Morgan Sullivan
08/16/2024 · High School
b) The equation \( 4 x^{2}+8 x-1=0 \) has roots \( \alpha \) and \( \beta \). (i) Find the values of \( \alpha^{3} \beta+\alpha \beta^{3} \) and \( \left(\alpha^{3} \beta\right) \times\left(\alpha \beta^{3}\right) \). (ii) Find an equation whose roots are \( \alpha^{3} \beta \) and \( \alpha \beta^{3} \).
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(i) \(\alpha^3 \beta + \alpha \beta^3 = -\frac{9}{8}\), \((\alpha^3 \beta)(\alpha \beta^3) = \frac{1}{256}\)
(ii) The equation is \(256x^2 + 288x + 1 = 0\)
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