Derive the quadratic equation.

The general form of a quadratic equation is:

ax^2 + bx + c = 0

Where a, b, and c are constants and x is the variable.

To derive the quadratic equation, we start with a general quadratic equation in the form:

y = ax^2 + bx + c

Next, we set y = 0 to find the roots of the equation:

0 = ax^2 + bx + c

This gives us the quadratic equation in the form:

ax^2 + bx + c = 0

Therefore, the quadratic equation is derived from setting the general quadratic equation equal to zero.

Step 1: Start with the general form of a quadratic equation:

```
ax² + bx + c = 0
```

Step 2: Complete the square:

To complete the square, we need to add and subtract the square of half the coefficient of the linear term, which is (b/2)².

```
ax² + bx + (b/2)² - (b/2)² + c = 0
```

Step 3: Group terms and factor:

```
a(x² + (b/a)x + (b/2a)²) - (b/2a)² + c = 0
```

```
a((x + b/2a)² - (b/2a)²) + c = 0
```

Step 4: Expand and simplify:

```
a(x + b/2a)² - (ab/4a²) + c = 0
```

```
a(x + b/2a)² = (ab/4a²) - c
```

Step 5: Divide both sides by 'a':

```
(x + b/2a)² = (ab - 4ac)/4a²
```

Step 6: Take the square root of both sides:

```
x + b/2a = ±√((ab - 4ac)/4a²)
```

Step 7: Solve for 'x':

```
x = -b/2a ± √((ab - 4ac)/4a²)
```

```
x = (-b ± √(b² - 4ac))/2a
```

Therefore, the quadratic equation is:

```
ax² + bx + c = 0
```

```
x = (-b ± √(b² - 4ac))/2a
```