To derive the quadratic formula using completing the square, we'll start with the quadratic equation in the form:
ax² + bx + c = 0
Step 1: Move the constant term (c) to the other side of the equation:
ax² + bx = -c
Step 2: Divide the entire equation by 'a' to simplify the equation and make the coefficient of x² equal to 1:
x² + (b/a)x = -c/a
Step 3: To complete the square, take half of the coefficient of 'x', square it, and add it to both sides of the equation. The left side will become a perfect square trinomial:
x² + (b/a)x + (b/2a)² = -c/a + (b/2a)²
Simplifying:
x² + (b/a)x + (b/2a)² = -c/a + (b/2a)²
Step 4: Simplify the right side of the equation:
x² + (b/a)x + (b/2a)² = (-4ac + b²)/(4a²)
Step 5: Factor the left side of the equation as a perfect square:
(x + b/2a)² = (-4ac + b²)/(4a²)
Step 6: Take the square root of both sides:
x + b/2a = ±√((-4ac + b²)/(4a²))
Step 7: Isolate x by subtracting b/2a from both sides:
x = (-b ± √(b² - 4ac))/(2a)
Therefore, the quadratic formula derived using completing the square is:
x = (-b ± √(b² - 4ac))/(2a)
HOPE THIS HELPS :)