35) The roots of the equation \(x^2 + 6x - 4 = 0\) are α, β. Find the quadratic equation whose roots are \(\alpha^2\) and \(\beta^2\).
Answer: If the roots are given, the quadratic equation is \(X^2 - (\text{sum of the roots})X + (\text{product of the roots}) = 0\).
For the given equation, \(x^2 + 6x - 4 = 0\):
Sum of roots: \(\alpha + \beta = -\frac{b}{a} = -6\)
Product of roots: \(\alpha\beta = \frac{c}{a} = -4\)
For the new equation with roots \(\alpha^2\) and \(\beta^2\):
New Sum of Roots:
\(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\)
\(= (-6)^2 - 2(-4) = 36 + 8 = 44\)
New Product of Roots:
\(\alpha^2\beta^2 = (\alpha\beta)^2 = (-4)^2 = 16\)
The required equation is:
\[ x^2 - (44)x + 16 = 0 \]
\[ x^2 - 44x + 16 = 0 \]