Solving $5x^2 - 20x - 60 = 0$ By Completing The Square A Step-by-Step Guide

$$

Introduction to Completing the Square

In this comprehensive guide, we will delve into the method of completing the square to solve the quadratic equation $5x^2 - 20x - 60 = 0$. Completing the square is a powerful technique in algebra that allows us to rewrite any quadratic equation in a form that makes it easy to find the solutions. This method is especially useful when the quadratic equation cannot be easily factored. Understanding completing the square not only provides a way to solve quadratic equations but also lays the foundation for understanding other algebraic concepts, such as the derivation of the quadratic formula.

Before we dive into the specifics of our equation, let's briefly discuss what a quadratic equation is and why completing the square is such a valuable tool. A quadratic equation is a polynomial equation of the second degree, generally written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a$ is not equal to zero. The solutions to a quadratic equation, also known as the roots or zeros, are the values of $x$ that satisfy the equation. These roots represent the points where the parabola defined by the quadratic equation intersects the x-axis.

While factoring is often the first method we try when solving quadratic equations, not all quadratics can be easily factored. This is where completing the square comes in handy. It's a systematic approach that transforms the quadratic equation into a perfect square trinomial, which can then be easily solved by taking the square root. In essence, completing the square allows us to rewrite the equation in the form $(x - h)^2 = k$, where $h$ and $k$ are constants. This form directly reveals the vertex of the parabola and simplifies the process of finding the roots.

In the following sections, we will break down the process of completing the square step-by-step, applying it specifically to the equation $5x^2 - 20x - 60 = 0$. By the end of this guide, you will have a solid understanding of how to use this technique to solve any quadratic equation, and you'll appreciate its power and versatility in algebraic problem-solving. We will also discuss common pitfalls and provide tips to ensure you master this crucial skill. So, let's embark on this journey to conquer quadratic equations using the method of completing the square!

Step-by-Step Solution for $5x^2 - 20x - 60 = 0$

Now, let's tackle the equation $5x^2 - 20x - 60 = 0$ using the completing the square method. We will go through each step meticulously to ensure a clear understanding of the process. This equation is a classic example where completing the square shines, as it might not be immediately obvious how to factor it directly.

Step 1: Divide by the Leading Coefficient

The first step in completing the square is to make sure the coefficient of the $x^2$ term is 1. In our equation, $5x^2 - 20x - 60 = 0$, the leading coefficient is 5. To make it 1, we divide the entire equation by 5. This gives us:

$$(5x^2 - 20x - 60) / 5 = 0 / 5$$

$$x^2 - 4x - 12 = 0$$

This simplified equation is much easier to work with. Dividing by the leading coefficient is a crucial step, as it sets the stage for the subsequent steps in completing the square. It transforms the equation into a form where we can readily apply the technique of adding and subtracting a constant to create a perfect square trinomial. Without this step, the process of completing the square becomes significantly more complex.

Step 2: Move the Constant Term to the Right Side

The next step involves isolating the terms with $x$ on one side of the equation and moving the constant term to the other side. In our equation, $x^2 - 4x - 12 = 0$, the constant term is -12. We move it to the right side by adding 12 to both sides of the equation:

$$x^2 - 4x - 12 + 12 = 0 + 12$$

$$x^2 - 4x = 12$$

This step sets up the equation for the core of the completing the square process: creating a perfect square trinomial on the left side. By isolating the $x^2$ and $x$ terms, we prepare the equation for the addition of a specific constant that will complete the square. This manipulation is a key part of the technique, allowing us to rewrite the left side as a squared binomial.

Step 3: Complete the Square

This is the heart of the completing the square method. We need to add a constant to both sides of the equation to make the left side a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the form $(x + a)^2$ or $(x - a)^2$. To find the constant we need to add, we take half of the coefficient of the $x$ term, square it, and add the result to both sides.

In our equation, $x^2 - 4x = 12$, the coefficient of the $x$ term is -4. Half of -4 is -2, and squaring -2 gives us 4. So, we add 4 to both sides of the equation:

$$x^2 - 4x + 4 = 12 + 4$$

$$x^2 - 4x + 4 = 16$$

Now, the left side, $x^2 - 4x + 4$, is a perfect square trinomial. It can be factored as $(x - 2)^2$. The right side simplifies to 16. This step is the essence of completing the square, transforming the equation into a form that is easily solvable by taking the square root.

Step 4: Factor and Simplify

As we established in the previous step, the left side of the equation, $x^2 - 4x + 4$, is a perfect square trinomial and can be factored as $(x - 2)^2$. Our equation now looks like this:

$$(x - 2)^2 = 16$$

This form is much simpler to solve. The left side is a squared binomial, and the right side is a constant. This is the goal of completing the square: to rewrite the equation in this form, where we can easily isolate $x$ by taking the square root.

Step 5: Take the Square Root

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

$$\sqrt{(x - 2)^2} = \pm\sqrt{16}$$

$$x - 2 = \pm 4$$

Remember to include both the positive and negative square roots, as both values will satisfy the equation. This is a crucial step in finding all possible solutions to the quadratic equation. Taking only the positive square root would lead to missing one of the solutions.

Step 6: Solve for $x$

Now we have two simple equations to solve for $x$:

1. $x - 2 = 4$
2. $x - 2 = -4$

Solving the first equation, we add 2 to both sides:

$$x - 2 + 2 = 4 + 2$$

$$x = 6$$

Solving the second equation, we add 2 to both sides:

$$x - 2 + 2 = -4 + 2$$

$$x = -2$$

So, we have found two solutions for $x$: 6 and -2.

Final Solutions

The solutions to the equation $5x^2 - 20x - 60 = 0$, obtained by completing the square, are $x = 6$ and $x = -2$. These are the values of $x$ that make the equation true. We can verify these solutions by substituting them back into the original equation and confirming that they satisfy it.

Completing the square provides a reliable method for solving quadratic equations, especially when factoring is not straightforward. This step-by-step approach ensures accuracy and a clear understanding of the underlying algebraic principles.

Verification of Solutions

To ensure the accuracy of our solutions, it's always a good practice to verify them. We will substitute $x = 6$ and $x = -2$ back into the original equation, $5x^2 - 20x - 60 = 0$, to confirm that they indeed satisfy the equation.

Verification for $x = 6$

Substitute $x = 6$ into the equation:

$$5(6)^2 - 20(6) - 60 = 0$$

$$5(36) - 120 - 60 = 0$$

$$180 - 120 - 60 = 0$$

$$60 - 60 = 0$$

$$0 = 0$$

Since the equation holds true, $x = 6$ is indeed a solution.

Verification for $x = -2$

Substitute $x = -2$ into the equation:

$$5(-2)^2 - 20(-2) - 60 = 0$$

$$5(4) + 40 - 60 = 0$$

$$20 + 40 - 60 = 0$$

$$60 - 60 = 0$$

$$0 = 0$$

Since the equation holds true, $x = -2$ is also a solution.

The verification process confirms that both $x = 6$ and $x = -2$ are valid solutions to the quadratic equation $5x^2 - 20x - 60 = 0$. This step reinforces the correctness of our completing the square method and provides confidence in our results.

Common Mistakes and How to Avoid Them

While completing the square is a powerful technique, there are common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate solutions. Let's discuss some of these common errors and how to prevent them.

Mistake 1: Forgetting to Divide by the Leading Coefficient

The first critical step in completing the square is to ensure that the coefficient of the $x^2$ term is 1. Forgetting to divide the entire equation by the leading coefficient can lead to incorrect results. Always remember to perform this step before proceeding with the rest of the method. In our example, we divided the equation $5x^2 - 20x - 60 = 0$ by 5 to get $x^2 - 4x - 12 = 0$.

Mistake 2: Incorrectly Calculating the Constant to Add

The heart of completing the square lies in finding the correct constant to add to both sides of the equation. This constant is calculated by taking half of the coefficient of the $x$ term and squaring it. A common mistake is to forget to square the result or to take half of the coefficient incorrectly. For example, in the equation $x^2 - 4x = 12$, the coefficient of the $x$ term is -4. Half of -4 is -2, and squaring -2 gives us 4. Adding 4 to both sides completes the square.

Mistake 3: Forgetting to Add the Constant to Both Sides

To maintain the balance of the equation, it's crucial to add the constant to both sides. Adding it only to one side will lead to an incorrect equation and, consequently, incorrect solutions. Always remember to add the calculated constant to both the left and right sides of the equation.

Mistake 4: Not Factoring the Perfect Square Trinomial Correctly

After completing the square, the left side of the equation should be a perfect square trinomial. This trinomial can be factored into the form $(x + a)^2$ or $(x - a)^2$. A common mistake is to factor it incorrectly. Remember that the value of $a$ is half of the coefficient of the $x$ term in the original quadratic expression. In our example, $x^2 - 4x + 4$ factors to $(x - 2)^2$, where -2 is half of -4.

Mistake 5: Forgetting the $\pm$ When Taking the Square Root

When taking the square root of both sides of the equation, it's essential to consider both the positive and negative roots. Forgetting the $\pm$ sign will result in missing one of the solutions. Remember that both the positive and negative square roots will satisfy the equation. For instance, when we have $(x - 2)^2 = 16$, taking the square root gives us $x - 2 = \pm 4$.

Mistake 6: Arithmetic Errors

Simple arithmetic errors can derail the entire process. Double-check your calculations, especially when dealing with fractions or negative numbers. A small mistake in arithmetic can lead to a completely wrong answer. Take your time and be meticulous with each step.

By being aware of these common mistakes and taking steps to avoid them, you can master the completing the square method and solve quadratic equations with confidence. Always double-check your work and verify your solutions to ensure accuracy.

Conclusion

In this comprehensive guide, we have thoroughly explored the method of completing the square to solve the quadratic equation $5x^2 - 20x - 60 = 0$. We have broken down the process into clear, manageable steps, starting from dividing by the leading coefficient and moving the constant term, to completing the square, factoring, taking the square root, and finally solving for $x$. The solutions we found are $x = 6$ and $x = -2$, which we verified by substituting them back into the original equation.

Completing the square is a powerful technique that allows us to solve any quadratic equation, regardless of whether it can be easily factored. It is a fundamental concept in algebra and provides a solid foundation for understanding more advanced topics. By mastering this method, you gain a deeper understanding of quadratic equations and their properties.

We also discussed common mistakes that students often make when completing the square and provided tips on how to avoid them. These include remembering to divide by the leading coefficient, correctly calculating the constant to add, adding the constant to both sides, factoring the perfect square trinomial accurately, considering both positive and negative square roots, and avoiding arithmetic errors.

By understanding and practicing the steps involved in completing the square, you can confidently solve quadratic equations and tackle more complex algebraic problems. This method not only provides a solution but also enhances your problem-solving skills and algebraic intuition.

Completing the square is more than just a technique; it's a way of thinking about quadratic equations. It allows us to rewrite the equation in a form that reveals its underlying structure and makes it easier to solve. So, continue to practice this method, and you will find yourself becoming more proficient and confident in your algebraic abilities. Remember, the key to mastering any mathematical concept is practice and a thorough understanding of the fundamental principles. Happy solving!