Marsavana Circuit Grade \( 1 / \) Mathematies Investigation INVESTIGATION \# 3 3.1 The solutions of a quadratic equation are given byx \( =\frac{-2 \pm \sqrt{2 m+5}}{7} \). For which value( \( s \) ) of \( m \) will this equation have? a) Two equal solutions? \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) (2) b) No real solutions \( \qquad \) \( \qquad \) \( \qquad \) 3.2 Show that \( k \) lies between -8 and 1 in the following inequality: \[ k^{2}+7 k \leq 8 \] \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) (4) TOTAL: 50

**3.1** The quadratic solutions are given by \[ x = \frac{-2 \pm \sqrt{2m+5}}{7}. \] The expression inside the square root, \(\sqrt{2m+5}\), represents the discriminant (up to a constant factor) of the quadratic equation. **a) Two equal solutions** For a quadratic equation to have two equal (repeated) solutions, the discriminant must be zero. Therefore, we set \[ 2m + 5 = 0. \] Solving for \(m\): \[ 2m = -5 \quad \Rightarrow \quad m = -\frac{5}{2}. \] **b) No real solutions** For the quadratic equation to have no real solutions, the expression under the square root must be negative. That is, \[ 2m + 5 < 0. \] Solving for \(m\): \[ 2m < -5 \quad \Rightarrow \quad m < -\frac{5}{2}. \] --- **3.2** We are given the inequality \[ k^{2} + 7k \leq 8. \] First, bring all terms to one side: \[ k^{2} + 7k - 8 \leq 0. \] To solve the inequality, factor the quadratic expression. We look for two numbers whose product is \(-8\) and whose sum is \(7\). These numbers are \(8\) and \(-1\), so we can factor as: \[ k^{2} + 7k - 8 = (k - 1)(k + 8). \] Thus, the inequality becomes: \[ (k - 1)(k + 8) \leq 0. \] The critical points (roots) are given by setting each factor equal to zero: \[ k - 1 = 0 \quad \Rightarrow \quad k = 1, \] \[ k + 8 = 0 \quad \Rightarrow \quad k = -8. \] For the product \((k - 1)(k + 8)\) to be less than or equal to zero, \(k\) must lie between the roots. Therefore, \[ -8 \leq k \leq 1. \] This shows that \(k\) lies between \(-8\) and \(1\).