Solving Quadratic Equations By Completing The Square A Comprehensive Guide

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In mathematics, quadratic equations play a fundamental role. They appear in various contexts, from physics to engineering, and mastering the techniques to solve them is crucial. One powerful method for solving quadratic equations is completing the square. This method not only provides solutions but also offers insights into the structure of quadratic expressions. In this comprehensive guide, we will delve into the process of solving quadratic equations by completing the square, providing a step-by-step approach with detailed explanations and examples. We will specifically address the equation x2−20x=−34x^2 - 20x = -34, demonstrating each step meticulously to ensure a clear understanding.

Understanding Quadratic Equations

Before diving into the method of completing the square, it's essential 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:

ax2+bx+c=0ax^2 + bx + c = 0

where a, b, and c are constants, and x represents the variable. The coefficient a cannot be zero, as that would make the equation linear rather than quadratic. The solutions to a quadratic equation, also known as roots or zeros, are the values of x that satisfy the equation. These solutions can be real or complex numbers.

Several methods exist to solve quadratic equations, including factoring, using the quadratic formula, and completing the square. Each method has its strengths and is suitable for different types of equations. Completing the square is particularly useful because it can be applied to any quadratic equation and it provides a pathway to derive the quadratic formula itself. This method transforms the quadratic equation into a perfect square trinomial, which can then be easily solved.

The Method of Completing the Square

The method of completing the square involves transforming a quadratic equation into the form:

(x+p)2=q(x + p)^2 = q

where p and q are constants. Once the equation is in this form, we can easily solve for x by taking the square root of both sides. The steps to complete the square are as follows:

  1. Ensure the coefficient of x2x^2 is 1: If the coefficient of x2x^2 is not 1, divide the entire equation by that coefficient.
  2. Move the constant term to the right side: Isolate the terms involving x on one side of the equation.
  3. Add a value to both sides to complete the square: Take half of the coefficient of the x term, square it, and add it to both sides of the equation. This step ensures that the left side becomes a perfect square trinomial.
  4. Factor the perfect square trinomial: The left side of the equation can now be factored into the form (x+p)2(x + p)^2.
  5. Solve for x: Take the square root of both sides of the equation and solve for x.

These steps provide a systematic way to solve any quadratic equation by completing the square. Let's apply these steps to our specific equation: x2−20x=−34x^2 - 20x = -34.

Solving x2−20x=−34x^2 - 20x = -34 by Completing the Square

Now, let's apply the method of completing the square to the equation x2−20x=−34x^2 - 20x = -34. We will follow each step meticulously to arrive at the solution.

Step 1: Ensure the coefficient of x2x^2 is 1

In our equation, x2−20x=−34x^2 - 20x = -34, the coefficient of x2x^2 is already 1, so we can proceed to the next step. This simplifies the process, as we don't need to divide the equation by any value.

Step 2: Move the constant term to the right side

The constant term is already on the right side of the equation, which is −34-34. So, we have:

x2−20x=−34x^2 - 20x = -34

This step is crucial because it isolates the terms with x on one side, preparing us to complete the square on the left side.

Step 3: Add a value to both sides to complete the square

To complete the square, we need to add a value to both sides of the equation that will make the left side a perfect square trinomial. This value is determined by taking half of the coefficient of the x term and squaring it. In our equation, the coefficient of the x term is −20-20. Half of −20-20 is −10-10, and (−10)2(-10)^2 is 100100. Therefore, we add 100100 to both sides of the equation:

x2−20x+100=−34+100x^2 - 20x + 100 = -34 + 100

Simplifying the right side, we get:

x2−20x+100=66x^2 - 20x + 100 = 66

This step is the heart of the method, as adding the correct value transforms the quadratic expression into a form that can be easily factored.

Step 4: Factor the perfect square trinomial

The left side of the equation, x2−20x+100x^2 - 20x + 100, is now a perfect square trinomial. It can be factored as follows:

(x−10)2=66(x - 10)^2 = 66

This factorization is based on the fact that (x−10)(x−10)(x - 10)(x - 10) expands to x2−20x+100x^2 - 20x + 100. The binomial (x−10)(x - 10) represents the square root of the perfect square trinomial, which is a critical step in solving the equation.

Step 5: Solve for xx

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

(x−10)2=±66\sqrt{(x - 10)^2} = \pm\sqrt{66}

This gives us:

x−10=±66x - 10 = \pm\sqrt{66}

Now, we add 1010 to both sides to isolate x:

x=10±66x = 10 \pm \sqrt{66}

So, the two solutions for x are:

x=10+66x = 10 + \sqrt{66}

x=10−66x = 10 - \sqrt{66}

These are the exact solutions to the quadratic equation x2−20x=−34x^2 - 20x = -34. We have successfully solved the equation by completing the square.

Verifying the Solutions

To ensure our solutions are correct, we can substitute them back into the original equation x2−20x=−34x^2 - 20x = -34 and verify that the equation holds true. Let's verify the first solution, x=10+66x = 10 + \sqrt{66}:

(10+66)2−20(10+66)=−34(10 + \sqrt{66})^2 - 20(10 + \sqrt{66}) = -34

Expanding and simplifying:

(100+2066+66)−(200+2066)=−34(100 + 20\sqrt{66} + 66) - (200 + 20\sqrt{66}) = -34

166+2066−200−2066=−34166 + 20\sqrt{66} - 200 - 20\sqrt{66} = -34

−34=−34-34 = -34

The equation holds true for the first solution. Now, let's verify the second solution, x=10−66x = 10 - \sqrt{66}:

(10−66)2−20(10−66)=−34(10 - \sqrt{66})^2 - 20(10 - \sqrt{66}) = -34

Expanding and simplifying:

(100−2066+66)−(200−2066)=−34(100 - 20\sqrt{66} + 66) - (200 - 20\sqrt{66}) = -34

166−2066−200+2066=−34166 - 20\sqrt{66} - 200 + 20\sqrt{66} = -34

−34=−34-34 = -34

Again, the equation holds true. Thus, both solutions are correct. Verifying the solutions is a crucial step to ensure accuracy, especially in mathematical problem-solving.

Advantages of Completing the Square

Completing the square offers several advantages over other methods for solving quadratic equations:

  1. Universality: It can be used to solve any quadratic equation, regardless of whether the equation can be easily factored or not.
  2. Derivation of the quadratic formula: The quadratic formula itself is derived by completing the square on the general quadratic equation ax2+bx+c=0ax^2 + bx + c = 0. Understanding completing the square provides a deeper understanding of the quadratic formula.
  3. Vertex form of a quadratic: Completing the square allows us to rewrite a quadratic equation in vertex form, which provides valuable information about the parabola's vertex and axis of symmetry.
  4. Conceptual understanding: The method enhances the understanding of quadratic expressions and their properties. It builds a solid foundation for more advanced mathematical concepts.

These advantages make completing the square a valuable tool in any mathematician's toolkit.

Real-World Applications

Quadratic equations, and therefore the method of completing the square, have numerous applications in various fields. Some notable examples include:

  1. Physics: Quadratic equations are used to describe projectile motion, where the path of an object through the air is modeled by a parabola. The solutions to these equations can determine the range, maximum height, and time of flight of the projectile.
  2. Engineering: In electrical engineering, quadratic equations are used to analyze circuits and determine the current and voltage in various components. In mechanical engineering, they are used in the design of structures and machines.
  3. Economics: Quadratic equations can model cost, revenue, and profit functions. They can be used to determine the break-even point, the maximum profit, and the optimal production level.
  4. Computer Graphics: Quadratic equations are used in computer graphics to create curves and surfaces. They are essential for rendering realistic images and animations.

These applications highlight the practical importance of understanding and solving quadratic equations.

Common Mistakes to Avoid

While completing the square is a powerful method, there are common mistakes that students often make. Being aware of these mistakes can help in avoiding them:

  1. Forgetting to divide by the coefficient of x2x^2: If the coefficient of x2x^2 is not 1, it is essential to divide the entire equation by this coefficient before proceeding with the steps. Forgetting this step will lead to incorrect solutions.
  2. Incorrectly calculating the value to add: The value to add to both sides is half of the coefficient of the x term, squared. It is crucial to calculate this value correctly. A common mistake is to forget to square the result or to take the wrong sign.
  3. Incorrectly factoring the perfect square trinomial: The perfect square trinomial should factor into the form (x+p)2(x + p)^2 or (x−p)2(x - p)^2. Ensure that the factorization matches the signs and values in the trinomial.
  4. Forgetting the ±\pm sign when taking the square root: When taking the square root of both sides of the equation, remember to include both the positive and negative roots. Forgetting the ±\pm sign will result in missing one of the solutions.
  5. Algebraic errors: Errors in basic algebra, such as adding or subtracting terms incorrectly, can lead to incorrect solutions. It is important to be meticulous and double-check each step.

By being mindful of these common mistakes, you can improve your accuracy and confidence in solving quadratic equations by completing the square.

Conclusion

In this guide, we have explored the method of completing the square for solving quadratic equations. We have provided a step-by-step approach, demonstrating each step with the equation x2−20x=−34x^2 - 20x = -34. We have also discussed the advantages of completing the square, its real-world applications, and common mistakes to avoid. Mastering this method is essential for anyone studying mathematics, as it provides a powerful tool for solving quadratic equations and a deeper understanding of quadratic expressions.

Completing the square is more than just a method for finding solutions; it's a way to transform and understand quadratic equations in a more profound way. By following the steps outlined in this guide and practicing regularly, you can confidently solve a wide range of quadratic equations and appreciate the elegance and utility of this mathematical technique. Remember, the key to mastering completing the square is practice and a thorough understanding of each step. With dedication and perseverance, you can become proficient in this valuable mathematical skill.