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If α and β are the roots of 7x2+12x4=0,...

If α and β are the roots of 7x2+12x4=0,find the value of \frac{αβ}{(α + β)^2}

A.

\frac{7}{36}

B.

- \frac{36}{7}

C.

\frac{36}{7}

D.

- \frac{7}{36}

Correct answer is D

The general form of a quadratic equation is:

x^2 -(sum of roots)x +(product of roots) = 0

7x^2+12x-4=0

Divide through by 7

=x^2+\frac{12}{7}x-\frac{4}{7}=0

=x^2-(-\frac{12}{7})x+(-\frac{4}{7})=0

\therefore sum of roots = -\frac{12}{7}, and products of roots =-\frac{4}{7}

α + β = -\frac{12}{7}, αβ = -\frac{4}{7}

\frac{αβ}{(α + β)^2} = \frac{\frac{-4}{7}}{(\frac{-12}{7})^2}

=\frac{\frac{-4}{7}}{\frac{144}{49}}=-\frac{4}{7}\times\frac{49}{144}

\therefore - \frac{7}{36}