Mastering Completing The Square Solving Quadratic Equations And Perfect Square Trinomials

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To truly master the technique of completing the square, a foundational skill lies in recognizing and constructing perfect square trinomials. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. This section delves into finding the constant term, denoted as 'c', which transforms a given quadratic expression into a perfect square trinomial. The core principle we'll employ is derived from the algebraic identity: (x + a)² = x² + 2ax + a². Our goal is to manipulate the given expressions to fit this pattern, thereby revealing the value of 'c'. This exploration is not just an exercise in algebra; it's a cornerstone for solving quadratic equations, simplifying complex expressions, and understanding the nature of quadratic functions. The ability to quickly and accurately determine the value of 'c' will significantly enhance your problem-solving capabilities in various mathematical contexts.

To determine the value of c that completes the perfect square trinomial, we need to understand the relationship between the coefficients of the quadratic and linear terms. In a perfect square trinomial of the form x² + bx + c, the constant term 'c' is equal to the square of half the coefficient of the linear term (b). Mathematically, this is expressed as c = (b/2)². This formula is the key to unlocking the value of 'c' in the given expressions. Let's apply this principle to the following examples, step by step, to solidify our understanding and ensure we can confidently tackle any similar problem. Each example will serve as a practical demonstration of the formula, allowing you to see how the value of 'c' is derived and how it completes the perfect square trinomial.

Consider the quadratic expression x² + 18x + c. Here, the coefficient of the linear term (b) is 18. Following our formula, we first find half of b, which is 18 / 2 = 9. Then, we square this value to find c: 9² = 81. Therefore, the value of c that completes the perfect square trinomial is 81. This means that x² + 18x + 81 is a perfect square trinomial, and it can be factored as (x + 9)². This process highlights how the value of 'c' is intrinsically linked to the coefficient of the linear term, ensuring that the trinomial can be expressed as the square of a binomial. Understanding this connection is crucial for mastering the technique of completing the square. In the next examples, we'll apply this same principle to different quadratic expressions, reinforcing our understanding and developing our proficiency.

Now, let's move on to the expression x² + 22x + c. In this case, the coefficient of the linear term (b) is 22. Applying the same method, we first find half of b: 22 / 2 = 11. Next, we square this result to determine c: 11² = 121. Consequently, the value of c that completes the perfect square trinomial is 121. This gives us the perfect square trinomial x² + 22x + 121, which can be factored as (x + 11)². This example further illustrates the consistent application of the formula c = (b/2)², regardless of the specific value of b. By working through these examples, you're not just finding the values of 'c'; you're also building a deeper understanding of the structure of perfect square trinomials and how they relate to binomial squares. This understanding will be invaluable as you progress to more complex problems involving quadratic equations and functions.

  1. x² + 18x + c

    • b = 18
    • c = (18/2)² = 9² = 81

  2. x² + 22x + c

    • b = 22
    • c = (22/2)² = 11² = 121

  3. x² - 17x + c

    • b = -17
    • c = (-17/2)² = 289/4

  4. x² + 5x + c

    • b = 5
    • c = (5/2)² = 25/4

  5. x² - 42x + c

    • b = -42
    • c = (-42/2)² = (-21)² = 441

Completing the square is a powerful technique for solving quadratic equations, particularly when factoring is not straightforward. This method transforms a quadratic equation into a perfect square trinomial on one side, allowing us to easily isolate the variable. In this section, we will apply this technique to find the solution sets of given quadratic equations. The fundamental idea behind completing the square is to manipulate the equation algebraically so that one side becomes a perfect square trinomial, which can then be expressed as the square of a binomial. This transformation simplifies the process of solving for the variable, as we can then take the square root of both sides and isolate the variable. This method is not only a valuable tool for finding solutions but also provides insights into the structure and properties of quadratic equations.

The process of solving quadratic equations by completing the square involves several key steps. First, we ensure that the coefficient of the x² term is 1. If it's not, we divide the entire equation by that coefficient. Next, we move the constant term to the right side of the equation. Then, we take half of the coefficient of the x term, square it, and add it to both sides of the equation. This step is crucial as it creates a perfect square trinomial on the left side. Once we have a perfect square trinomial, we can factor it as the square of a binomial. Finally, we take the square root of both sides of the equation, solve for x, and obtain the solution set. This step-by-step approach ensures that we can systematically solve any quadratic equation using the method of completing the square. Let's apply this method to the given equations to see how it works in practice.

Consider the equation x² + 16x + 39 = 0. To solve this by completing the square, we first move the constant term to the right side: x² + 16x = -39. Next, we take half of the coefficient of the x term (which is 16), square it (8² = 64), and add it to both sides: x² + 16x + 64 = -39 + 64. This simplifies to x² + 16x + 64 = 25. Now, we factor the left side as a perfect square: (x + 8)² = 25. Taking the square root of both sides gives us x + 8 = ±5. Solving for x, we get two possible solutions: x = -8 + 5 = -3 and x = -8 - 5 = -13. Therefore, the solution set for the equation x² + 16x + 39 = 0 is {-3, -13}. This example demonstrates the power of completing the square in transforming a seemingly complex quadratic equation into a solvable form. By systematically applying the steps, we can find the roots of the equation and understand its behavior.

Now, let's tackle the equation x² - 14x - 72 = 0 using the same method. First, we move the constant term to the right side: x² - 14x = 72. Next, we take half of the coefficient of the x term (which is -14), square it ((-7)² = 49), and add it to both sides: x² - 14x + 49 = 72 + 49. This simplifies to x² - 14x + 49 = 121. We then factor the left side as a perfect square: (x - 7)² = 121. Taking the square root of both sides yields x - 7 = ±11. Solving for x, we find two solutions: x = 7 + 11 = 18 and x = 7 - 11 = -4. Thus, the solution set for the equation x² - 14x - 72 = 0 is {18, -4}. This example further reinforces the application of the completing the square technique, showcasing its effectiveness in solving quadratic equations with different coefficients and constant terms. By mastering this method, you gain a versatile tool for tackling a wide range of quadratic problems.

  1. x² + 16x + 39 = 0

    • x² + 16x = -39
    • x² + 16x + (16/2)² = -39 + (16/2)²
    • x² + 16x + 64 = -39 + 64
    • (x + 8)² = 25
    • x + 8 = ±5
    • x = -8 ± 5
    • x = -3, -13
    • Solution Set: {-3, -13}

  2. x² - 14x - 72 = 0

    • x² - 14x = 72
    • x² - 14x + (-14/2)² = 72 + (-14/2)²
    • x² - 14x + 49 = 72 + 49
    • (x - 7)² = 121
    • x - 7 = ±11
    • x = 7 ± 11
    • x = 18, -4
    • Solution Set: {18, -4}