What is the ratio of sum of squares of roots to the product of the roo

1.	What is the ratio of sum of squares of roots to the product of the roots of the equation 7x2 + 12x + 18 = 0?
Answer» Let α, β be the roots of the equation 7x² + 12x + 18 = 0. \(\bigg[\)For a quadratic equation ax² + bx + c = 0, sum of roots = \(-\frac{a}{b}\), product of roots = + \(\frac{c}{a}\)\(\bigg]\) ∴ α + β = \(\frac{12}{7}\) and αβ = \(\frac{18}{7}\) ⇒ (α + β)² = \(\bigg(\frac{-12}{7}\bigg)^2\) ⇒ α² + β²+ 2αβ = \(\frac{144}{49}\) ⇒ α² + β² = \(\frac{144}{49}\) - \(\frac{36}{7}\) = \(\frac{-108}{49}\) ∴ Required ratio = α² + β² : αβ = \(\frac{\frac{-108}{49}}{\frac{18}{7}}\) = \(-\frac{6}{7}\) = – 6 : 7.

What is the ratio of sum of squares of roots to the product of the roots of the equation 7x2 + 12x + 18 = 0?

Answer»

Let α, β be the roots of the equation 7x² + 12x + 18 = 0.

\(\bigg[\)For a quadratic equation ax² + bx + c = 0, sum of roots = \(-\frac{a}{b}\), product of roots = + \(\frac{c}{a}\)\(\bigg]\)

∴ α + β = \(\frac{12}{7}\) and αβ = \(\frac{18}{7}\)

⇒ (α + β)² = \(\bigg(\frac{-12}{7}\bigg)^2\) ⇒ α² + β²+ 2αβ = \(\frac{144}{49}\)

⇒ α² + β² = \(\frac{144}{49}\) - \(\frac{36}{7}\) = \(\frac{-108}{49}\)

∴ Required ratio = α² + β² : αβ = \(\frac{\frac{-108}{49}}{\frac{18}{7}}\) = \(-\frac{6}{7}\) = – 6 : 7.