Completing the square comes before the formula, because the formula is what it produces.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The method
For x² + 6x, half of 6 is 3, and 3² is 9. So x² + 6x + 9 is (x + 3)².
The formula
Completing the square on ax² + bx + c = 0 in general gives x = (−b ± √(b² − 4ac)) ÷ 2a.
The part under the root
b² − 4ac decides everything: positive gives two roots, zero gives one, negative gives none.
Making a perfect square
x² + 6x becomes a perfect square by adding 9, giving (x + 3)². The number to add is always the square of half the coefficient of x. Adding it and subtracting it again keeps the expression unchanged.
It produces vertex form
Completing the square turns y = x² + 6x + 5 into y = (x + 3)² − 4, so the vertex is at (−3, −4). The technique is not only for solving — it is how you find a parabola's turning point exactly.
The quadratic formula comes from it
Completing the square on the general ax² + bx + c = 0 produces x = (−b ± √(b² − 4ac))/(2a). The formula is not a separate fact — it is this technique carried out once, in general, so you never have to repeat it.
When to use which method
Factor when it factors easily. Use the formula when it does not. Complete the square when you need the vertex or are deriving something. Having three methods and choosing sensibly is the point.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Maya Complete the Square on x² + 6x + 5
Maya rewrites a quadratic in vertex form.
- Step 1
She takes half of 6, which is 3, and squares it to get 9.
Step 4: Your Turn
Practice makes it stick.
The Discriminant
Problem 1 of 2
x² − 4x + 3. What is b² − 4ac?
How Many Roots
Problem 2 of 2
x² + 2x + 5. What is b² − 4ac?
Complete and Decide
1 of 4
x² + 8x + ? completes the square. What number?
2 of 4
x² + 10x + ? completes the square. What number?
3 of 4
x² − 6x + 9. What is b² − 4ac?
4 of 4
x² + 3x + 1. What is b² − 4ac?
Step 5: Quick Check
Show what you know.
Question 1 of 2
x² − 2x − 8. What is b² − 4ac?
Question 2 of 2
A negative discriminant means what?
What You Learned
- Completing the square rewrites a quadratic in vertex form.
- The quadratic formula is what completing the square gives in general.
- The discriminant b² − 4ac decides how many real roots there are.