How To Factor Completing The Square

Completing the square is a powerful algebraic technique used to solve quadratic equations, rewrite them in vertex form, and even simplify complex expressions. Mastering this method unlocks a deeper understanding of quadratic functions and their applications. This guide provides a comprehensive breakdown of how to factor by completing the square, offering step-by-step instructions, examples, and insights to solidify your grasp of this essential skill.

Understanding the Foundation: What is Completing the Square?

Completing the square transforms a quadratic expression of the form ax² + bx + c into the form a(x - h)² + k. The beauty of this transformation lies in creating a perfect square trinomial within the expression. A perfect square trinomial is a trinomial that can be factored into the square of a binomial, such as (x + 2)² or (x - 3)². By completing the square, we essentially force a quadratic expression to fit this pattern, allowing us to rewrite it in a more manageable and informative way. This process reveals the vertex of the parabola represented by the quadratic equation, making it invaluable for graphing and optimization problems.

The Step-by-Step Guide to Completing the Square

Let's break down the process into manageable steps with illustrative examples. We'll start with the basic case where a = 1 and then move on to cases where a ≠ 1.

Case 1: When a = 1 (The coefficient of x² is 1)

This is the simplest scenario. Here's how to complete the square for a quadratic expression like x² + bx + c:

Step 1: Prepare the Expression

• Identify the coefficients b and c.

• Move the constant term (c) to the right side of the equation (if you're solving an equation) or isolate it by leaving space after the bx term (if you're simply rewriting the expression).

  Example: Let's say we have the expression x² + 6x + 5. We rewrite it as x² + 6x + ____ + 5. We've left a blank space to add a value that will complete the square.

Step 2: Calculate the Completing Term

• Take half of the coefficient of the x term (b), square it, and add it to both sides of the equation (if you're solving an equation) or add and subtract it within the expression (if you're simply rewriting the expression). This value is * (b/2)²*.

  Example: b = 6. Half of 6 is 3, and 3 squared is 9. So, we add and subtract 9: x² + 6x + 9 - 9 + 5.

Step 3: Factor the Perfect Square Trinomial

• The first three terms now form a perfect square trinomial. Factor this trinomial into the form (x + b/2)² or (x - b/2)², depending on the sign of b.

  Example: x² + 6x + 9 factors into (x + 3)². Our expression now looks like this: (x + 3)² - 9 + 5.

Step 4: Simplify

• Combine the constant terms.

  Example: -9 + 5 = -4. Therefore, the completed square form is (x + 3)² - 4.

Summary of Case 1

• Original Expression: x² + bx + c
• Completing Term: (b/2)²
• Completed Square Form: *(x + b/2)² + (c - (b/2)²) *

Example 1: Completing the Square for x² + 8x + 12

1. Prepare: x² + 8x + ____ + 12
2. Calculate: (8/2)² = 4² = 16. So, x² + 8x + 16 - 16 + 12
3. Factor: (x + 4)² - 16 + 12
4. Simplify: (x + 4)² - 4

Example 2: Completing the Square for x² - 10x + 21

1. Prepare: x² - 10x + ____ + 21
2. Calculate: (-10/2)² = (-5)² = 25. So, x² - 10x + 25 - 25 + 21
3. Factor: (x - 5)² - 25 + 21
4. Simplify: (x - 5)² - 4

Case 2: When a ≠ 1 (The coefficient of x² is not 1)

This case requires an extra step to ensure the coefficient of the x² term is 1 before completing the square.

Step 1: Factor out 'a'

• Factor out the coefficient a from the x² and x terms only. Leave the constant term c outside the parentheses.

  Example: Let's say we have the expression 2x² + 8x + 6. We factor out 2 from the first two terms: 2(x² + 4x) + 6.

Step 2: Complete the Square Inside the Parentheses

• Follow the steps from Case 1 to complete the square inside the parentheses. Identify the new b coefficient (the coefficient of x inside the parentheses), calculate (b/2)², and add and subtract it inside the parentheses.

  Example: Inside the parentheses, b = 4. (4/2)² = 2² = 4. So, we add and subtract 4 inside the parentheses: 2(x² + 4x + 4 - 4) + 6.

Step 3: Factor the Perfect Square Trinomial Inside the Parentheses

• Factor the perfect square trinomial inside the parentheses.

  Example: x² + 4x + 4 factors into (x + 2)². Our expression now looks like this: 2((x + 2)² - 4) + 6.

Step 4: Distribute 'a'

• Distribute the a value (the value you factored out in Step 1) to both terms inside the parentheses.

  Example: 2((x + 2)² - 4) = 2(x + 2)² - 8. Our expression now looks like this: 2(x + 2)² - 8 + 6.

Step 5: Simplify

• Combine the constant terms.

  Example: -8 + 6 = -2. Therefore, the completed square form is 2(x + 2)² - 2.

Summary of Case 2

• Original Expression: ax² + bx + c
• Factoring out 'a': a(x² + (b/a)x) + c
• Completing Term (inside parentheses): * (b/2a)²*
• Completed Square Form: *a(x + b/2a)² + (c - a(b/2a)²) *

Example 3: Completing the Square for 3x² - 12x + 5

1. Factor out 'a': 3(x² - 4x) + 5
2. Complete the Square (inside parentheses): (-4/2)² = (-2)² = 4. So, 3(x² - 4x + 4 - 4) + 5
3. Factor: 3((x - 2)² - 4) + 5
4. Distribute: 3(x - 2)² - 12 + 5
5. Simplify: 3(x - 2)² - 7

Example 4: Completing the Square for -2x² + 16x - 24

1. Factor out 'a': -2(x² - 8x) - 24
2. Complete the Square (inside parentheses): (-8/2)² = (-4)² = 16. So, -2(x² - 8x + 16 - 16) - 24
3. Factor: -2((x - 4)² - 16) - 24
4. Distribute: -2(x - 4)² + 32 - 24
5. Simplify: -2(x - 4)² + 8

Completing the Square When Solving Quadratic Equations

Completing the square is particularly useful for solving quadratic equations, especially when factoring is difficult or impossible. Here's how it works:

Step 1: Rewrite the Equation

• Start with the quadratic equation in the form ax² + bx + c = 0.

Step 2: Complete the Square

• Follow the steps outlined above (Case 1 or Case 2) to complete the square on the left side of the equation.

Step 3: Isolate the Squared Term

• Isolate the squared term (x - h)² on one side of the equation.

Step 4: Take the Square Root

• Take the square root of both sides of the equation. Remember to include both the positive and negative square roots.

Step 5: Solve for x

• Solve for x by isolating it. You'll typically get two solutions.

Example 5: Solving x² + 6x + 5 = 0 by Completing the Square

1. Rewrite: The equation is already in the correct form.

2. Complete the Square: (We already did this in a previous example) (x + 3)² - 4 = 0

3. Isolate: (x + 3)² = 4

4. Square Root: x + 3 = ±√4 => x + 3 = ±2

5. Solve:

   • x + 3 = 2 => x = -1
   • x + 3 = -2 => x = -5

   Therefore, the solutions are x = -1 and x = -5.

Example 6: Solving 2x² - 8x + 6 = 0 by Completing the Square

1. Rewrite: The equation is already in the correct form.

2. Complete the Square: (We already found the completed square form earlier) 2(x - 2)² - 2 = 0

3. Isolate: 2(x - 2)² = 2 => (x - 2)² = 1

4. Square Root: x - 2 = ±√1 => x - 2 = ±1

5. Solve:

   • x - 2 = 1 => x = 3
   • x - 2 = -1 => x = 1

   Therefore, the solutions are x = 3 and x = 1.

Why Completing the Square Matters: Applications and Benefits

While completing the square might seem like a purely algebraic exercise, it has significant applications in various areas:

• Solving Quadratic Equations: As demonstrated above, it provides a reliable method for finding the roots of quadratic equations, even when factoring is challenging.
• Finding the Vertex of a Parabola: The completed square form, a(x - h)² + k, directly reveals the vertex of the parabola represented by the quadratic equation. The vertex is the point (h, k). This is crucial for graphing quadratic functions and understanding their behavior.
• Optimization Problems: Knowing the vertex allows you to determine the maximum or minimum value of a quadratic function. This is essential for solving optimization problems in various fields, such as physics, engineering, and economics.
• Deriving the Quadratic Formula: The quadratic formula itself is derived by completing the square on the general quadratic equation ax² + bx + c = 0. This demonstrates the fundamental nature of the technique.
• Calculus Applications: Completing the square can simplify integrals and other calculus problems involving quadratic expressions.
• Understanding Conic Sections: The standard form equations for ellipses and hyperbolas are derived using techniques similar to completing the square.

Common Mistakes to Avoid

• Forgetting to Factor out 'a': When a ≠ 1, failing to factor it out before completing the square is a common error. Remember to factor a from the x² and x terms only.
• Only Adding the Completing Term: When rewriting an expression (not solving an equation), you must both add and subtract the completing term to maintain the expression's value.
• Incorrectly Calculating the Completing Term: Ensure you correctly calculate (b/2)². Pay attention to the sign of b.
• Forgetting the ± Sign: When taking the square root of both sides of an equation, remember to include both the positive and negative square roots. This ensures you find all possible solutions.
• Distributing 'a' Incorrectly: After completing the square inside the parentheses (when a ≠ 1), be sure to distribute a to both terms inside the parentheses.

Advanced Tips and Tricks

• Fractions: Don't be intimidated by fractions! Completing the square works just the same with fractional coefficients.
• Practice: The key to mastering completing the square is consistent practice. Work through numerous examples with varying coefficients.
• Visual Representation: Visualize the process geometrically. Completing the square can be thought of as rearranging areas to form a perfect square.

FAQs About Completing the Square

• Is completing the square always the best method for solving quadratic equations? Not always. If the quadratic equation can be easily factored, factoring is usually the quicker method. However, completing the square always works, regardless of whether the equation is factorable. The quadratic formula is also a good alternative, especially when the coefficients are complex.
• Can completing the square be used with complex numbers? Yes, the principles of completing the square apply to quadratic expressions with complex coefficients.
• What if I have a quadratic expression with only an x² term and a constant term (ax² + c)? You don't need to complete the square in this case. You can directly solve for x by isolating the x² term and taking the square root.
• How is completing the square related to the vertex form of a quadratic equation? The process of completing the square directly transforms a quadratic equation into its vertex form, a(x - h)² + k, where (h, k) represents the vertex of the parabola.
• Can I use a calculator to help with completing the square? Yes, calculators can help with the arithmetic calculations involved, especially when dealing with fractions or decimals. However, it's important to understand the underlying process and not rely solely on the calculator.

Conclusion: Mastering the Art of Completing the Square

Completing the square is a fundamental technique in algebra with far-reaching applications. By understanding the steps involved and practicing consistently, you can master this skill and unlock a deeper understanding of quadratic functions and their properties. From solving equations to finding the vertex of a parabola, completing the square empowers you to tackle a wide range of mathematical problems with confidence. So, embrace the challenge, practice diligently, and unlock the power of completing the square!