What is Completing the Square?
Completing the square is an algebraic technique used to turn any quadratic expression into a perfect square trinomial.
A perfect square trinomial is an expression that factors into two identical parentheses:
By adding the missing constant term to both sides of an equation, we can solve quadratics directly using square roots, without needing the quadratic formula!
The Magic Formula: (b2)2
For any quadratic expression starting with x2 + bx:
When you add this number:
Notice that the number inside the parentheses is always exactly half of b!
- •If b = 6: half is 3, square is 9 ⟹ (x + 3)2
- •If b = -8: half is -4, square is 16 ⟹ (x - 4)2
- •If b = 5: half is 52, square is 254 ⟹ (x + 52)2
Worked Example 1: Standard Problem (a = 1)
Problem:
Solve the quadratic equation by completing the square:
x2 + 6x - 16 = 0
Step 1: Move the constant term to the right side
Add 16 to both sides:
Step 2: Calculate the magic number (b2)2
Here, b = 6:
Step 3: Add 9 to BOTH sides of the equation
Step 4: Factor the left side as a binomial squared
Step 5: Take the square root of both sides
Remember the plus-or-minus (±) sign!
Now split into two simple equations:
Final Answer: The solutions are x = 2 and x = -8.
Worked Example 2: When Leading Coefficient a > 1
If the equation has a number in front of x2 (a ≠ 1), you must divide every single term by a before completing the square.
Problem:
Solve: 2x2 - 12x - 14 = 0
Step 1: Divide everything by 2
Step 2: Move the constant to the right
Step 3: Find the magic number (b2)2
Here b = -6:
Step 4: Add 9 to both sides
Step 5: Take the square root of both sides
Final Answer: The solutions are x = 7 and x = -1.
Converting to Vertex Form (y = a(x - h)2 + k)
Completing the square is also the fastest way to find the vertex of a parabola:
- Group the x terms: y = (x2 - 8x) + 19
- Magic number for -8: half is -4, squared is +16.
- Add and subtract 16 on the same side:
The vertex of the parabola is at (h, k) = (4, 3)!
Quick Reference Summary
| Step | Action | Watch Out For |
|---|---|---|
| 1. Isolate | Move constant to right: x2 + bx = c | Divide by a first if a ≠ 1 |
| 2. Half & Square | Compute (b2)2 | Squaring a negative gives a positive |
| 3. Balance | Add to BOTH sides | Don't forget the right side! |
| 4. Factor | (x + b2)2 = Total | Sign inside matches sign of b |
| 5. Square Root | x + b2 = ±\sqrt{Total} | Always write ± for two roots |
Don't Forget the Plus/Minus Sign!
If (x + 3)2 = 25, writing only x + 3 = 5 misses half the problem! Always include ± 5 so you capture both positive and negative roots.