01 · THE METHOD
Why Complete the Square?
Not all quadratic equations are easy to factorise by splitting the middle term. Completing the square converts ax² + bx + c = 0 into:
(x + b/2a)² = (b² - 4ac) / 4a²This algebraic transformation forms the foundation for deriving the Quadratic Formula!
The Golden Rule
To complete the square for x² + px, add and subtract (p/2)², which is square of half the coefficient of x!
02 · ALGORITHMIC STEPS
4 Steps to Complete the Square
EXAMPLE 1
Solve 2x² - 5x + 3 = 0 by Completing the Square
Step 1: Divide by 'a' (make coefficient of x² equal to 1):
x² - (5/2)x + (3/2) = 0
Step 2: Move constant term to RHS:
x² - (5/2)x = -3/2
Step 3: Add (p/2)² = (-5/4)² = 25/16 to both sides:
x² - (5/2)x + 25/16 = -3/2 + 25/16
(x - 5/4)² = (-24 + 25) / 16
(x - 5/4)² = 1 / 16
(x - 5/4)² = (-24 + 25) / 16
(x - 5/4)² = 1 / 16
Step 4: Take square root of both sides:
x - 5/4 = ± 1/4
x = 5/4 + 1/4 = 6/4 = 3/2
x = 5/4 - 1/4 = 4/4 = 1
x = 5/4 + 1/4 = 6/4 = 3/2
x = 5/4 - 1/4 = 4/4 = 1
Roots are x = 3/2 and x = 1.
03 · QUICK SUMMARY
Key Takeaways
01
Always ensure the coefficient of x² is 1 before completing the square.
02
Add (p/2)² (half the coefficient of x squared) to both sides of the equation.
03
Take square roots on both sides taking care of the ± sign to obtain both roots.