How To Rewrite A Function By Completing The Square

Completing the square is a powerful algebraic technique that allows us to rewrite quadratic expressions, and consequently, functions, into a more insightful form. This technique is invaluable for solving quadratic equations, finding the vertex of a parabola, and simplifying expressions in calculus and other areas of mathematics. Understanding how to rewrite a function by completing the square not only enhances your algebraic skills but also provides a deeper understanding of the properties of quadratic functions.

Why Complete the Square?

Before diving into the "how," let's briefly address the "why." Completing the square transforms a quadratic function from its standard form, f(x) = ax² + bx + c, into vertex form, f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. This vertex form makes it immediately apparent where the parabola reaches its minimum or maximum value (depending on the sign of a), and it also simplifies the process of graphing the function. Furthermore, this technique can be adapted to solve quadratic equations even when factoring is difficult or impossible.

The Core Idea: Creating a Perfect Square Trinomial

At its heart, completing the square involves manipulating a quadratic expression to create a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial, such as x² + 2x + 1 = (x + 1)². The goal is to take an incomplete quadratic expression and add a constant term to "complete" the square.

Step-by-Step Guide to Completing the Square

Let's break down the process into manageable steps using a general quadratic function and then illustrate with specific examples.

1. Ensure the Leading Coefficient is 1:

If the coefficient of the x² term (a) is not 1, factor it out from the x² and x terms. This is crucial because the subsequent steps rely on having a leading coefficient of 1.

General Form: f(x) = ax² + bx + c Modified Form: f(x) = a(x² + (b/a)x) + c

2. Calculate the Value to Complete the Square:

Take half of the coefficient of the x term (the term inside the parentheses), square it, and add it inside the parentheses. This value is what "completes" the square.

Coefficient of x: (b/a) Half of the coefficient: (b/2a) Value to add: (b/2a)² = b²/4a²

3. Add and Subtract to Maintain Balance:

Add the calculated value inside the parentheses. However, since you're adding something to the expression, you must also subtract an equivalent value outside the parentheses to maintain the overall value of the function. Remember to account for the factor a that you factored out in step 1. You are essentially adding 0 to the equation.

Add inside: a(x² + (b/a)x + b²/4a²) Subtract outside: c - a(b²/4a²) which simplifies to c - b²/4a

4. Rewrite as a Perfect Square and Simplify:

Rewrite the expression inside the parentheses as the square of a binomial. This binomial will be of the form (x + b/2a). Also, simplify the constant term outside the parentheses.

Perfect Square: a(x + b/2a)² Simplified Constant: c - b²/4a

5. The Vertex Form:

You now have the quadratic function in vertex form: f(x) = a(x + b/2a)² + (c - b²/4a). From this form, you can directly identify the vertex as (-b/2a, c - b²/4a). Remember that h = -b/2a and k = c - b²/4a.

Examples to Solidify Understanding

Let's work through a few examples to illustrate the process.

Example 1: f(x) = x² + 6x + 5

1. Leading Coefficient: The leading coefficient is already 1.

2. Calculate: Half of 6 is 3, and 3 squared is 9.

3. Add and Subtract: f(x) = (x² + 6x + 9) + 5 - 9

4. Rewrite and Simplify: f(x) = (x + 3)² - 4

5. Vertex Form: The vertex form is f(x) = (x + 3)² - 4. The vertex is at (-3, -4).

Example 2: f(x) = 2x² - 8x + 10

1. Leading Coefficient: Factor out the 2: f(x) = 2(x² - 4x) + 10

2. Calculate: Half of -4 is -2, and -2 squared is 4.

3. Add and Subtract: f(x) = 2(x² - 4x + 4) + 10 - 2(4)

4. Rewrite and Simplify: f(x) = 2(x - 2)² + 10 - 8 = 2(x - 2)² + 2

5. Vertex Form: The vertex form is f(x) = 2(x - 2)² + 2. The vertex is at (2, 2).

Example 3: f(x) = -x² + 4x - 7

1. Leading Coefficient: Factor out the -1: f(x) = -(x² - 4x) - 7

2. Calculate: Half of -4 is -2, and -2 squared is 4.

3. Add and Subtract: f(x) = -(x² - 4x + 4) - 7 - (-1)(4)

4. Rewrite and Simplify: f(x) = -(x - 2)² - 7 + 4 = -(x - 2)² - 3

5. Vertex Form: The vertex form is f(x) = -(x - 2)² - 3. The vertex is at (2, -3).

Common Mistakes to Avoid

• Forgetting to Factor Out 'a': This is perhaps the most common mistake. If a is not 1, you must factor it out before completing the square.
• Incorrectly Subtracting: When you add a value inside the parentheses, you must subtract an equivalent value outside the parentheses, remembering to account for the factored-out a.
• Sign Errors: Be meticulous with your signs, especially when dealing with negative coefficients.
• Rushing the Process: Completing the square requires careful attention to detail. Take your time and double-check each step.

Completing the Square to Solve Quadratic Equations

While completing the square is excellent for rewriting functions, it's also a method for solving quadratic equations. The process is similar, but with a slight twist. Let's consider the equation ax² + bx + c = 0.

1. Move the Constant Term: Move the constant term c to the right side of the equation: ax² + bx = -c.

2. Ensure Leading Coefficient is 1: If a is not 1, divide both sides of the equation by a: x² + (b/a)x = -c/a.

3. Complete the Square: Take half of the coefficient of the x term, square it, and add it to both sides of the equation: x² + (b/a)x + (b/2a)² = -c/a + (b/2a)².

4. Rewrite and Simplify: Rewrite the left side as a squared binomial and simplify the right side: (x + b/2a)² = -c/a + b²/4a². Further simplification gives (x + b/2a)² = (b² - 4ac) / 4a².

5. Take the Square Root: Take the square root of both sides of the equation: x + b/2a = ±√(b² - 4ac) / 2a.

6. Isolate x: Isolate x to find the solutions: x = -b/2a ± √(b² - 4ac) / 2a. This is the quadratic formula! Completing the square is, in fact, how the quadratic formula is derived.

Example: Solve x² + 4x - 5 = 0 by Completing the Square

1. Move Constant: x² + 4x = 5

2. Leading Coefficient: Already 1.

3. Complete the Square: Half of 4 is 2, and 2 squared is 4: x² + 4x + 4 = 5 + 4

4. Rewrite and Simplify: (x + 2)² = 9

5. Take Square Root: x + 2 = ±3

6. Isolate x: x = -2 ± 3. Therefore, x = 1 or x = -5.

The Underlying Mathematics: Why Does This Work?

Completing the square relies on the algebraic identity (x + a)² = x² + 2ax + a². The key is recognizing that a quadratic expression of the form x² + bx is "almost" a perfect square. By adding the term (b/2)², we create the perfect square trinomial (x + b/2)². The act of adding and subtracting the same value ensures that we're not changing the fundamental value of the expression, only its form.

Furthermore, when the leading coefficient a is not 1, factoring it out allows us to focus on completing the square for the x² + (b/a)x portion of the expression. The subsequent multiplication by a and subtraction ensures the overall expression remains equivalent.

Applications Beyond Quadratics

While completing the square is primarily associated with quadratic functions, its underlying principles have applications in other areas of mathematics:

• Conic Sections: Completing the square is used extensively when working with conic sections (circles, ellipses, parabolas, and hyperbolas) to rewrite their equations in standard form, making it easier to identify key features like the center, radius, foci, and vertices.
• Calculus: Completing the square can simplify integrals involving quadratic expressions in the denominator.
• Optimization Problems: The vertex form obtained through completing the square directly provides the maximum or minimum value of a quadratic function, which is useful in optimization problems.

Alternative Methods: When to Use What

While completing the square is a versatile technique, other methods exist for dealing with quadratic functions and equations. It’s helpful to understand when each method is most appropriate:

• Factoring: If the quadratic expression can be easily factored, this is often the quickest method for solving quadratic equations. However, not all quadratic expressions are factorable with integer coefficients.
• Quadratic Formula: The quadratic formula provides a general solution for any quadratic equation. It's particularly useful when factoring is difficult or impossible, and when you only need the solutions without rewriting the function.
• Graphing: Graphing the quadratic function can visually reveal the solutions (x-intercepts) and the vertex. This is useful for gaining a visual understanding, but may not provide exact solutions.

Completing the square shines when you need to rewrite the quadratic function in vertex form to find the vertex or understand its behavior, or when you want to derive the quadratic formula.

Conclusion: Mastering Completing the Square

Completing the square is more than just a mechanical process; it's a gateway to a deeper understanding of quadratic functions and their properties. By mastering this technique, you gain a powerful tool for solving equations, analyzing graphs, and simplifying expressions in various mathematical contexts. While it may seem challenging at first, practice and a clear understanding of the underlying principles will make completing the square a valuable asset in your mathematical toolkit. Remember to pay attention to detail, avoid common mistakes, and appreciate the elegance of this method in transforming quadratic expressions into a more insightful and useful form.