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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

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