1. Derive the quadratic formula from the standard form (ax2 + bx + c = 0) of a quadratic equation by following the steps below.

2. Divide all terms in the equation by a.

3. Subtract the constant (the term without an x) from both sides.

4. Add a constant (in terms of a and b) that will complete the square.

5. Take the square root of both sides of the equation.

6. Solve for x.

We are given that:

a x^2 + b x + c = 0

Divide all terms with c / a:

x^2 + (b / a) x + (c / a) = 0

Subtract c from both sides:

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

x^2 + (b / a) x = - c / a

Add a constant k to complete the square:

where k = ((b / a) / 2) ^2 = (b / 2a) ^2

x^2 + (b / a) x + k = - c / a + k

x^2 + (b / a) x + (b/2a) ^2 = - c / a + (b / 2a) ^2

So the perfect square trinomial is:

(x + b/2a) ^2 = - c / a + (b / 2a) ^2

Taking the square root of both sides:

x + b/2a = sqrt [ - c / a + (b / 2a) ^2]

x = sqrt [ (-c/a) + (b / 2a) ^2] - (b/2a)