Completing the Square: A Comprehensive Guide for Mathematical Equations

Completing the square is a powerful algebraic technique used to solve quadratic equations, transform algebraic expressions, and find zeros of polynomials. Let's explore this technique in depth with examples and practical applications.

Understanding Quadratic Equations

A quadratic equation in standard form looks like ax2 bx c 0, where a, b, c are constants and a ≠ 0. Completing the square involves transforming this equation into a perfect square trinomial, which makes it easier to solve. However, not all equations require this process, as some can be solved by simpler methods.

Perfect Square Trinomials

Sometimes, a quadratic equation can be rewritten as a perfect square trinomial immediately, eliminating the need for the completing the square method. An example given is the equation x2 - 125 0.

x2 - 125 is already in the form (x - a)2 0 where a 5 and -125 a2. Therefore, we can directly solve it as:

(x - 5 0) and boxed{x 5}.

General Steps for Completing the Square

Even though the equation in question is a perfect square, let's discuss the general process for solving X^2 Bx C 0 using completing the square:

Move the constant term to the other side of the equation:

(x^2 Bx -C)

Determine the value that completes the square: (left(frac{B}{2}right)^2)

Add (left(frac{B}{2}right)^2) to both sides of the equation:

(x^2 Bx left(frac{B}{2}right)^2 -C left(frac{B}{2}right)^2)

This yields the perfect square trinomial on the left:

(left(x frac{B}{2}right)^2 -C left(frac{B}{2}right)^2)

Solve for (x) by taking the square root of both sides:

(x frac{B}{2} pm sqrt{-C left(frac{B}{2}right)^2})

(x -frac{B}{2} pm sqrt{-C left(frac{B}{2}right)^2})

Example: x2 - 125 0

Our initial equation (x^2 - 125 0) can be transformed directly into the form of a perfect square:

(x^2 - 2 cdot x cdot 5 5^2 5^2 - 125)

((x - 5)^2 0)

By taking the square root of both sides:

(sqrt{(x - 5)^2} sqrt{0})

(x - 5 0)

Thus, the solution is:

boxed{x 5}.

Practical Applications

The process of completing the square is not just theoretical; it has practical applications in various disciplines, including physics, engineering, and statistics. For example, it can be used to find the vertex of a parabola or to solve optimization problems.

In conclusion, while the equation (x^2 - 125 0) does not require completing the square, understanding this technique can help in solving more complex equations. Completing the square is a valuable tool for algebraic manipulation and problem-solving in mathematics and beyond.