Worksheet On Completing The Square

Completing the Square: A Comprehensive Worksheet and Guide

Completing the square is a crucial algebraic technique used to solve quadratic equations, simplify expressions, and even graph parabolas. This worksheet provides a step-by-step guide, numerous examples, and practice problems to solidify your understanding of this essential mathematical concept. Whether you're struggling with the basics or aiming to master advanced applications, this comprehensive resource will help you conquer completing the square.

Introduction: Understanding Quadratic Equations and Their Forms

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. These equations can be solved using various methods, including factoring, the quadratic formula, and completing the square. Completing the square is particularly useful when factoring isn't straightforward or when dealing with equations that don't have easily identifiable integer factors. It's also vital for understanding the vertex form of a parabola, which is expressed as y = a(x - h)² + k, where (h, k) represents the vertex of the parabola.

Steps to Completing the Square

The process of completing the square involves manipulating a quadratic expression to create a perfect square trinomial, which can then be factored easily. Here’s a step-by-step guide:

1. Ensure the coefficient of x² is 1: If the coefficient of x² (the 'a' in ax² + bx + c) is not 1, divide the entire equation by that coefficient. This ensures that the process works correctly.

2. Move the constant term to the right side: Isolate the terms containing x on the left side of the equation. Move the constant term (c) to the right side.

3. Find half of the coefficient of x and square it: Identify the coefficient of x (the 'b' in ax² + bx + c). Divide this coefficient by 2 and then square the result. This number will be added to both sides of the equation to maintain balance.

4. Add the result to both sides: Add the value calculated in step 3 to both sides of the equation. This completes the square on the left side.

5. Factor the perfect square trinomial: The left side of the equation should now be a perfect square trinomial—a trinomial that can be factored into the square of a binomial (x + p)².

6. Solve for x (if applicable): If the equation is set equal to zero, you can now solve for x by taking the square root of both sides and isolating x.

Examples: Completing the Square in Action

Let's work through a few examples to solidify the steps:

Example 1: A Simple Case

Solve x² + 6x + 5 = 0 by completing the square.

1. The coefficient of x² is already 1.

2. Move the constant term: x² + 6x = -5

3. Half of the coefficient of x (6) is 3; 3² = 9.

4. Add 9 to both sides: x² + 6x + 9 = -5 + 9 => x² + 6x + 9 = 4

5. Factor: (x + 3)² = 4

6. Solve for x: x + 3 = ±√4 => x + 3 = ±2 => x = -1 or x = -5

Example 2: Dealing with a non-unit coefficient

Solve 2x² - 8x + 5 = 0 by completing the square.

1. Divide by 2: x² - 4x + 5/2 = 0

2. Move the constant term: x² - 4x = -5/2

3. Half of -4 is -2; (-2)² = 4

4. Add 4 to both sides: x² - 4x + 4 = -5/2 + 4 => x² - 4x + 4 = 3/2

5. Factor: (x - 2)² = 3/2

6. Solve for x: x - 2 = ±√(3/2) => x = 2 ± √(3/2)

Example 3: Completing the Square to Find the Vertex

Find the vertex of the parabola y = x² - 10x + 21 by completing the square.

1. We'll work with the expression on the right side to put it in vertex form, y = a(x-h)² + k.

2. Rewrite as: y = (x² - 10x) + 21

3. Half of -10 is -5; (-5)² = 25

4. Add and subtract 25 within the parenthesis: y = (x² - 10x + 25 - 25) + 21

5. Factor and simplify: y = (x - 5)² - 25 + 21 => y = (x - 5)² - 4

The vertex is (5, -4). Remember, the vertex form is y = a(x - h)² + k, and the vertex is (h, k).

Explanation of the underlying mathematical principles

Completing the square relies heavily on the concept of perfect square trinomials and their factorization. A perfect square trinomial is a trinomial that can be expressed as the square of a binomial. For example, x² + 6x + 9 is a perfect square trinomial because it factors to (x + 3)². The key is recognizing that the constant term (9 in this case) is always the square of half the coefficient of the x term (half of 6 is 3, and 3² = 9). This is why we add (b/2)² to both sides of the equation during the completing the square process: we're deliberately creating this perfect square trinomial.

The process essentially transforms the quadratic equation from its standard form (ax² + bx + c = 0) into its vertex form (a(x - h)² + k = 0). This vertex form reveals crucial information about the parabola represented by the quadratic equation, most notably its vertex (h, k). The vertex represents the minimum or maximum point of the parabola.

Frequently Asked Questions (FAQs)

• What if the coefficient of x² is negative? If the coefficient of x² is negative, factor out the negative sign before completing the square. This will make the process easier to manage.

• Can completing the square be used for any quadratic equation? Yes, it can be used to solve any quadratic equation, even those with complex roots.

• Why is completing the square important? Beyond solving equations, completing the square is essential for deriving the quadratic formula, understanding conic sections, and simplifying various algebraic expressions. It underpins many advanced mathematical concepts.

• What if I make a mistake in the calculations? Carefully review each step, paying close attention to signs and calculations. It is helpful to double check your work at each stage.

Worksheet Exercises

Now it's time to practice! Solve the following quadratic equations by completing the square:

1. x² + 8x + 15 = 0
2. x² - 10x + 24 = 0
3. x² + 6x - 7 = 0
4. 2x² + 12x + 10 = 0
5. 3x² - 18x + 27 = 0
6. x² - 4x + 1 = 0 (Find the approximate solutions, rounding to two decimal places)
7. -x² + 6x - 5 = 0

Find the vertex of the following parabolas by completing the square:

8. y = x² + 4x - 5
9. y = x² - 6x + 10
10. y = -2x² + 8x - 6

Conclusion: Mastering Completing the Square

Completing the square is a versatile and powerful algebraic technique. While it might initially seem complex, consistent practice will make you proficient in its application. This worksheet provides a solid foundation, allowing you to build confidence and master this essential skill. Remember to break down the process step by step, double check your work, and practice regularly. With enough practice, completing the square will become second nature, opening up further avenues in your mathematical studies. Remember to seek help if needed and don't be afraid to ask questions. Mathematical understanding is built through practice and perseverance.