Solving $x^2-12x+23=0$ By Completing The Square A Step-by-Step Guide

Solving quadratic equations can sometimes feel like navigating a maze, but with the right tools and techniques, it becomes a much smoother journey. One powerful method for tackling these equations is completing the square. This technique not only helps you find the solutions (also known as roots) but also provides a deeper understanding of the structure of quadratic equations. In this article, we'll dive deep into the method of completing the square, using the equation $x^2 - 12x + 23 = 0$ as our primary example. We'll break down each step, explain the logic behind it, and ensure you're equipped to solve similar problems with confidence. So, let's get started and unlock the secrets of completing the square!

Understanding Quadratic Equations

Before we jump into the method, let's take a moment to understand what a quadratic equation is. 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. The solutions to a quadratic equation are the values of $x$ that satisfy the equation, also known as the roots of the equation. These roots can be real or complex numbers.

There are several methods to solve quadratic equations, including factoring, using the quadratic formula, and, of course, completing the square. Each method has its strengths and weaknesses, and the best approach often depends on the specific equation you're dealing with. Completing the square is particularly useful because it can be applied to any quadratic equation, and it provides a systematic way to rewrite the equation into a form that is easy to solve. It's also a foundational concept for deriving the quadratic formula itself, so mastering this technique gives you a solid understanding of quadratic equations.

When we look at our example equation, $x^2 - 12x + 23 = 0$, we can identify that $a = 1$, $b = -12$, and $c = 23$. Notice that the coefficient of the $x^2$ term is 1, which simplifies the process of completing the square. If the coefficient were not 1, we would need to divide the entire equation by that coefficient as an initial step. Now that we have a basic understanding of quadratic equations and our specific example, let's move on to the heart of the matter: the steps involved in completing the square.

Steps to Solve $x^2 - 12x + 23 = 0$ by Completing the Square

The method of completing the square transforms a quadratic equation into a perfect square trinomial, which can then be easily solved. Let's walk through the steps using our equation, $x^2 - 12x + 23 = 0$, as a guide. The key idea behind completing the square is to manipulate the equation so that one side becomes a perfect square trinomial, which is a trinomial that can be factored into the form $(x + k)^2$ or $(x - k)^2$. This allows us to easily solve for $x$ by taking the square root of both sides.

Step 1: Move the Constant Term to the Right Side

The first step in completing the square is to isolate the $x^2$ and $x$ terms on one side of the equation. We do this by moving the constant term (in our case, 23) to the right side of the equation. To do this, we subtract 23 from both sides of the equation:

$x^2 - 12x + 23 - 23 = 0 - 23$

This simplifies to:

$x^2 - 12x = -23$

Now, we have the $x^2$ and $x$ terms on the left side, and the constant term on the right side. This sets us up for the next step, which involves finding the value that completes the square.

Step 2: Complete the Square

This is the core of the method. To complete the square, we need to add a constant to both sides of the equation that will make the left side a perfect square trinomial. The constant we need to add is determined by taking half of the coefficient of the $x$ term (which is -12 in our case), squaring it, and adding the result to both sides. Half of -12 is -6, and (-6) squared is 36. So, we add 36 to both sides of the equation:

$x^2 - 12x + 36 = -23 + 36$

Step 3: Factor the Left Side

The left side of the equation is now a perfect square trinomial. This means it can be factored into the form $(x - k)^2$. In our case, $x^2 - 12x + 36$ can be factored as $(x - 6)^2$. The right side of the equation simplifies to 13. So, our equation now looks like this:

$(x - 6)^2 = 13$

This step is crucial because it transforms the quadratic equation into a form that is easy to solve for $x$. We have successfully created a perfect square on one side, which allows us to isolate $x$ by taking the square root.

Step 4: Take the Square Root of Both Sides

To solve for $x$, 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:

$\sqrt{(x - 6)^2} = \pm \sqrt{13}$

This simplifies to:

$x - 6 = \pm \sqrt{13}$

Step 5: Solve for $x$

Finally, we isolate $x$ by adding 6 to both sides of the equation:

$x = 6 \pm \sqrt{13}$

This gives us two solutions for $x$: $x = 6 + \sqrt{13}$ and $x = 6 - \sqrt{13}$. These are the roots of the quadratic equation $x^2 - 12x + 23 = 0$.

Verifying the Solutions

It's always a good practice to verify our solutions to ensure they are correct. We can do this by substituting each value of $x$ back into the original equation and checking if it holds true.

Verification for $x = 6 + \sqrt{13}$

Substitute $x = 6 + \sqrt{13}$ into the original equation:

$(6 + \sqrt{13})^2 - 12(6 + \sqrt{13}) + 23 = 0$

Expand and simplify:

$(36 + 12\sqrt{13} + 13) - (72 + 12\sqrt{13}) + 23 = 0$

$49 + 12\sqrt{13} - 72 - 12\sqrt{13} + 23 = 0$

$0 = 0$

Verification for $x = 6 - \sqrt{13}$

Substitute $x = 6 - \sqrt{13}$ into the original equation:

$(6 - \sqrt{13})^2 - 12(6 - \sqrt{13}) + 23 = 0$

Expand and simplify:

$(36 - 12\sqrt{13} + 13) - (72 - 12\sqrt{13}) + 23 = 0$

$49 - 12\sqrt{13} - 72 + 12\sqrt{13} + 23 = 0$

$0 = 0$

Both solutions satisfy the original equation, confirming that our solutions are correct.

Why Completing the Square Works

The beauty of completing the square lies in its systematic approach to transforming a quadratic equation into a solvable form. The method works because it leverages the structure of perfect square trinomials. By adding a specific constant to both sides of the equation, we create a perfect square on one side, which can then be factored into the form $(x + k)^2$ or $(x - k)^2$. This form is incredibly useful because taking the square root of both sides allows us to isolate $x$ and find the solutions.

Imagine you have a puzzle with missing pieces. Completing the square is like finding those missing pieces and fitting them in to create a complete picture. In this case, the complete picture is a perfect square trinomial, which is much easier to handle than the original quadratic expression. The method provides a clear pathway to the solutions, making it a valuable tool in your mathematical toolkit.

Common Mistakes to Avoid

While completing the square is a powerful technique, it's essential to be aware of common mistakes that can lead to incorrect solutions. Here are some pitfalls to watch out for:

1. Forgetting to Add the Constant to Both Sides: When you add a constant to complete the square on one side of the equation, you must add the same constant to the other side to maintain the equality. Failing to do so will throw off your solution.
2. Incorrectly Calculating the Constant: The constant to add is half of the coefficient of the $x$ term, squared. Make sure you perform this calculation accurately. Double-check your arithmetic to avoid errors.
3. Forgetting the $\pm$ Sign When Taking the Square Root: When you take the square root of both sides of the equation, remember that there are two possible roots: a positive root and a negative root. Failing to consider both roots will lead to missing one of the solutions.
4. Algebraic Errors: Be careful with your algebraic manipulations, especially when expanding and simplifying expressions. A small error can propagate through the rest of the solution, leading to an incorrect answer. Double-check each step to ensure accuracy.
5. Not Simplifying Radicals: After finding the solutions, simplify any radicals if possible. This presents the answers in their simplest form and demonstrates a complete understanding of the problem.

By being aware of these common mistakes and taking the time to double-check your work, you can increase your accuracy and confidence in solving quadratic equations by completing the square.

Practice Problems

To truly master completing the square, practice is key. Here are a few practice problems you can try:

1. $x^2 + 6x + 5 = 0$
2. $x^2 - 8x + 12 = 0$
3. $x^2 + 10x - 11 = 0$
4. $2x^2 + 8x - 10 = 0$ (Hint: Divide by 2 first)
5. $3x^2 - 12x + 9 = 0$ (Hint: Divide by 3 first)

Work through these problems step-by-step, applying the method we've discussed. Check your answers by substituting them back into the original equations. The more you practice, the more comfortable and confident you'll become with completing the square.

Conclusion

Completing the square is a valuable technique for solving quadratic equations. It provides a systematic approach to finding the solutions and offers a deeper understanding of the structure of these equations. By following the steps we've outlined and practicing regularly, you can master this method and confidently tackle quadratic equations. Remember to move the constant term, complete the square by adding the appropriate constant to both sides, factor the perfect square trinomial, take the square root, and solve for $x$. And don't forget to verify your solutions!

So, the correct solutions to the equation $x^2 - 12x + 23 = 0$ are $x = 6 + \sqrt{13}$ and $x = 6 - \sqrt{13}$. Keep practicing, and you'll become a quadratic equation-solving pro in no time!