Solving Quadratic Equations By Completing The Square Find A And B

Solve the equation $2 x^2+12 x=66$ by completing the square. Find the values of $a$ and $b$ in the solutions $x=a-\sqrt{b}$ and $x=a+\sqrt{b}$.

The quadratic equation, a fundamental concept in algebra, takes the general form of ax² + bx + c = 0, where a, b, and c are constants, and x represents the unknown variable. Mastering the techniques to solve these equations is crucial for various mathematical applications. Among the methods available, completing the square stands out as a powerful and versatile approach. This method not only provides solutions but also offers valuable insights into the structure and properties of quadratic equations.

In this comprehensive guide, we will delve into the method of completing the square, providing a step-by-step explanation alongside a detailed example to solidify your understanding. By the end of this guide, you will be well-equipped to tackle quadratic equations with confidence and precision. Our focus will be on solving the equation $2x^2 + 12x = 66$ by completing the square and determining the values of a and b in the solutions $x = a - √b$ and $x = a + √b$.

Understanding the Method of Completing the Square

Completing the square is a technique used to rewrite a quadratic equation in a form that allows for easy extraction of the solutions. The core idea is to manipulate the equation into the form (x + h)² = k, where h and k are constants. This form is particularly useful because we can directly solve for x by taking the square root of both sides.

The process involves transforming the quadratic expression into a perfect square trinomial, which is a trinomial that can be factored into the square of a binomial. The general steps for completing the square are as follows:

1. Divide by the leading coefficient: If the coefficient of the $x^2$ term (i.e., a) is not 1, divide the entire equation by a. This step ensures that the coefficient of $x^2$ is 1, which is necessary for completing the square.
2. Move the constant term: Move the constant term (c) to the right side of the equation. This isolates the terms containing x on one side.
3. Complete the square: Take half of the coefficient of the x term (i.e., b), square it, and add the result to both sides of the equation. This step creates a perfect square trinomial on the left side.
4. Factor the perfect square trinomial: Factor the left side of the equation as the square of a binomial. This should be in the form $(x + h)^2$.
5. Solve for x: Take the square root of both sides of the equation, remembering to consider both the positive and negative roots. Then, solve for x.

By following these steps, we can convert any quadratic equation into a solvable form. Let’s apply this method to the given equation, $2x^2 + 12x = 66$.

Step-by-Step Solution for $2x^2 + 12x = 66$

To effectively illustrate the method of completing the square, we'll meticulously dissect the equation $2x^2 + 12x = 66$. This step-by-step breakdown ensures clarity and provides a solid understanding of the process.

1. Divide by the Leading Coefficient

The initial step in completing the square involves ensuring the coefficient of the $x^2$ term is 1. In our equation, $2x^2 + 12x = 66$, the leading coefficient is 2. To achieve our goal, we divide every term in the equation by 2:

rac{2x^2}{2} + rac{12x}{2} = rac{66}{2}

This simplifies to:

$x^2 + 6x = 33$

Now, the coefficient of the $x^2$ term is indeed 1, setting the stage for the next step in our process.

2. Move the Constant Term

In this particular equation, the constant term is already isolated (or non-existent) on the right side, which simplifies this step for us. The equation remains:

$x^2 + 6x = 33$

If we had a constant term on the left side, we would subtract it from both sides to move it to the right side. However, in this case, we can proceed directly to the most crucial step of completing the square.

3. Complete the Square

Completing the square is the heart of this method. It involves transforming the left side of the equation into a perfect square trinomial. To achieve this, we take half of the coefficient of the x term, square it, and add the result to both sides of the equation.

In our equation, the coefficient of the x term is 6. Half of 6 is 3, and squaring 3 gives us 9. Thus, we add 9 to both sides of the equation:

$x^2 + 6x + 9 = 33 + 9$

This simplifies to:

$x^2 + 6x + 9 = 42$

By adding 9, we have successfully created a perfect square trinomial on the left side of the equation.

4. Factor the Perfect Square Trinomial

The next step is to factor the perfect square trinomial. By definition, a perfect square trinomial can be factored into the square of a binomial. In this case, $x^2 + 6x + 9$ can be factored as $(x + 3)^2$. This can be confirmed by expanding $(x + 3)^2$, which indeed equals $x^2 + 6x + 9$.

Our equation now looks like this:

$(x + 3)^2 = 42$

This form is crucial because it allows us to easily solve for x by taking the square root of both sides.

5. Solve for x

Now we solve for x. Taking the square root of both sides of the equation $(x + 3)^2 = 42$ gives us:

$x + 3 = ±√42$

This equation gives us two possible solutions for x, one where we add the square root of 42 and one where we subtract it. To isolate x, we subtract 3 from both sides:

$x = -3 ± √42$

Thus, we have two solutions:

$x = -3 - √42$ $x = -3 + √42$

Comparing these solutions with the given form $x = a - √b$ and $x = a + √b$, we can identify the values of a and b. In this case, a = -3 and b = 42.

Determining the Values of a and b

Having solved the quadratic equation by completing the square, we have arrived at the solutions:

$x = -3 - √42$ $x = -3 + √42$

The problem asks us to express these solutions in the form:

$x = a - √b$ $x = a + √b$

By directly comparing our solutions with the given form, we can easily identify the values of a and b. In both solutions, the term without the square root is -3, and the term under the square root is 42. Therefore:

$a = -3$ $b = 42$

These values satisfy both forms of the solution, confirming our result. Thus, the values of a and b that complete the solutions are -3 and 42, respectively.

Conclusion

In this comprehensive guide, we have successfully demonstrated how to solve a quadratic equation by completing the square. We meticulously walked through each step, from dividing by the leading coefficient to factoring the perfect square trinomial and ultimately solving for x. Through this process, we determined the solutions to the equation $2x^2 + 12x = 66$ and expressed them in the form $x = a - √b$ and $x = a + √b$. By comparing our solutions, we accurately identified the values of a and b as -3 and 42, respectively.

Completing the square is a valuable technique in algebra, providing not only a method to solve quadratic equations but also a deeper understanding of their structure. This method is particularly useful when the quadratic equation cannot be easily factored. By mastering this technique, you gain a powerful tool in your mathematical toolkit, enabling you to tackle a wide range of quadratic equations with confidence and precision.

Furthermore, the process of completing the square is fundamental in deriving the quadratic formula, a universal formula for solving any quadratic equation. Thus, understanding completing the square not only solves equations directly but also provides a foundation for more advanced concepts in algebra. Keep practicing, and you'll find this method becomes second nature, enhancing your problem-solving abilities in mathematics.