The quadratic function：练习题

Find the values of the constant \(k\) for which the quadratic equation \(x^2 + (k-2)x + 9 = 0\) has two equal real roots.

The roots of the quadratic equation \(x^2 - 4x + 2 = 0\) are \(\alpha\) and \(\beta\).
Find a quadratic equation with integer coefficients whose roots are \(\frac{\alpha}{\beta}\) and \(\frac{\beta}{\alpha}\).

The roots of the quadratic equation \(ax^2 + bx + c = 0\) (where \(a \neq 0\)) are in the ratio \(2:3\). Which of the following expressions correctly relates the constants \(a\), \(b\), and \(c\)?

Find the coordinates of the minimum point of the curve \(y = x^2 - 8x + 19\) by completing the square.

Find the set of values of \(k\) for which the quadratic equation \(x^2 + kx + (k+3) = 0\) has no real roots.

The quadratic equation \(x^2 + 6x + k = 0\) has no real roots. Find the range of possible values for the constant \(k\).

The roots of the equation \(x^2 - 5x + 2 = 0\) are \(\alpha\) and \(\beta\).
Without solving the equation, form a quadratic equation with integer coefficients whose roots are \(\frac{1}{\alpha + 1}\) and \(\frac{1}{\beta + 1}\).

The roots of the equation \( x^2 + kx + (k+3) = 0 \) are \( \alpha \) and \( \beta \). Given that \( \alpha^2 + \beta^2 = 7 \), find the possible values of the constant \( k \).

(a) By completing the square, express \(x^2 - 8x + 19\) in the form \((x - p)^2 + q\), where \(p\) and \(q\) are integers.
(b) Hence, write down the coordinates of the minimum point of the curve with equation \(y = x^2 - 8x + 19\).

The roots of the equation \(x^2 - 7x + 2 = 0\) are \(\alpha\) and \(\beta\).
Without solving the equation,
(a) write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\),
(b) find the value of \(\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\),
(c) form a quadratic equation with integer coefficients whose roots are \(\frac{1}{\alpha + 1}\) and \(\frac{1}{\beta + 1}\).