Completing the Square
Master the technique of completing the square to solve quadratic equations and rewrite them in vertex form.
Definition
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Worked Examples
Complete the square:
Identify the coefficient of x
→ Coefficient is 8
Take half of b
→ Half is 4
Square the result
→ Need to add 16
Add and subtract this value
→ Balance maintained
Factor the perfect square trinomial
→ Completed square form
Answer:
Common Mistakes
Forgetting to add to both sides when solving an equation
Why it's wrong: When completing the square in an equation, whatever you add to one side must be added to the other side to maintain equality.
Correct: After adding to the left, add the same value to the right.
Not factoring out the leading coefficient first
Why it's wrong: When in , completing the square directly gives wrong results.
Correct: First factor out from the and terms, then complete the square inside.
Using instead of to complete the square
Why it's wrong: The formula requires half of the coefficient of , squared.
Correct: Always use , not .
Forgetting to multiply when distributing
Why it's wrong: When is factored out, the value subtracted inside gets multiplied by .
Correct: In , the becomes when distributed: .
Interactive Visual
Linear Function Explorer
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Interactive Sandbox
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y = 2x + 1
m=2, b=1
Expression Calculator
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Practice Problems
16 problemsWhat value completes the square for ?
Why It Matters
- Solves any quadratic equation (even when factoring doesn't work)
- Reveals the vertex of a parabola directly from the equation
- Derives the quadratic formula (which comes from completing the square!)
- Transforms circles from general to standard form in geometry
Real World Applications
Projectile Motion
Engineers use completing the square to find the maximum height of projectiles.
Example:
The height of a ball is feet. Completing the square: . Maximum height is 69 feet at seconds.
A rocket's height is given by meters.
What is the maximum height?
Step 1: Write the mathematical expression
Complete the square to find the vertex:
Architecture and Design
Architects use completing the square to design parabolic arches and determine their highest points.
Example:
A bridge arch follows . Converting to vertex form: . The arch reaches 100 meters high at its center.
A parabolic antenna dish is modeled by .
At what x-coordinate is the dish deepest?
Step 1: Write the mathematical expression
Find the vertex x-coordinate:
Economics and Profit Optimization
Businesses use quadratic models to find optimal pricing for maximum profit.
Example:
A company's profit is dollars where is the price. Vertex form: . Maximum profit of 50 dollars occurs at price 10 dollars.
Revenue is modeled by where is units sold in thousands.
How many units maximize revenue?
Step 1: Write the mathematical expression
Find the vertex:
Key Takeaways
- 1Completing the square transforms into
- 2The key value to add is always
- 3When , factor out from the -terms first
- 4Vertex form reveals the vertex directly
- 5This technique can solve any quadratic equation, even when factoring fails
Frequently Asked Questions
Glossary
- Completing the square
- A method to rewrite a quadratic expression as a perfect square plus or minus a constant
- Perfect square trinomial
- A trinomial that factors as or , such as
- Vertex form
- The form where is the vertex of the parabola
- Standard form
- The form for a quadratic expression
Formula Card
Completing the Square Formula
The core transformation for completing the square
Value to Complete
The value you add and subtract to create a perfect square trinomial
Vertex Form
The vertex of the parabola is at point (h, k)
Perfect Square Patterns
Expanding a squared binomial creates a perfect square trinomial