Use the discriminant to determine the number of solutions to a quadratic equation.

The quadratic equation $2x^2 - 5x + k = 0$ has exactly one real solution. Which of the following could be the value of $k$?

A quadratic equation is given by (k - 2)x² + 6x + (3k + 1) = 0, where k is a real constant. (a) Determine the values of k for which this equation has exactly two distinct real solutions. (3 marks) (b) Determine all values of k for which the equation has exactly one real solution, and find the solution corresponding to each value of k. (2 marks)

A parabola has equation \( y = mx^2 - 4x + (m+3) \), where \( m \) is a real constant. (a) Determine the values of \( m \) for which the parabola intersects the \( x \)-axis at exactly one point. (b) Hence, determine the set of values of \( m \) for which the parabola does not intersect the \( x \)-axis. (c) Verify algebraically that your answer to part (a) is reasonable by substituting one value of \( m \) into the original equation and evaluating the discriminant.

Use the discriminant to determine the number of real solutions to the quadratic equation $2x^2 + kx + 5 = 0$, where $k$ is a real constant. Hence, find the range of values of $k$ for which this equation has at least one real solution.