Solve The Equation X^2 + 6x - 4 = 0 By Completing The Square. What Is The Value Of X?

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Completing the square is a powerful technique used to solve quadratic equations and rewrite them in a more insightful form. This method allows us to transform a quadratic equation into a perfect square trinomial, making it easier to find the solutions. In this comprehensive guide, we will delve into the step-by-step process of solving the equation x² + 6x - 4 = 0 by completing the square. We will explore the underlying principles, provide detailed explanations, and offer examples to solidify your understanding. Let's embark on this journey to master the art of completing the square.

Understanding Quadratic Equations

Before diving into the method of completing the square, it's crucial to have a solid grasp of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is:

ax² + bx + c = 0

where a, b, and c are constants, and a ≠ 0. The solutions to a quadratic equation are also known as its roots or zeros. These are the values of x that make the equation true.

Quadratic equations can be solved using various methods, including factoring, using the quadratic formula, and, of course, completing the square. Each method has its advantages and disadvantages, and the choice of method often depends on the specific equation at hand.

The Essence of Completing the Square

Completing the square is a technique that transforms a quadratic expression into a perfect square trinomial, which is a trinomial that can be factored into the square of a binomial. For example, x² + 2x + 1 is a perfect square trinomial because it can be factored as (x + 1)². The key idea behind completing the square is to manipulate the quadratic equation by adding and subtracting a specific constant term to create this perfect square trinomial.

To understand why this works, let's consider the general form of a perfect square trinomial:

(x + k)² = x² + 2kx + k²

Notice that the constant term k² is the square of half the coefficient of the x term (which is 2k). This relationship is the cornerstone of the completing the square method. We aim to manipulate our original quadratic equation to fit this form, allowing us to easily solve for x.

Step-by-Step Solution for x² + 6x - 4 = 0

Now, let's apply the method of completing the square to solve the equation x² + 6x - 4 = 0. We'll break down the process into manageable steps.

Step 1: Isolate the Constant Term

The first step is to move the constant term to the right side of the equation. This is done by adding the constant term to both sides. In our case, the constant term is -4, so we add 4 to both sides:

x² + 6x - 4 + 4 = 0 + 4

This simplifies to:

x² + 6x = 4

Step 2: Complete the Square

This is the heart of the method. We need to add a constant to both sides of the equation that will make the left side a perfect square trinomial. To find this constant, we take half of the coefficient of the x term, square it, and add the result to both sides.

The coefficient of the x term is 6. Half of 6 is 3, and 3 squared is 9. So, we add 9 to both sides of the equation:

x² + 6x + 9 = 4 + 9

Step 3: Factor the Perfect Square Trinomial

The left side of the equation is now a perfect square trinomial. It can be factored as the square of a binomial. In this case, x² + 6x + 9 factors into (x + 3)²:

(x + 3)² = 13

Step 4: Take the Square Root of Both Sides

To isolate the x term, we take the square root of both sides of the equation. Remember that when taking the square root, we need to consider both the positive and negative roots:

√(x + 3)² = ±√13

This gives us:

x + 3 = ±√13

Step 5: Solve for x

Finally, we isolate x by subtracting 3 from both sides:

x = -3 ± √13

Therefore, the solutions to the equation x² + 6x - 4 = 0 are x = -3 + √13 and x = -3 - √13.

Summarizing the Solution

In summary, we have solved the equation x² + 6x - 4 = 0 by completing the square. The solutions are:

x = -3 ± √13

This can be expressed as two separate solutions:

  • x₁ = -3 + √13
  • x₂ = -3 - √13

Advantages of Completing the Square

Completing the square is not just a method for solving quadratic equations; it's a powerful technique with several advantages:

  • Deriving the Quadratic Formula: The quadratic formula, a general solution for quadratic equations, is derived by completing the square on the general quadratic equation ax² + bx + c = 0. This makes completing the square a fundamental concept in understanding the quadratic formula.
  • Finding the Vertex of a Parabola: Completing the square allows us to rewrite a quadratic equation in vertex form, which directly reveals the vertex of the parabola represented by the equation. The vertex is the minimum or maximum point of the parabola, depending on the sign of the leading coefficient.
  • Solving Equations That Cannot Be Easily Factored: While factoring is a quick method for solving some quadratic equations, it is not always applicable. Completing the square provides a reliable method for solving any quadratic equation, regardless of whether it can be factored.
  • Understanding Transformations of Functions: Completing the square is used in the study of conic sections and transformations of functions. It helps to rewrite equations in a standard form that makes it easier to analyze and graph the function.

Completing the Square with a ≠ 1

So far, we have focused on completing the square when the coefficient of the x² term (a) is 1. However, the method can be extended to quadratic equations where a ≠ 1. The key difference is that we first need to divide the entire equation by a to make the coefficient of x² equal to 1. Let's illustrate this with an example.

Consider the equation:

2x² + 8x - 10 = 0

Step 1: Divide by a

Divide both sides of the equation by 2 (since a = 2):

x² + 4x - 5 = 0

Step 2: Isolate the Constant Term

Add 5 to both sides:

x² + 4x = 5

Step 3: Complete the Square

Take half of the coefficient of the x term (which is 4), square it (2² = 4), and add it to both sides:

x² + 4x + 4 = 5 + 4

Step 4: Factor the Perfect Square Trinomial

Factor the left side:

(x + 2)² = 9

Step 5: Take the Square Root of Both Sides

√(x + 2)² = ±√9

x + 2 = ±3

Step 6: Solve for x

Subtract 2 from both sides:

x = -2 ± 3

This gives us two solutions:

  • x₁ = -2 + 3 = 1
  • x₂ = -2 - 3 = -5

Therefore, the solutions to the equation 2x² + 8x - 10 = 0 are x = 1 and x = -5.

Common Mistakes to Avoid

While completing the square is a straightforward method, it's easy to make mistakes if you're not careful. Here are some common pitfalls to avoid:

  • Forgetting to Divide by a: If a ≠ 1, remember to divide the entire equation by a before completing the square.
  • Adding the Constant to Only One Side: Make sure to add the constant term you're using to complete the square to both sides of the equation. This maintains the equality.
  • Incorrectly Factoring the Perfect Square Trinomial: Double-check that you have factored the perfect square trinomial correctly. It should be in the form (x + k)² or (x - k)², where k is half the coefficient of the x term.
  • Forgetting the ± Sign: When taking the square root of both sides, remember to include both the positive and negative roots. This is crucial for finding all solutions to the quadratic equation.
  • Arithmetic Errors: Simple arithmetic mistakes can lead to incorrect solutions. Take your time and carefully perform each calculation.

Practice Problems

To solidify your understanding of completing the square, try solving these practice problems:

  1. x² - 4x + 1 = 0
  2. 3x² + 12x - 15 = 0
  3. x² + 5x + 6 = 0
  4. 2x² - 6x + 1 = 0

By working through these problems, you'll gain confidence in your ability to complete the square and solve quadratic equations.

Conclusion

Completing the square is a versatile and powerful technique for solving quadratic equations. It not only provides a method for finding solutions but also deepens your understanding of the structure and properties of quadratic equations. By mastering this method, you'll be well-equipped to tackle a wide range of mathematical problems and gain a deeper appreciation for the beauty and elegance of algebra. Remember to practice regularly, pay attention to detail, and you'll become proficient in completing the square. This method is more than just a tool for solving equations; it's a gateway to a more profound understanding of mathematics.