二次方程的根：練習題

Given that the roots of the quadratic equation \(2x^2 - 5x + 3 = 0\) are \(\alpha\) and \(\beta\), calculate the value of \(\alpha + \beta + \alpha\beta\).

The quadratic equation \(x^2 + px + q = 0\) has roots \(\alpha\) and \(\beta\), where \(q \neq 0\). Which of the following is the quadratic equation with roots \(\frac{1}{\alpha}\) and \(\frac{1}{\beta}\)?

Let \( \alpha \) and \( \beta \) be the roots of the quadratic equation \( x^2 - x + 1 = 0 \). Determine the quadratic equation with integer coefficients whose roots are \( \alpha^3 \) and \( \beta^3 \).

Given that the roots of the quadratic equation \(3x^2 + 4x - 5 = 0\) are \(\alpha\) and \(\beta\), find the value of \(\alpha^2 + \beta^2\).

The quadratic equation \(2x^2 - 5x + 7 = 0\) has roots \(\alpha\) and \(\beta\). Determine the value of \(\alpha^2 + \beta^2\).

The quadratic equation \(4x^2 - kx + 9 = 0\) has roots \(\alpha\) and \(\beta\). Given that \(\alpha + \beta = 3\), find the value of the constant \(k\).

The roots of the quadratic equation \(x^2 + 3x + 1 = 0\) are \(\alpha\) and \(\beta\). Find the value of \(\alpha^3 + \beta^3\) and use it to form a quadratic equation with roots \(\alpha^3\) and \(\beta^3\).

The quadratic equation \(3x^2 - 5x + 1 = 0\) has roots \(\alpha\) and \(\beta\).
(a) Write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\).
(b) Find a quadratic equation with integer coefficients that has roots \(\frac{1}{\alpha}\) and \(\frac{1}{\beta}\).