Find the quadratic equation whose roots are equal to the sum and product of the roots of the quadratic equation \(2x^2 + 4x - 1 = 0\).
\(2x^2 + 5x + 2 = 0\)
For \(2x^2 + 4x - 1 = 0\), sum of roots \(= -\tfrac{b}{a} = -\tfrac{4}{2} = -2\) and product of roots \(= \tfrac{c}{a} = -\tfrac{1}{2}\).
New roots are \(-2\) and \(-\tfrac{1}{2}\); their sum \(= -\tfrac{5}{2}\) and product \(= 1\). The required equation is \(x^2 - (\text{sum})x + \text{product} = 0\), giving \(x^2 + \tfrac{5}{2}x + 1 = 0\).
Multiplying both sides by \(2\): \(2x^2 + 5x + 2 = 0\).
Hence, \(2x^2 + 5x + 2 = 0\) is correct.
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