Completing the square is a technique that converts a quadratic from standard form to vertex form. The SAT tests it directly and also uses it in circle equation questions.
The Method
To complete the square for :
- Take half of :
- Square it:
- Add and subtract this value
Converting to Vertex Form
Example 1: Write in vertex form.
Half of 6 is 3. Square it: .
Vertex form: . The vertex is .
Example 2: Write in vertex form.
Half of is . Square it: .
Vertex: .
When the Leading Coefficient Isn't 1
Example 3: Write in vertex form.
Factor out the 2 from the first two terms:
Complete the square inside the parentheses:
Vertex: .
Solving Equations by Completing the Square
Example 4: Solve
Application: Circle Equations
The standard form of a circle is with center and radius .
Example 5: Find the center and radius of
Group and complete the square for each variable:
Center: , radius: .
Practice Problems
Problem 1: Write in vertex form.
Solution
Vertex:
Problem 2: Solve by completing the square.
Solution
or
Problem 3: Find the center of the circle .
Solution
Center:
Key Takeaways
- Take half the coefficient of , then square it — that's the number you add
- Vertex form immediately gives you the vertex
- If the leading coefficient isn't 1, factor it out first
- This technique also works for finding circle centers and radii
- On the SAT, completing the square is often faster than the quadratic formula for finding the vertex
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