Complete The Expression So It Forms A Perfect-square Trinomial.

Completing the Square: A Comprehensive Guide to Perfect Square Trinomials

Completing the square is a fundamental algebraic technique used to solve quadratic equations and rewrite quadratic expressions in a more manageable form. It's particularly useful when dealing with equations that can't be easily factored. Understanding this method unlocks a deeper understanding of parabolas, their vertices, and the solutions to quadratic equations. This comprehensive guide will walk you through the process, explain the underlying mathematics, and provide numerous examples to solidify your understanding.

Introduction: What is a Perfect Square Trinomial?

A perfect square trinomial is a trinomial (a polynomial with three terms) that can be factored into the square of a binomial. In other words, it's the result of squaring a binomial expression. The general form of a perfect square trinomial is:

a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²

Notice the pattern: the first and last terms are perfect squares (a² and b²), and the middle term is twice the product of the square roots of the first and last terms (2ab or -2ab). This pattern is key to understanding how to complete the square.

Steps to Complete the Square

Let's break down the process of completing the square step-by-step. We'll use the general quadratic expression:

ax² + bx + c

While completing the square works for any quadratic, it's most commonly used when 'a' (the coefficient of x²) is 1. If 'a' is not 1, you must first factor out 'a' from the x² and x terms.

1. Ensure the Leading Coefficient is 1: If the coefficient of x² (a) is not 1, factor it out from the x² and x terms. Let's assume for now that a=1. Our expression becomes:

   x² + bx + c

2. Focus on the x and x² Terms: Ignore the constant term (c) for now. We'll concentrate on the x² and x terms.

3. Find Half of the Coefficient of x: Take the coefficient of x (which is b), divide it by 2, and square the result: (b/2)².

4. Add and Subtract (b/2)²: Add (b/2)² to the expression. To maintain the equality of the expression, we also subtract (b/2)². This step is crucial: we're not changing the value of the expression, only its form. Our expression now looks like this:

   x² + bx + (b/2)² - (b/2)² + c

5. Factor the Perfect Square Trinomial: The first three terms now form a perfect square trinomial: x² + bx + (b/2)² This can be factored as (x + b/2)². Our expression simplifies to:

   (x + b/2)² - (b/2)² + c

6. Simplify: Combine the constant terms, -(b/2)² and c.

Now you've completed the square! The expression is now in vertex form: (x + b/2)² + [c - (b/2)²]

Detailed Examples

Let's work through some examples to solidify this process:

Example 1: x² + 6x + 5

1. Leading Coefficient is 1: Already the case.

2. Focus on x and x² Terms: x² + 6x

3. Half of the Coefficient of x: (6/2)² = 3² = 9

4. Add and Subtract 9: x² + 6x + 9 - 9 + 5

5. Factor the Perfect Square Trinomial: (x + 3)² - 9 + 5

6. Simplify: (x + 3)² - 4

Therefore, x² + 6x + 5 = (x + 3)² - 4

Example 2: x² - 8x + 10

1. Leading Coefficient is 1: Already the case.

2. Focus on x and x² Terms: x² - 8x

3. Half of the Coefficient of x: (-8/2)² = (-4)² = 16

4. Add and Subtract 16: x² - 8x + 16 - 16 + 10

5. Factor the Perfect Square Trinomial: (x - 4)² - 16 + 10

6. Simplify: (x - 4)² - 6

Therefore, x² - 8x + 10 = (x - 4)² - 6

Example 3: 2x² + 12x - 5

1. Leading Coefficient is NOT 1: Factor out the 2: 2(x² + 6x) - 5

2. Focus on x and x² Terms (inside the parenthesis): x² + 6x

3. Half of the Coefficient of x: (6/2)² = 3² = 9

4. Add and Subtract 9 (inside the parenthesis): 2(x² + 6x + 9 - 9) - 5

5. Factor the Perfect Square Trinomial (inside the parenthesis): 2((x + 3)² - 9) - 5

6. Simplify: 2(x + 3)² - 18 - 5 = 2(x + 3)² - 23

Therefore, 2x² + 12x - 5 = 2(x + 3)² - 23

The Significance of Completing the Square

Completing the square isn't just a mechanical process; it has significant applications:

• Solving Quadratic Equations: Setting the completed square equal to zero allows you to easily solve for x. For instance, from Example 1, (x + 3)² - 4 = 0 leads to (x + 3)² = 4, meaning x + 3 = ±2, thus x = -1 or x = -5.

• Finding the Vertex of a Parabola: The vertex form of a quadratic equation, achieved through completing the square, directly reveals the vertex of the parabola. The vertex is (-b/2a, c - (b/2a)²). In Example 1, the vertex is (-3, -4).

• Graphing Quadratic Functions: Completing the square facilitates easier graphing of quadratic functions by providing the vertex and the axis of symmetry.

• Deriving the Quadratic Formula: The quadratic formula itself can be derived by completing the square on the general quadratic equation ax² + bx + c = 0.

Frequently Asked Questions (FAQ)

• Why do we add and subtract the same number? This maintains the equality of the expression. We are essentially adding zero, which doesn't change the value, but allows us to manipulate the expression into a perfect square trinomial.

• What if the coefficient of x² is negative? Factor out the negative coefficient first. For example, -x² + 4x + 5 becomes - (x² - 4x) + 5. Then proceed with completing the square within the parenthesis.

• Can I complete the square with fractions? Absolutely! The process remains the same; just be careful with your arithmetic.

• Is completing the square always the best method for solving quadratic equations? No, for simple quadratic equations that can be easily factored, factoring is often quicker. However, completing the square is essential for equations that are not easily factorable and is fundamental for understanding other mathematical concepts.

Conclusion: Mastering Completing the Square

Completing the square is a powerful algebraic technique with wide-ranging applications in solving quadratic equations, understanding parabolas, and simplifying expressions. While the initial steps may seem complex, with consistent practice, you'll master this technique and appreciate its significance in various mathematical contexts. Remember the key steps: ensure a leading coefficient of 1, find half the coefficient of x, square it, add and subtract, factor the perfect square trinomial, and simplify. By understanding and mastering this technique, you will build a stronger foundation in algebra and unlock a deeper understanding of quadratic functions. Remember to practice regularly with different examples to solidify your understanding and build confidence in this valuable algebraic skill.