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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.

Sep 20, 2022

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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 $$\color{brown}\boxed{x^2 - 14x + 121}$$

Sep 20, 2022