Solve By Completing The Square: A Step-by-Step Guide

by Rajiv Sharma 53 views

Hey guys! Today, we're diving deep into a powerful technique for solving quadratic equations: completing the square. This method is super useful, especially when factoring doesn't quite cut it, and it's the foundation for understanding the quadratic formula. We'll break down the process step-by-step, using a specific example to make sure you've got it down pat. Let's get started!

Understanding Completing the Square

Completing the square is a method used to solve quadratic equations by transforming a quadratic expression into a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. For example, $x^2 + 6x + 9$ is a perfect square trinomial because it can be factored into $(x + 3)^2$. The magic of this technique lies in its ability to convert any quadratic equation into a form where you can easily isolate the variable and find the solutions.

The core idea behind completing the square is to manipulate the quadratic equation into the form $(x + a)^2 = b$, where 'a' and 'b' are constants. Once you have the equation in this form, you can simply take the square root of both sides and solve for 'x'.

Why Use Completing the Square?

You might be wondering, "Why bother with completing the square when we have the quadratic formula?" That's a fair question! While the quadratic formula is a fantastic tool, completing the square provides a deeper understanding of the structure of quadratic equations. It also serves as the basis for deriving the quadratic formula itself. Plus, it comes in handy in various mathematical contexts, such as when dealing with circles and other conic sections.

The Steps Involved

Completing the square involves a few key steps:

  1. Ensure the coefficient of the $x^2$ term is 1: If it's not, divide the entire equation by that coefficient.
  2. Move the constant term to the right side of the equation: This isolates the terms with 'x' on the left side.
  3. Complete the square: Take half of the coefficient of the 'x' term, square it, and add it to both sides of the equation. This is the crucial step that creates the perfect square trinomial.
  4. Factor the perfect square trinomial: The left side should now be factorable into the form $(x + a)^2$.
  5. Take the square root of both sides: Remember to consider both positive and negative square roots.
  6. Solve for 'x': Isolate 'x' to find the solutions to the equation.

Solving $x^2 - 12x = 14$ by Completing the Square

Let's walk through the example equation $x^2 - 12x = 14$ step-by-step to see how completing the square works in practice.

Step 1: Ensure the coefficient of the $x^2$ term is 1

In our equation, $x^2 - 12x = 14$, the coefficient of the $x^2$ term is already 1, so we can skip this step. Awesome!

Step 2: Move the constant term to the right side of the equation

The constant term is already on the right side, so we're good to go here too. Our equation remains: $x^2 - 12x = 14$.

Step 3: Complete the square

This is where the magic happens! We need to figure out what number to add to both sides to create a perfect square trinomial on the left. Here's how:

  1. Take half of the coefficient of the 'x' term. In our case, the coefficient of 'x' is -12. Half of -12 is -6.
  2. Square the result. $( -6)^2 = 36$.
  3. Add this number (36) to both sides of the equation.

So, we add 36 to both sides: $x^2 - 12x + 36 = 14 + 36$. This simplifies to $x^2 - 12x + 36 = 50$.

Key Point: We added 36 to both sides because it's the number that completes the square on the left side. This ensures that the left side can be factored into a perfect square.

Step 4: Factor the perfect square trinomial

The left side, $x^2 - 12x + 36$, is now a perfect square trinomial. It can be factored as $(x - 6)^2$. So, our equation becomes: $(x - 6)^2 = 50$.

Why (x - 6)? Remember that we took half of the coefficient of the 'x' term (-12) and got -6. This -6 is the constant term in the binomial that we square. It's a neat little shortcut!

Step 5: Take the square root of both sides

Now, we take the square root of both sides of the equation: $\sqrt{(x - 6)^2} = \pm \sqrt{50}$. This gives us $x - 6 = \pm \sqrt{50}$.

Important: Don't forget the $\pm$ (plus or minus) sign! When you take the square root of a number, there are always two possible solutions: a positive and a negative one.

Step 6: Solve for 'x'

We're almost there! To isolate 'x', we add 6 to both sides: $x = 6 \pm \sqrt{50}$.

Now, let's simplify the $\sqrt{50}$. We can rewrite 50 as $25 * 2$, so $\sqrt{50} = \sqrt{25 * 2} = \sqrt{25} * \sqrt{2} = 5\sqrt{2}$.

Therefore, our solutions are: $x = 6 \pm 5\sqrt{2}$.

The Answer

So, going back to the original question, we determined that the number we needed to add to both sides of the equation was 36, and the solutions are $6 \pm 5\sqrt{2}$. This corresponds to option C in the original problem.

Practice Makes Perfect

Completing the square might seem a bit tricky at first, but with practice, you'll become a pro! The key is to understand the underlying principle of creating a perfect square trinomial and then following the steps systematically. Try working through more examples on your own, and you'll be solving quadratic equations like a boss in no time!

Additional Tips and Tricks

  • Always double-check your work: It's easy to make a small mistake, especially when dealing with square roots and signs. Take a moment to review each step to ensure accuracy.
  • Simplify radicals whenever possible: Simplifying radicals not only makes your answer look cleaner but also helps in comparing it with other solutions.
  • Use completing the square to derive the quadratic formula: This is a great exercise to deepen your understanding of both methods.
  • Don't be afraid to ask for help: If you're stuck, reach out to a teacher, tutor, or online resources. There are plenty of people who are happy to help you learn!

Conclusion

Completing the square is a fundamental technique in algebra with applications beyond just solving quadratic equations. By mastering this method, you'll gain a deeper understanding of quadratic expressions and their properties. So, keep practicing, and you'll be well on your way to conquering quadratic equations like a champ! Keep up the great work, and remember, math can be fun!