Solving Quadratic Equations By Completing The Square

by Tim Redaksi 53 views
Iklan Headers

Hey guys! Let's dive into the world of quadratic equations and learn a super useful technique called "completing the square." This method is a fantastic way to find the solutions (also known as roots) of these equations. We'll go through some examples together, so you can totally master it. This method might seem a bit tricky at first, but trust me, with a little practice, you'll be solving quadratic equations like a pro! It's all about transforming the equation into a special form that makes finding the solutions a breeze. Remember, practice makes perfect, so let's get started and have some fun with it!

What is Completing the Square?

Before we jump into the examples, let's quickly understand what completing the square is all about. Basically, it's a method where we manipulate a quadratic equation to create a perfect square trinomial on one side. A perfect square trinomial is a trinomial that can be factored into the square of a binomial, like (x + a)² or (x - a)². By doing this, we can easily isolate the variable and solve for its value. The whole point is to rewrite the equation in a form that makes it super easy to solve. This technique is especially helpful when dealing with equations that are difficult or impossible to factor directly. It provides a reliable and systematic approach to find the solutions.

Completing the square involves a few key steps. First, we need to ensure the coefficient of the x² term is 1. If it's not, we'll divide the entire equation by that coefficient. Next, we move the constant term to the right side of the equation. Then comes the clever part: we take half of the coefficient of the x term, square it, and add it to both sides of the equation. This crucial step transforms the left side into a perfect square trinomial. Finally, we factor the perfect square trinomial, simplify, and solve for x. This method is incredibly versatile and guarantees a solution, even when factoring seems impossible. So, let's get our hands dirty and start solving some equations!

Step-by-step solutions

a. x² + 12x + 3 = 0

Alright, let's solve our first equation, x² + 12x + 3 = 0. Here's how we can do it step-by-step:

  1. Move the constant term: First things first, move the constant term (3) to the right side of the equation. This gives us: x² + 12x = -3.
  2. Complete the square: Now, we'll complete the square. Take half of the coefficient of the x term (which is 12), square it ((12/2)² = 6² = 36), and add it to both sides of the equation. So we have: x² + 12x + 36 = -3 + 36.
  3. Factor and simplify: The left side is now a perfect square trinomial. Factor it as (x + 6)². Simplify the right side: (x + 6)² = 33.
  4. Solve for x: Take the square root of both sides: x + 6 = ±√33. Then, isolate x: x = -6 ± √33. So, the solutions are x = -6 + √33 and x = -6 - √33. Nice work, guys! We've successfully solved our first equation using completing the square. Notice how we cleverly manipulated the equation to isolate x. The solutions represent the points where the parabola of the quadratic equation intersects the x-axis. Using this method ensures we can find these points, even when factoring is challenging.

b. 7x² - 4x - 3 = 0

Let's tackle the equation 7x² - 4x - 3 = 0. Here's how to complete the square:

  1. Divide by the coefficient of x²: Since the coefficient of x² is 7 (not 1), divide the entire equation by 7: x² - (4/7)x - (3/7) = 0.
  2. Move the constant term: Move the constant term to the right side: x² - (4/7)x = 3/7.
  3. Complete the square: Take half of the coefficient of the x term (-4/7), square it ((-4/14)² = 16/196 = 4/49), and add it to both sides: x² - (4/7)x + 4/49 = 3/7 + 4/49.
  4. Factor and simplify: Factor the left side and simplify the right side: (x - 2/7)² = (21 + 4)/49, thus (x - 2/7)² = 25/49.
  5. Solve for x: Take the square root of both sides: x - 2/7 = ±5/7. Isolate x: x = 2/7 ± 5/7. Thus, the solutions are x = (2 + 5)/7 = 1 and x = (2 - 5)/7 = -3/7. Awesome, you did it! See how we had to first adjust the equation to suit the completing the square method? This involves careful manipulation to find the roots of the equation. Understanding this method is key to unlocking complex problems that might seem impossible to solve using other techniques.

c. 3x² + 2x - 7 = 0

Let's get to the equation 3x² + 2x - 7 = 0. Here’s the deal:

  1. Divide to get x² alone: Divide everything by 3: x² + (2/3)x - 7/3 = 0.
  2. Move the constant: Move the -7/3 to the other side: x² + (2/3)x = 7/3.
  3. Complete the square (again!): Take half of 2/3, which is 1/3, square it to get 1/9, and add it to both sides: x² + (2/3)x + 1/9 = 7/3 + 1/9.
  4. Factor and simplify: Factor the left side: (x + 1/3)² = (21 + 1)/9, which simplifies to (x + 1/3)² = 22/9.
  5. Solve for x: Take the square root: x + 1/3 = ±√(22/9). Solve for x: x = -1/3 ± √22/3. So, the solutions are x = (-1 + √22)/3 and x = (-1 - √22)/3. Fantastic work, everyone! See how each step brings us closer to isolating 'x'? This process of completing the square is a powerful tool. It transforms the equation step by step, which ensures that we find the correct solutions.

d. 3x² + 8x + 4 = 0

Alright, let's tackle 3x² + 8x + 4 = 0:

  1. Make the x² coefficient 1: Divide everything by 3: x² + (8/3)x + 4/3 = 0.
  2. Move the constant: x² + (8/3)x = -4/3.
  3. Complete the square: Half of 8/3 is 4/3. Square it to get 16/9. Add to both sides: x² + (8/3)x + 16/9 = -4/3 + 16/9.
  4. Factor and simplify: (x + 4/3)² = (-12 + 16)/9, which becomes (x + 4/3)² = 4/9.
  5. Solve for x: Take the square root: x + 4/3 = ±2/3. Isolate x: x = -4/3 ± 2/3. So, the solutions are x = -4/3 + 2/3 = -2/3 and x = -4/3 - 2/3 = -2. You're all doing amazing! This systematic approach to solving quadratic equations is both effective and reliable. It allows us to reach the solution step by step without any hidden complexity. Completing the square is not just about finding answers; it's about understanding the core structure of these equations.

e. 8x² = 18x - 9

Let's tackle our last equation: 8x² = 18x - 9. Here’s what we do:

  1. Rearrange the equation: Rewrite it in standard form: 8x² - 18x + 9 = 0.
  2. Divide to get x² coefficient = 1: Divide by 8: x² - (18/8)x + 9/8 = 0. Simplify: x² - (9/4)x + 9/8 = 0.
  3. Move the constant term: x² - (9/4)x = -9/8.
  4. Complete the square: Half of -9/4 is -9/8. Square it to get 81/64. Add it to both sides: x² - (9/4)x + 81/64 = -9/8 + 81/64.
  5. Factor and simplify: (x - 9/8)² = (-72 + 81)/64, so (x - 9/8)² = 9/64.
  6. Solve for x: Take the square root: x - 9/8 = ±3/8. Isolate x: x = 9/8 ± 3/8. Thus, x = 9/8 + 3/8 = 12/8 = 3/2 and x = 9/8 - 3/8 = 6/8 = 3/4. Well done, guys! You've solved all the equations! This completes the series of examples. Each step is designed to make the equations easier to solve. We've explored different forms, ensuring we're equipped to handle any quadratic equation. Completing the square helps you to see the true nature of these equations and understand their underlying structure.

Conclusion

Great job everyone! You've now learned how to solve quadratic equations by completing the square. Remember, this method gives you a reliable way to find the solutions, even when factoring is tough. Practice these steps with more examples, and you'll become super confident in solving any quadratic equation. Keep up the awesome work, and keep exploring the amazing world of mathematics! Understanding this technique significantly expands your mathematical toolkit and enhances your problem-solving skills. So keep practicing and you'll become a pro in no time! Keep exploring and keep having fun with math!