Completing the Square Calculator
Convert quadratic expressions to vertex form
What is Completing the Square?
Completing the square is an algebraic technique used to convert a quadratic expression from standard form (ax² + bx + c) to vertex form (a(x - h)² + k). This method is useful for finding the vertex of a parabola, solving quadratic equations, and understanding the transformations of quadratic functions.
How to Complete the Square
Follow these steps to complete the square:
- Factor out the coefficient of x² if it's not 1
- Move the constant term to the right side of the equation
- Take half of the coefficient of x and square it
- Add and subtract this value inside the parentheses
- Factor the perfect square trinomial inside the parentheses
- Simplify the expression
Why Complete the Square?
Completing the square is useful for:
- Finding the vertex of a parabola
- Converting between standard and vertex forms
- Solving quadratic equations
- Deriving the quadratic formula
- Understanding function transformations
- Finding maximum or minimum values
Examples
Example 1: Simple Case
- Original: x² + 6x + 5
- Half of 6 is 3, and 3² = 9
- Add and subtract 9: x² + 6x + 9 - 9 + 5
- Factor perfect square: (x + 3)² - 4
Example 2: With Coefficient
- Original: 2x² - 12x + 7
- Factor out 2: 2(x² - 6x) + 7
- Half of -6 is -3, and (-3)² = 9
- 2(x² - 6x + 9 - 9) + 7
- Final: 2(x - 3)² - 11
Common Mistakes to Avoid
- Forgetting to factor out the coefficient of x²
- Not halving the coefficient of x correctly
- Forgetting to add and subtract the same value
- Making sign errors when moving terms
- Not simplifying the constant term correctly
Tips for Success
- Always start by factoring out the coefficient of x²
- Double-check your arithmetic when finding (b/2)²
- Keep track of signs carefully
- Verify your answer by expanding it back
- Practice with simple examples first