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\(\text{Let $a$ and $b$ be the roots of the quadratic equation } 2x^2-10x+5=0. \text{What is the value of }(2a-3)(4b-6)?\)

 Jun 11, 2024
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Expanding the desired expression, we get \((2a-3)(4b-6)=8ab-12a-12b+18=8ab-12(a+b)+18\). This implies that we need the sum and product of the roots of the given equation, which are \(10/2=5\) and \(5/2\), respectively. Thus, the desired expression equals \(\)\(\left(8\cdot \frac{5}{2}\right) - (12 \cdot 5) + 18 = \boxed{-22}\).

 Jun 11, 2024

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