – 5x + 7 = 0, then α + β = – (– \(\frac{5}{6}\)) = \(\frac{5}{6}\), ab = \(\frac{7}{6}\).
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