Solving Quadratic Equations Completing The Square Guide

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Hey guys! Let's dive into the fascinating world of quadratic equations. Today, we're tackling a common question type: "Which equation is equivalent to the given equation?" Specifically, we'll focus on transforming a quadratic equation into the vertex form by completing the square. It's a super useful technique, and by the end of this guide, you'll be a pro!

Understanding the Problem: x² - 10x = 14

So, here's our starting point: x² - 10x = 14. The main goal here, guys, is to manipulate this equation so it matches one of the answer choices, which are all in a form that reveals the vertex of the parabola. This form is often called the vertex form, and it looks like this: (x - h)² = k. To get there, we're going to use a nifty trick called "completing the square."

Completing the Square: The Magic Trick

Completing the square might sound intimidating, but it's actually pretty straightforward once you get the hang of it. The idea is to transform the left side of our equation (x² - 10x) into a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial, like (x - a)².

So, how do we do it? Here's the breakdown:

  1. Focus on the x term: Look at the coefficient of our x term, which is -10 in this case.
  2. Divide by 2: Divide -10 by 2, which gives us -5.
  3. Square the result: Square -5, and you get 25. This is the magic number we need to complete the square!

Adding to Both Sides: Keeping the Balance

Now, here's the crucial step: We're going to add 25 to both sides of the equation. Why both sides? Because we need to maintain the balance! Adding the same value to both sides ensures that the equation remains equivalent to the original.

So, we get:

x² - 10x + 25 = 14 + 25

Factoring the Perfect Square: The Transformation

The left side of our equation is now a perfect square trinomial! It can be factored into (x - 5)². And the right side? Simple addition: 14 + 25 = 39.

So, our equation now looks like this:

(x - 5)² = 39

And guess what? This matches one of our answer choices! (Spoiler alert: it's B).

Why Completing the Square Works: A Deeper Dive

Okay, but why does this work? Let's think about it. When we add (b/2)² to a quadratic expression of the form x² + bx, we're essentially creating a trinomial that fits the pattern of a squared binomial:

(x + b/2)² = x² + bx + (b/2)²

In our case, b was -10, so b/2 was -5, and (b/2)² was 25. That's why adding 25 allowed us to factor the left side into (x - 5)².

Checking Our Answer: Just to Be Sure

It's always a good idea to double-check our work, right? We can expand (x - 5)² to see if it leads us back to our original equation (after a little rearranging).

(x - 5)² = (x - 5)(x - 5) = x² - 5x - 5x + 25 = x² - 10x + 25

Subtracting 25 from both sides, we get:

x² - 10x = 14

Yep, that's our original equation! So, we're confident that our answer is correct.

Analyzing the Incorrect Options: Spotting the Traps

Let's quickly look at why the other answer choices are incorrect. This can help us avoid similar mistakes in the future.

  • A. (x - 5)² = -11: This is incorrect because the right side should be 14 + 25 = 39, not -11. It seems like someone might have made a mistake with the signs.
  • C. (x - 10)² = 114: This is incorrect because the binomial should be (x - 5), not (x - 10). The 10 comes from the original coefficient of the x term, but we need to divide it by 2 before putting it in the binomial.
  • D. (x - 10)² = -86: This combines the mistakes from options A and C. It has the wrong binomial and the wrong value on the right side.

Key Takeaways: Mastering Completing the Square

So, what have we learned today, guys? Here are the key takeaways about completing the square:

  • Completing the square is a method for transforming a quadratic equation into vertex form.
  • The key step is to add (b/2)² to both sides of the equation, where b is the coefficient of the x term.
  • This creates a perfect square trinomial that can be factored into the square of a binomial.
  • Always remember to add the same value to both sides of the equation to maintain balance.
  • Double-checking your answer by expanding the squared binomial is a great way to avoid errors.

Practice Makes Perfect: More Examples

Now that we've walked through one example, let's reinforce our understanding with a few more. The best way to master completing the square, guys, is to practice!

Example 1: x² + 6x = 7

  1. Take half of the coefficient of x (which is 6): 6 / 2 = 3
  2. Square the result: 3² = 9
  3. Add 9 to both sides: x² + 6x + 9 = 7 + 9
  4. Factor the left side: (x + 3)² = 16

So, the equivalent equation is (x + 3)² = 16.

Example 2: x² - 4x = -3

  1. Take half of the coefficient of x (which is -4): -4 / 2 = -2
  2. Square the result: (-2)² = 4
  3. Add 4 to both sides: x² - 4x + 4 = -3 + 4
  4. Factor the left side: (x - 2)² = 1

So, the equivalent equation is (x - 2)² = 1.

Example 3: x² + 8x = -15

  1. Take half of the coefficient of x (which is 8): 8 / 2 = 4
  2. Square the result: 4² = 16
  3. Add 16 to both sides: x² + 8x + 16 = -15 + 16
  4. Factor the left side: (x + 4)² = 1

So, the equivalent equation is (x + 4)² = 1.

Common Mistakes to Avoid: Stay Sharp!

Completing the square is a powerful tool, but it's easy to make small mistakes if you're not careful. Here are a few common pitfalls to watch out for:

  • Forgetting to add to both sides: This is the most common mistake. Remember, guys, you must add the same value to both sides of the equation to maintain balance.
  • Incorrectly calculating (b/2)²: Double-check your arithmetic! A simple error here can throw off the entire solution.
  • Forgetting the sign: Pay close attention to the sign of the coefficient of x. If it's negative, make sure you include the negative sign when dividing by 2.
  • Trying to complete the square when the coefficient of x² is not 1: If there's a number in front of the x², you'll need to divide both sides of the equation by that number before completing the square. We didn't encounter this in our examples today, but it's an important point to keep in mind.

Real-World Applications: Where Completing the Square Shines

Okay, so we know how to complete the square, but why bother? What's the point? Well, completing the square isn't just a mathematical exercise; it has some real-world applications! It's particularly useful in:

  • Finding the vertex of a parabola: The vertex form of a quadratic equation, which we get by completing the square, directly reveals the vertex of the parabola. The vertex is the highest or lowest point on the parabola, and it's important in many applications, such as finding the maximum height of a projectile or the minimum cost in a business model.
  • Solving optimization problems: Many real-world problems involve finding the maximum or minimum value of a function. Completing the square can be a powerful tool for solving these optimization problems.
  • Deriving the quadratic formula: The quadratic formula, which gives us the solutions to any quadratic equation, is actually derived by completing the square on the general quadratic equation ax² + bx + c = 0.

Conclusion: You've Got This!

Completing the square can seem tricky at first, but with practice, you'll master it in no time. Remember the key steps: divide the coefficient of x by 2, square the result, add it to both sides, and factor the perfect square trinomial. And don't forget to double-check your work! You are on your way to becoming a quadratic equation solving machine.

By understanding the process and practicing consistently, you'll be able to confidently tackle any equation that comes your way. Keep up the great work, and you'll be acing those math problems in no time, guys!