Solve each of the following equations by using the formula :
(i) `5x^(2)-2x-3=0`
(ii) `x^(2)=18x-77`

Answer

To solve the given quadratic equations using the quadratic formula, we will follow these steps:

### (i) Solve \(5x^2 - 2x - 3 = 0\)

1. **Identify coefficients**:
- Here, \(a = 5\), \(b = -2\), and \(c = -3\).

2. **Calculate the discriminant \(D\)**:
\[ D = b^2 - 4ac = (-2)^2 - 4 \cdot 5 \cdot (-3) \]
\[ D = 4 + 60 = 64 \]

3. **Apply the quadratic formula**:
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
Substituting the values:
\[ x = \frac{-(-2) \pm \sqrt{64}}{2 \cdot 5} \]
\[ x = \frac{2 \pm 8}{10} \]

4. **Calculate the two possible values for \(x\)**:
- First solution:
\[ x_1 = \frac{2 + 8}{10} = \frac{10}{10} = 1 \]
- Second solution:
\[ x_2 = \frac{2 - 8}{10} = \frac{-6}{10} = -\frac{3}{5} \]

5. **Final solutions**:
\[ x = 1 \quad \text{or} \quad x = -\frac{3}{5} \]