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

To solve these quadratic equations using the quadratic formula, we use the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). (i) For the equation \( 5x^2 + 2x + 1 = 0 \), we have \( a = 5 \), \( b = 2 \), and \( c = 1 \). The discriminant \( b^2 - 4ac = 2^2 - 4(5)(1) = 4 - 20 = -16 \). Since the discriminant is negative, the solutions will be complex: \[ x = \frac{-2 \pm \sqrt{-16}}{10} = \frac{-2 \pm 4i}{10} = \frac{-1 \pm 2i}{5}. \] (ii) For the equation \( 7x^2 + 9x + 1 = 0 \), we have \( a = 7 \), \( b = 9 \), and \( c = 1 \). The discriminant \( b^2 - 4ac = 9^2 - 4(7)(1) = 81 - 28 = 53 \). Since it's positive, the solutions are real: \[ x = \frac{-9 \pm \sqrt{53}}{14}. \] Therefore, the simplified surd forms of the solutions are: (i) \( x = \frac{-1 \pm 2i}{5} \) and (ii) \( x = \frac{-9 \pm \sqrt{53}}{14} \).