quadratic equation

Question

1. Derive the Quadratic Formula by completing the square on the standard form of a quadratic equation.

Answer 1

The standard form is:

ax² + bx + c = 0

x² + bx/a + c/a = 0

If we add (b/2a)² - c/a to both sides, we get:

x² + bx/a + (b/2a)² = (b/2a)² - c/a

(x + b/2a)² = (b/2a)² - c/a

(x + b/2a)² = (b² - 4ac)/4a²

x + b/2a = sqrt(b² - 4ac)/2a

x = (-b ± sqrt(b² - 4ac))/2a

which is the Quadratic Formula

Answer 2

Gladly!

ax^2+bx+c = 0, a is not zero (otherwise it is a linear equation)

x^2 + (b/a)x+ (c/a) = 0 <--- divides by non-zero a

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

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

(x + b/(2a))(x + b/(2a)) = (b/(2a))^2 - c/a <--- VERIFY THIS STEP WITH FOIL !!!!!

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

The right side is b^2/(4a^2) - c/a

= b^2/(4a^2) - 4ac/(4a^2) <-- the common denominator is 4a^2

=( b^2-4ac)/(4a^2)

taking the square root of both sides:

x + b/(2a) = sqrt( b^2 - 4ac) / 2a

subtracting b/(2a) completes the proof of the quadratic formula