The roots of the quadratic equation \(2x^{2} - 5x + m = 0\) are \(\alpha\) and \(\beta\), where m is a constant. Find \(\alpha^{2} + \beta^{2}\) in terms of...
The roots of the quadratic equation \(2x^{2} - 5x + m = 0\) are \(\alpha\) and \(\beta\), where m is a constant. Find \(\alpha^{2} + \beta^{2}\) in terms of m.
Answer Details
To find \(\alpha^{2} + \beta^{2}\), we need to use the following identities:
\[\alpha + \beta = \frac{-b}{a}\quad\text{and}\quad \alpha\beta = \frac{c}{a}\]
where a, b, and c are the coefficients of the quadratic equation \(ax^{2} + bx + c = 0\).
In this case, the quadratic equation is \(2x^{2} - 5x + m = 0\), so a = 2, b = -5, and c = m.
Using the identity \(\alpha + \beta = \frac{-b}{a}\), we have:
\[\alpha + \beta = \frac{-(-5)}{2} = \frac{5}{2}\]
Using the identity \(\alpha\beta = \frac{c}{a}\), we have:
\[\alpha\beta = \frac{m}{2}\]
Now, we can use the identity:
\[\alpha^{2} + \beta^{2} = (\alpha + \beta)^{2} - 2\alpha\beta\]
Substituting the values we found earlier, we get:
\[\alpha^{2} + \beta^{2} = \left(\frac{5}{2}\right)^{2} - 2\left(\frac{m}{2}\right)\]
Simplifying, we get:
\[\alpha^{2} + \beta^{2} = \frac{25}{4} - m\]
Therefore, the answer is \(\frac{25}{4} - m\), which is.