Example Question - expression transformation

Algebraic Expression Transformation

<p>令 \( y + \frac{1}{y} = b \)</p> <p>要表达 \( y^{2} + \frac{1}{y^{2}} \)，首先将原方程两边平方:</p> <p>\( (y + \frac{1}{y})^{2} = b^{2} \)</p> <p>展开左边，应用平方差公式得到:</p> <p>\( y^{2} + 2 \cdot y \cdot \frac{1}{y} + \frac{1}{y^{2}} = b^{2} \)</p> <p>简化得到:</p> <p>\( y^{2} + 2 + \frac{1}{y^{2}} = b^{2} \)</p> <p>现在将方程两边同时减去2:</p> <p>\( y^{2} + \frac{1}{y^{2}} = b^{2} - 2 \)</p> <p>因此，\( y^{2} + \frac{1}{y^{2}} \) 表达为 \( b \) 的项是:</p> <p>\( y^{2} + \frac{1}{y^{2}} = b^{2} - 2 \)</p>