Let a and b be the roots of the quadratic x^2 - 5x + 3 = 2x^2 - 11x + 14. Find the quadratic whose roots are a^2 and b^2.
+2668 There is a way to solve this without finding the roots of the quadratic.
The equation simplifies to \(0 = x^2 -6x + 11\). The sum of the roots is \(-{b \over a} = 6\) and the product is \({c \over a} = 11\)
The new quadratic would be \((x-a^2)(x-b^2)\), which simplifies to \(x^2 - (a^2 + b^2)x + a^2b^2\)
Note that \(a^2 + b^2 = (a+b)^2 - 2ab = 36 - 2 \times 11 = 14\) and \(a^2 b^2 = (a\times b)^2 = 11^2 = 121\).
So the quadratic is \(\color{brown}\boxed{x^2 - 14x + 121}\)
