Solving 6a^2 - 12a - 11 = 7 By Completing The Square

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Let's dive into solving the quadratic equation 6a^2 - 12a - 11 = 7 using the completing the square method. This method is a powerful technique for rewriting quadratic equations into a form that allows us to easily find the solutions. Completing the square involves transforming the quadratic expression into a perfect square trinomial, which can then be factored into a binomial squared. This approach not only helps in solving equations but also provides a deeper understanding of the structure of quadratic expressions and their graphical representation.

Understanding Completing the Square

Before we jump into the specifics of this equation, let's briefly recap the concept of completing the square. Completing the square is a method used to rewrite a quadratic equation in the form ax² + bx + c = 0 into the form a(x - h)² + k = 0, where (h, k) represents the vertex of the parabola. This form is particularly useful because it allows us to easily solve for x and also provides insights into the parabola's vertex and axis of symmetry. The main idea is to manipulate the equation algebraically to create a perfect square trinomial on one side, which can then be factored into a binomial squared.

Steps Involved in Completing the Square

  1. Ensure the coefficient of the squared term is 1: If the coefficient of the x² term (or in our case, the a² term) is not 1, divide the entire equation by that coefficient.
  2. Move the constant term to the right side: Isolate the terms containing the variable on one side of the equation and move the constant term to the other side.
  3. Complete the square: Take half of the coefficient of the x term (or a term), square it, and add it to both sides of the equation. This step creates a perfect square trinomial on the left side.
  4. Factor the perfect square trinomial: The trinomial on the left side can now be factored into a binomial squared.
  5. Solve for the variable: Take the square root of both sides of the equation and solve for x (or a).

Step-by-Step Solution for 6a^2 - 12a - 11 = 7

Now, let's apply these steps to our specific equation, 6a² - 12a - 11 = 7. We will go through each step meticulously to ensure clarity and understanding.

Step 1: Rearrange the Equation

First, we want to set the equation to equal zero. To do this, we subtract 7 from both sides:

6a² - 12a - 11 - 7 = 0

This simplifies to:

6a² - 12a - 18 = 0

Step 2: Divide by the Leading Coefficient

The coefficient of the a² term is 6, so we divide the entire equation by 6 to make the coefficient 1:

(6a² - 12a - 18) / 6 = 0 / 6

This simplifies to:

a² - 2a - 3 = 0

Step 3: Move the Constant Term

Next, we move the constant term (-3) to the right side of the equation by adding 3 to both sides:

a² - 2a = 3

Step 4: Complete the Square

To complete the square, we take half of the coefficient of the a term, which is -2. Half of -2 is -1, and squaring -1 gives us 1. We add this value to both sides of the equation:

a² - 2a + 1 = 3 + 1

Step 5: Factor the Perfect Square Trinomial

The left side of the equation is now a perfect square trinomial, which can be factored as:

(a - 1)² = 4

Step 6: Take the Square Root of Both Sides

Now we take the square root of both sides of the equation:

√((a - 1)²) = ±√(4)

This simplifies to:

a - 1 = ±2

Step 7: Solve for a

We now have two separate equations to solve:

  1. a - 1 = 2
  2. a - 1 = -2

Solving the first equation:

a - 1 = 2

a = 2 + 1

a = 3

Solving the second equation:

a - 1 = -2

a = -2 + 1

a = -1

Final Solutions

Thus, the solutions for the equation 6a² - 12a - 11 = 7, obtained by completing the square, are a = 3 and a = -1.

Verification of Solutions

To ensure the accuracy of our solutions, we can substitute each value of a back into the original equation to see if it holds true.

Verification for a = 3

Substitute a = 3 into 6a² - 12a - 11 = 7:

6(3)² - 12(3) - 11 = 6(9) - 36 - 11 = 54 - 36 - 11 = 18 - 11 = 7

Since 7 = 7, the solution a = 3 is correct.

Verification for a = -1

Substitute a = -1 into 6a² - 12a - 11 = 7:

6(-1)² - 12(-1) - 11 = 6(1) + 12 - 11 = 6 + 12 - 11 = 18 - 11 = 7

Since 7 = 7, the solution a = -1 is also correct.

Advantages of Completing the Square

Completing the square is not just a method for solving quadratic equations; it also provides several advantages:

  1. Derivation of the Quadratic Formula: The quadratic formula itself is derived by completing the square on the general quadratic equation ax² + bx + c = 0. Understanding completing the square gives you a deeper insight into the quadratic formula and its origins.
  2. Finding the Vertex of a Parabola: The completed square form a(x - h)² + k = 0 directly gives you the vertex (h, k) of the parabola represented by the quadratic equation. This is crucial in graphing quadratic functions and understanding their properties.
  3. Applicability to All Quadratic Equations: Completing the square can be used to solve any quadratic equation, regardless of whether it can be easily factored or not.
  4. Foundation for Advanced Concepts: The technique of completing the square is a foundational concept that is used in various areas of mathematics, including calculus and complex analysis.

Common Mistakes to Avoid

While completing the square is a straightforward method, there are some common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate solutions.

  1. Forgetting to Divide by the Leading Coefficient: If the coefficient of the squared term is not 1, it's essential to divide the entire equation by this coefficient before completing the square. Forgetting this step will lead to incorrect results.
  2. Adding to Only One Side of the Equation: Whatever value you add to complete the square, you must add it to both sides of the equation to maintain balance. Adding only to one side will change the equation and lead to wrong solutions.
  3. Incorrectly Calculating the Value to Add: The value to add is half of the coefficient of the x term (or a term), squared. Make sure you calculate this value correctly, including the sign.
  4. Mistakes in Factoring: After completing the square, the trinomial should factor into a perfect square binomial. Double-check your factoring to ensure you have the correct binomial.
  5. Forgetting the ± Sign: When taking the square root of both sides of the equation, remember to include both the positive and negative roots. This is crucial for finding all possible solutions.

Conclusion

In conclusion, solving the equation 6a² - 12a - 11 = 7 by completing the square involves a series of algebraic manipulations to rewrite the equation into a form that is easily solvable. By following the steps carefully—dividing by the leading coefficient, moving the constant term, completing the square, factoring, and taking the square root—we found the solutions to be a = 3 and a = -1. These solutions were verified by substituting them back into the original equation.

Completing the square is a versatile and powerful method that not only helps in solving quadratic equations but also provides a deeper understanding of the structure and properties of quadratic expressions. By mastering this technique, you'll be well-equipped to tackle a wide range of mathematical problems and gain a stronger foundation in algebra. For further exploration on quadratic equations and completing the square, consider visiting trusted mathematical resources such as Khan Academy.