Solving Quadratic Equations By Completing The Square

In the realm of algebra, quadratic equations hold a prominent position. They frequently appear in various mathematical and scientific contexts. Mastering the techniques to solve them is paramount. One such technique is completing the square, a method that transforms a quadratic equation into a perfect square trinomial, making it readily solvable. In this comprehensive guide, we will delve into the process of solving the quadratic equation $2x^2 + 12x = 66$ by completing the square, step by step, to find the values of $a$ and $b$ in the solutions $x = a - \sqrt{b}$ and $x = a + \sqrt{b}$.

Understanding Quadratic Equations and the Importance of Completing the Square

Before diving into the specifics, let's establish a firm understanding of quadratic equations. A quadratic equation is a polynomial equation of the second degree, generally expressed in the standard form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a \ne 0$. The solutions to a quadratic equation, also known as roots or zeros, are the values of $x$ that satisfy the equation.

There are several methods to solve quadratic equations, including factoring, using the quadratic formula, and completing the square. Completing the square is particularly valuable because it provides a systematic approach to solve any quadratic equation, regardless of whether it can be factored easily or not. Moreover, it forms the basis for deriving the quadratic formula itself.

Step-by-Step Solution: Completing the Square for $2x^2 + 12x = 66$

Let's embark on the journey of solving the given quadratic equation $2x^2 + 12x = 66$ by completing the square. We will meticulously follow each step to arrive at the solutions.

1. Make the Coefficient of $x^2$ Equal to 1

The first step in completing the square is to ensure that the coefficient of the $x^2$ term is 1. In our equation, the coefficient of $x^2$ is 2. To make it 1, we divide both sides of the equation by 2:

$$\frac{2x^2 + 12x}{2} = \frac{66}{2}$$

This simplifies to:

$$x^2 + 6x = 33$$

Now, the coefficient of $x^2$ is indeed 1, setting the stage for the next step.

2. Move the Constant Term to the Right Side

Next, we need to isolate the $x^2$ and $x$ terms on the left side of the equation. In our current equation, the constant term is already on the right side, which is 33. So, we don't need to perform any additional operations in this step. Our equation remains:

$$x^2 + 6x = 33$$

3. Complete the Square

This is the heart of the method. To complete the square, we need to add a constant term to both sides of the equation that will make the left side a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial, such as $(x + k)^2$ or $(x - k)^2$.

To determine the constant term to add, we take half of the coefficient of the $x$ term (which is 6), square it, and add the result to both sides. Half of 6 is 3, and squaring 3 gives us 9. So, we add 9 to both sides of the equation:

$$x^2 + 6x + 9 = 33 + 9$$

Now, the left side is a perfect square trinomial, and the equation becomes:

$$x^2 + 6x + 9 = 42$$

4. Factor the Perfect Square Trinomial

The left side of the equation, $x^2 + 6x + 9$, is a perfect square trinomial and can be factored as $(x + 3)^2$. So, we rewrite the equation as:

$$(x + 3)^2 = 42$$

5. Take the Square Root of Both Sides

To eliminate the square on the left side, we take the square root of both sides of the equation. Remember to consider both positive and negative square roots:

$$\sqrt{(x + 3)^2} = \pm\sqrt{42}$$

This gives us:

$$x + 3 = \pm\sqrt{42}$$

6. Isolate $x$

To solve for $x$, we subtract 3 from both sides of the equation:

$$x = -3 \pm \sqrt{42}$$

7. Express the Solutions in the Form $x = a - \sqrt{b}$ and $x = a + \sqrt{b}$

Now, we have the solutions in the form $x = -3 \pm \sqrt{42}$. Comparing this with the given forms $x = a - \sqrt{b}$ and $x = a + \sqrt{b}$, we can identify the values of $a$ and $b$:

• $a = -3$
• $b = 42$

Therefore, the solutions are:

• $x = -3 - \sqrt{42}$
• $x = -3 + \sqrt{42}$

Conclusion: Mastering Completing the Square

In this detailed guide, we have successfully solved the quadratic equation $2x^2 + 12x = 66$ by completing the square. We meticulously followed each step, from making the coefficient of $x^2$ equal to 1 to isolating $x$ and expressing the solutions in the desired form. We found that $a = -3$ and $b = 42$. Completing the square is a powerful technique that empowers you to solve any quadratic equation. By mastering this method, you gain a deeper understanding of quadratic equations and their solutions. This skill is invaluable in various mathematical and scientific disciplines.

Remember, practice is key to proficiency. Work through numerous examples to solidify your understanding and build confidence in your ability to solve quadratic equations by completing the square. With dedication and perseverance, you will master this essential algebraic technique.

Additional Tips for Completing the Square

• Double-Check Your Work: Always verify your steps to minimize errors. A small mistake can lead to an incorrect solution.
• Simplify Radicals: If possible, simplify the square root term in your solutions. This can make your answers more concise.
• Practice Regularly: Consistent practice is crucial for mastering any mathematical technique. Solve a variety of quadratic equations using completing the square to enhance your skills.
• Understand the Concept: Don't just memorize the steps. Strive to understand the underlying principles of completing the square. This will enable you to apply the technique effectively in different contexts.

Applications of Completing the Square

Completing the square is not merely an abstract mathematical technique. It has practical applications in various fields, including:

• Deriving the Quadratic Formula: The quadratic formula, a widely used method for solving quadratic equations, is derived by completing the square on the general quadratic equation $ax^2 + bx + c = 0$.
• Graphing Quadratic Functions: Completing the square helps to rewrite a quadratic function in vertex form, which reveals the vertex (maximum or minimum point) of the parabola and aids in graphing the function.
• Optimization Problems: In calculus and optimization problems, completing the square can be used to find the maximum or minimum value of a quadratic expression.
• Physics and Engineering: Quadratic equations and completing the square appear in various physics and engineering applications, such as projectile motion and circuit analysis.

Conclusion: Embrace the Power of Completing the Square

Completing the square is a versatile and essential tool in the arsenal of anyone studying mathematics or related fields. It not only provides a method for solving quadratic equations but also enhances your understanding of algebraic manipulations and problem-solving strategies. By mastering this technique, you unlock a gateway to more advanced mathematical concepts and applications.

So, embrace the power of completing the square, practice diligently, and watch your mathematical abilities soar!