If \(\alpha\) and \(\beta\) are the roots of \(x^{2} + x - 2 = 0\), find the value of \(\frac{1}{\alpha^{2}} + \frac{1}{\beta^{2}}\).
Question 1 Report
If \(\alpha\) and \(\beta\) are the roots of \(x^{2} + x - 2 = 0\), find the value of \(\frac{1}{\alpha^{2}} + \frac{1}{\beta^{2}}\).
Answer Details
To find the value of \(\frac{1}{\alpha^{2}} + \frac{1}{\beta^{2}}\), we first need to find the values of \(\alpha\) and \(\beta\).
Given that \(x^{2} + x - 2 = 0\), we can factorize the quadratic equation as \((x-1)(x+2)=0\).
Thus, the roots are \(\alpha = 1\) and \(\beta = -2\).
Now, we can substitute these values into the expression to get:
\[\frac{1}{\alpha^{2}} + \frac{1}{\beta^{2}} = \frac{1}{1^{2}} + \frac{1}{(-2)^{2}} = 1 + \frac{1}{4} = \frac{5}{4}\]
Therefore, the answer is \(\frac{5}{4}\).
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