Determine complex conjugate solutions of real quadratic equations with real coefficients using factorisation, completing the square and the quadratic formula, with and without technology.

Consider the quadratic equation \(2x^2 - 6x + 11 = 0\). (a) Determine the discriminant of this equation. (1 mark) (b) Hence determine the complex conjugate solutions of the equation using the quadratic formula. Express your answer in the form \(a \pm bi\), where \(a\) and \(b\) are real numbers. (2 marks)

Consider the quadratic equation \(2x^2 - 8x + 17 = 0\). (a) Determine the solutions using the quadratic formula, giving your answer in the form \(a \pm bi\). (2 marks) (b) Verify that your solutions satisfy the original equation. (1 mark)

A projectile's height (m) above ground is modelled by \(h(t) = -5t^2 + 6t + 7\), where \(t\) is time (s) after launch. (a) Determine the values of \(t\) for which \(h(t) = 10\) by completing the square. Express your answer in exact form. (2 marks) (b) Interpret the nature of your solutions from part (a) in the context of the projectile's motion. (1 mark)