Forcing a Factor
What if a trinomial doesn't factor neatly? We can "force" it to become a perfect square by adding a specific number. This is called Completing the Square.
If you have \(x^2 + bx\), add \((\frac{b}{2})^2\) to make it \((x + \frac{b}{2})^2\).
Worked Examples
Example 1: Finding the Constant
What number completes the square for \(x^2 + 6x\)?
- Take half of 6: \(3\).
- Square it: \(3^2 = 9\).
- Expression: \(x^2 + 6x + 9 = (x + 3)^2\).
- Result: 9
Example 2: Solving an Equation
Solve \(x^2 + 4x - 5 = 0\) by completing the square.
- Move constant: \(x^2 + 4x = 5\).
- Complete square: Add \((\frac{4}{2})^2 = 4\) to both sides.
- \(x^2 + 4x + 4 = 5 + 4\).
- Simplify: \((x + 2)^2 = 9\).
- Take square root: \(x + 2 = \pm 3\).
- Solutions: \(x = -2 + 3 = 1\) and \(x = -2 - 3 = -5\).
- Result: \(\{-5, 1\}\)
The Bridge to Quantum Mechanics
In Quantum Mechanics, the probability of finding a particle often looks like a "Gaussian Bell Curve": \(P(x) \propto e^{-ax^2 + bx}\). To find the center of this curve (the most likely spot for the particle), we must Complete the Square in the exponent. This tells us the "Expectation Value" of the position. Without this algebraic trick, we couldn't accurately describe the "fuzziness" of a quantum particle's location.