(b) Solve the following quadratic equations by using the quadratic formula, giving the solu simplified Surd form. \( \begin{array}{ll}\text { (i) } 5 x^{2}+2 x+1=0 & \text { (ii) } 7 x^{2}+9 x+1=0\end{array} \)

Certainly! Let's solve each quadratic equation using the quadratic formula and express the solutions in simplified surd form. The **quadratic formula** is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \( ax^2 + bx + c = 0 \). --- ### (i) \( 5x^{2} + 2x + 1 = 0 \) **Step 1: Identify the coefficients** \[ a = 5, \quad b = 2, \quad c = 1 \] **Step 2: Calculate the discriminant (\( D \))** \[ D = b^2 - 4ac = (2)^2 - 4(5)(1) = 4 - 20 = -16 \] **Step 3: Analyze the discriminant** - Since \( D < 0 \), the equation has **two complex conjugate solutions**. **Step 4: Apply the quadratic formula** \[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{-16}}{2 \times 5} = \frac{-2 \pm 4i}{10} \] **Step 5: Simplify the solutions** \[ x = \frac{-2}{10} \pm \frac{4i}{10} = -\frac{1}{5} \pm \frac{2i}{5} \] **Final Solutions:** \[ x = -\frac{1}{5} + \frac{2i}{5} \quad \text{and} \quad x = -\frac{1}{5} - \frac{2i}{5} \] --- ### (ii) \( 7x^{2} + 9x + 1 = 0 \) **Step 1: Identify the coefficients** \[ a = 7, \quad b = 9, \quad c = 1 \] **Step 2: Calculate the discriminant (\( D \))** \[ D = b^2 - 4ac = (9)^2 - 4(7)(1) = 81 - 28 = 53 \] **Step 3: Analyze the discriminant** - Since \( D > 0 \) and is not a perfect square, the equation has **two distinct real irrational solutions**. **Step 4: Apply the quadratic formula** \[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{53}}{2 \times 7} = \frac{-9 \pm \sqrt{53}}{14} \] **Final Solutions:** \[ x = \frac{-9 + \sqrt{53}}{14} \quad \text{and} \quad x = \frac{-9 - \sqrt{53}}{14} \] --- These are the solutions to the given quadratic equations expressed in simplified surd form.