Completing the Square in Algebra
This page covers the concept of completing the square in algebra, a fundamental technique for solving quadratic equations and understanding quadratic functions. It provides several examples and key points about the method.
The page begins with basic examples of completing the square, such as expanding (x + 1)². It then progresses to more complex problem-solving scenarios.
Example: (x + 1)² = x² + x + x + 1
This example demonstrates the expansion of a perfect square binomial, which is the basis for understanding the reverse process of completing the square.
The page includes several solved examples of completing the square to find solutions to quadratic equations:
Example: Solve for x: (x + 5)² = 24
Solution: x + 5 = ±√24, so x = -5 ± √24
This example shows how to solve a quadratic equation that's already in the form of a perfect square equal to a constant.
Another example demonstrates the process of completing the square when the equation isn't initially in perfect square form:
Example: Solve x² + 10x + 1 = 0
Solution: (x + 5)² = 24, then x + 5 = ±√24, so x = -5 ± √24
The page also covers how to use completing the square to find the minimum or maximum of a quadratic function:
Highlight: For a quadratic function in the form y = a(x - h)² + k, the vertex is at (h, k). If a > 0, it's a minimum; if a < 0, it's a maximum.
Example: Write y = x² - 10x + 3 in the form y = a(x + p)² + q
Solution: y = (x - 5)² - 22
This example illustrates how to rewrite a quadratic function in vertex form, which is useful for identifying the minimum or maximum point of the parabola.
The page concludes with notes on identifying whether a quadratic function has a minimum or maximum based on the sign of the leading coefficient:
Vocabulary: Minimum - occurs when the coefficient of x² is positive
Vocabulary: Maximum - occurs when the coefficient of x² is negative
These concepts are crucial for understanding the behavior of quadratic functions and are essential in 8th grade algebra completing the square notes.
