Show proof of deriving the Quadratic Formula from the Quadratic Equation ax^2 + bx + c = O

factor out an a

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

compete the square and don't change the value
a(x^2 + b/a + b^2/4a^2) - a(b^2/4a^2) + c = 0

a(x + b/2a)^2 - b^2/4a + c = 0
(x + b/2a)^2 - b^2/4a^2 + c/a = 0
(x+b/2a)^2 = (b^2 - 4ac)/4a^2
x + b/2a = ±√(b^2-4ac) / 2a
x = (-b ± √(b^2-4ac))/2a

To derive the quadratic formula, let's start with the quadratic equation: ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable.

Step 1: Divide the entire equation by a to simplify it:
(ax^2 + bx + c) / a = 0 / a
x^2 + (b/a)x + c/a = 0

Step 2: Move the constant term to the right side of the equation:
x^2 + (b/a)x = -c/a

Step 3: Complete the square on the left side of the equation. To do this, add the square of half the coefficient of x, which is (b/2a)^2:
x^2 + (b/a)x + (b/2a)^2 = -c/a + (b/2a)^2

Step 4: Simplify the right side of the equation:
x^2 + (b/a)x + (b^2/4a^2) = (-4ac + b^2)/(4a^2)

Step 5: Factor the left side of the equation as a perfect square:
(x + (b/2a))^2 = (-4ac + b^2)/(4a^2)

Step 6: Take the square root of both sides of the equation:
sqrt[(x + (b/2a))^2] = sqrt((-4ac + b^2)/(4a^2))

Step 7: Solve for x:
x + (b/2a) = ±sqrt((-4ac + b^2)/(4a^2))

Step 8: Simplify the right-hand side:
x + (b/2a) = ±[sqrt(-4ac + b^2)] / (2a)

Step 9: Subtract (b/2a) from both sides of the equation:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)

Therefore, the quadratic formula is:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)

This formula can be used to find the solutions (or roots) of any quadratic equation of the form ax^2 + bx + c = 0.