Solving $0=9(x^2+6x)-18$ By Completing The Square A Step-by-Step Guide

Jul 11, 2025 by ADMIN 71 views

$Solving Quadratic Equations by Completing the Square A Step-by-Step Guide to \$0=9(x^2+6x)-18\$$

Understanding quadratic equations and mastering methods to solve them is a cornerstone of algebra. Among the various techniques available, completing the square stands out as a powerful and versatile approach. This article delves into the step-by-step process of solving the quadratic equation $0=9(x^2+6x)-18$ by completing the square, providing clarity and insight into each stage. Before we dive into the specifics, let’s first understand the broader concept of quadratic equations and why completing the square is such a valuable technique.

A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a$ is not equal to zero. These equations appear in various fields of science, engineering, and mathematics, modeling phenomena like projectile motion, electrical circuits, and financial growth. Solving quadratic equations involves finding the values of $x$ that satisfy the equation, commonly known as the roots or solutions.

There are several methods to solve quadratic equations, each with its strengths and weaknesses. Factoring, the quadratic formula, and graphical methods are among the common approaches. However, completing the square holds a unique position due to its ability to transform any quadratic equation into a form where solutions can be easily extracted. This method is not just a problem-solving tool; it provides a deeper understanding of the structure of quadratic equations and serves as the basis for deriving the quadratic formula itself.

Completing the square involves rewriting a quadratic equation in the form $(x + p)^2 = q$, where $p$ and $q$ are constants. This form allows us to solve for $x$ by taking the square root of both sides, leading to a straightforward solution process. The key to this method lies in adding a specific constant to the quadratic expression to create a perfect square trinomial, which can then be factored into a binomial squared. This constant is derived from the coefficient of the $x$ term, making the process systematic and applicable to any quadratic equation.

Step-by-Step Solution of $0=9(x^2+6x)-18$ by Completing the Square

Now, let's apply the method of completing the square to solve the given equation, $0=9(x^2+6x)-18$. We will break down the process into manageable steps, ensuring a clear understanding of each action and its purpose. This detailed walkthrough will not only provide the solutions but also equip you with the skills to tackle similar problems confidently.

Step 1: Simplify the Equation

The initial equation is $0=9(x^2+6x)-18$. Our first step is to simplify the equation by distributing the 9 across the terms inside the parenthesis:

$0 = 9x^2 + 54x - 18$

Next, to make the completing the square process easier, we want the coefficient of the $x^2$ term to be 1. We can achieve this by dividing the entire equation by 9:

$0 = x^2 + 6x - 2$

This simplified form sets the stage for the core steps of completing the square.

Step 2: Isolate the Quadratic and Linear Terms

To proceed with completing the square, we need to isolate the quadratic ($x^2$) and linear ($6x$) terms on one side of the equation. We can do this by adding 2 to both sides:

$2 = x^2 + 6x$

This rearrangement prepares the equation for the addition of the constant term that will create the perfect square trinomial.

Step 3: Complete the Square

This is the heart of the method. To complete the square, we need to add a constant term to both sides of the equation that will make the left side a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the form $(x + p)^2$. The constant we need to add is determined by taking half of the coefficient of the $x$ term (which is 6 in this case), squaring it, and adding the result to both sides.

Half of the coefficient of $x$ is $\frac{6}{2} = 3$. Squaring this value gives us $3^2 = 9$. So, we add 9 to both sides of the equation:

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

This gives us:

$11 = x^2 + 6x + 9$

The right side of the equation is now a perfect square trinomial.

Step 4: Factor the Perfect Square Trinomial

The next step is to factor the perfect square trinomial. The expression $x^2 + 6x + 9$ can be factored as $(x + 3)^2$. So, our equation becomes:

$11 = (x + 3)^2$

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

Step 5: Solve for $x$

To solve for $x$, we take the square root of both sides of the equation:

$\sqrt{11} = \sqrt{(x + 3)^2}\$

This yields:

$\pm\sqrt{11} = x + 3$

Now, we isolate $x$ by subtracting 3 from both sides:

$x = -3 \pm \sqrt{11}$

Thus, we have two solutions for $x$:

$x = -3 + \sqrt{11}$ and $x = -3 - \sqrt{11}$

These are the roots of the quadratic equation $0=9(x^2+6x)-18$.

Analyzing the Incorrect Options

Now, let's examine the incorrect options provided in the original question and understand why they do not correctly represent the steps in completing the square for the given equation.

The options presented were:

$18+81=9(x^2+6x+9)$
$18+9=9(x^2+6x+9)$
$\sqrt{342}=(x+6)^2$
$\sqrt{99}=(x+3)^2$

Option 1: $18+81=9(x^2+6x+9)$

This option attempts to add a constant to both sides of the equation but does so incorrectly. Starting from the original equation $0=9(x^2+6x)-18$, we can add 18 to both sides to get $18 = 9(x^2 + 6x)$. To complete the square inside the parenthesis, we need to add $(\frac{6}{2})^2 = 9$ inside the parenthesis. However, because this 9 is inside the parenthesis being multiplied by 9, we are effectively adding $9 \times 9 = 81$ to the right side of the equation. Therefore, to balance the equation, we must add 81 to the left side as well. This leads to the correct transformation:

$18 + 81 = 9(x^2 + 6x + 9)$

This step is correct in its arithmetic. However, it's presented out of context, without the prior steps of isolating the quadratic and linear terms and dividing through by the leading coefficient. While the arithmetic is accurate, it's not a complete or clearly presented step in the process.

Option 2: $18+9=9(x^2+6x+9)$

This option contains a critical error. It correctly identifies that 9 needs to be added inside the parenthesis to complete the square but fails to account for the factor of 9 outside the parenthesis. As explained above, adding 9 inside the parenthesis is equivalent to adding 81 (9 * 9) to the right side of the equation. Therefore, the left side should also reflect the addition of 81, not 9. The correct equation should be $18 + 81 = 9(x^2 + 6x + 9)$, not $18 + 9 = 9(x^2 + 6x + 9)$. This error stems from a misunderstanding of how the distributive property affects the constant added to complete the square.

Option 3: $\sqrt{342}=(x+6)^2$

This option is incorrect because it jumps to an incorrect conclusion about the factored form and the square root. To understand the error, let's trace back from the correct steps. After completing the square correctly, we arrived at $11 = (x + 3)^2$. This option incorrectly states that $\sqrt{342}=(x+6)^2$. There's no direct path from the original equation, through the correct steps of completing the square, to this statement. The error likely arises from a confusion in factoring or a miscalculation in taking the square root.

Option 4: $\sqrt{99}=(x+3)^2$

This option is partially correct but misses a crucial step. It correctly identifies the factored form $(x + 3)^2$ but makes an error in the value on the other side of the equation. From our correct steps, we have $11 = (x + 3)^2$. Taking the square root would lead to $\sqrt{11} = x + 3$, not $\sqrt{99} = (x + 3)^2$. The 99 likely comes from an attempt to combine the initial constant term (18) with the term added to complete the square (81), but this calculation is misplaced in the equation. The correct form involves the square root of 11, not 99.

Key Takeaways

Simplify First: Always simplify the equation before attempting to complete the square.
Account for Coefficients: Be mindful of coefficients outside the parenthesis when adding constants to complete the square.
Factor Correctly: Ensure the perfect square trinomial is factored accurately.
Solve Methodically: Follow the steps of taking the square root and isolating $x$ carefully.

By understanding these key points and practicing diligently, you can master the technique of completing the square and confidently solve a wide range of quadratic equations. Completing the square is not just a method; it’s a journey into the heart of quadratic equations, revealing their structure and paving the way for deeper mathematical understanding.

Conclusion

In conclusion, solving quadratic equations by completing the square involves a systematic process of simplifying, isolating terms, creating a perfect square trinomial, factoring, and solving for the variable. The equation $0=9(x^2+6x)-18$ can be solved by first simplifying to $0 = x^2 + 6x - 2$, then isolating the quadratic and linear terms, completing the square by adding 9 to both sides, factoring the perfect square trinomial, and finally, solving for $x$ to obtain the solutions $x = -3 + \sqrt{11}$ and $x = -3 - \sqrt{11}$. Understanding each step and the underlying principles is crucial for mastering this technique and applying it to various mathematical problems.