Where does the Quadratic Formula come from?

Answer 1

The quadratic equation comes from a method called completing the square and then solving for x:

1. Step 1: Start with the generic quadratic equation:

ax² + bx + c = 0

1. Step 2: Divide everything by "a" so the coefficient in front of x² is 1:

x² + (b/a)x + (c/a) = 0

1. Step 3: Subtract (c/a) from both sides so all "x" values are on one side:

x² + (b/a)x = -(c/a)

1. Step 4: Now, we complete the square of the left half of the equation by adding (b/2a)² to it, and then add the same to the right side of the equation. This makes the left half of the equation a perfect square:

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

We can now re-write this as

(x + (b/2a))² = -(c/a) + (b²/4a²)

because we made it a perfect square

We can also merge the right hand of the equation together by multiplying (-c/a) by (4a/4a), which is the same as multiplying it by 1:

(x + (b/2a))² = (-4ac + b²)/4a² or (x + (b/2a))² = (b² - 4ac)/4a²

1. Step 5: Take the square root of both sides:

x + (b/2a) = +/- sqrt((b² - 4ac)/4a²)

= +/- sqrt(b² - 4ac) / 2a

1. Step 6: Now solve for x by subtracting (b/2a) from both sides and then simplify:

x = (-b/2a) +/- sqrt(b² - 4ac)/2a

x = (-b +/- sqrt(b² - 4ac))/2a <-- We've derived the quadratic equation