Given the quadratic equation \(x^2 - 3ax + a^2 = 0\), the sum and product of the roots \(\alpha\) and \(\beta\) are related to the coefficients of the quadratic equation:
\[ \alpha + \beta = 3a \]
\[ \alpha \beta = a^2 \]
We are also given that \(\alpha^2 + \beta^2 = 74\).
Now, let's express \(\alpha^2 + \beta^2\) in terms of the sum and product of the roots:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta \]
Substitute the values from the given information:
\[ 74 = (3a)^2 - 2a^2 \]
\[ 74 = 9a^2 - 2a^2 \]
Combine like terms:
\[ 74 = 7a^2 \]
Now, solve for \(a\):
\[ 7a^2 = 74 \]
\[ a^2 = \frac{74}{7} \]
\[ a = \pm \sqrt{\frac{74}{7}} \]
Therefore, the values of \(a\) that satisfy the given condition are \(a = \sqrt{\frac{74}{7}}\) and \(a = -\sqrt{\frac{74}{7}}\).
0 टिप्पणियाँ:
Post a Comment