Solving Quadratic Equations A Step-by-Step Guide To 3(x+4)^2=39

Introduction to Quadratic Equations

Understanding quadratic equations is crucial in mathematics, as they appear in various fields, from physics to engineering and economics. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a$ is not equal to zero. These equations can be solved using several methods, including factoring, completing the square, and the quadratic formula. In this article, we will delve into solving a specific quadratic equation: $3(x+4)^2=39$. This example will allow us to explore different techniques and understand the nuances of solving such equations effectively.

Importance of Solving Quadratic Equations

Why is it so important to understand how to solve quadratic equations? The answer lies in their widespread applications. Quadratic equations model parabolic trajectories in physics, describe curves in engineering designs, and are used in economic models to predict growth and decay. Mastering the methods to solve these equations opens doors to understanding and solving complex real-world problems. For instance, engineers might use quadratic equations to calculate the trajectory of a projectile, while economists might use them to model supply and demand curves. Therefore, the ability to manipulate and solve quadratic equations is a fundamental skill in many disciplines. This article aims to equip you with the necessary knowledge to tackle such problems confidently.

Problem Statement: $3(x+4)^2=39$

Let's focus on the specific quadratic equation we aim to solve: $3(x+4)^2=39$. This equation might look intimidating at first glance, but it can be solved systematically using algebraic techniques. The key to solving this equation is to isolate the variable $x$. This involves a series of steps, including simplifying the equation, taking the square root, and solving for $x$. Before diving into the solution, it's essential to understand the structure of the equation. We have a squared term, $(x+4)^2$, which indicates that we are dealing with a quadratic relationship. The coefficient $3$ and the constant $39$ further define the specific nature of this equation. By carefully manipulating these elements, we can find the values of $x$ that satisfy the equation. Understanding the underlying principles will not only help in solving this particular equation but also in tackling other quadratic equations in the future.

Step-by-Step Solution

Step 1: Simplify the Equation

The first step in solving the quadratic equation $3(x+4)^2=39$ is to simplify it. To do this, we divide both sides of the equation by 3. This gives us:

$(x+4)^2 = 13$

This simplification makes the equation easier to work with. By reducing the coefficients, we've cleared the path for the next steps in the solution. This is a crucial step in solving any algebraic equation – simplifying it as much as possible before proceeding. Simplifying not only makes the equation more manageable but also reduces the chances of making errors in subsequent steps. In this case, dividing by 3 isolates the squared term, which is essential for the next operation.

Step 2: Take the Square Root

Now that we have $(x+4)^2 = 13$, the next step is to take the square root of both sides of the equation. When taking the square root, it's important to remember that we must consider both the positive and negative roots. This gives us:

$x+4 = ±\sqrt{13}$

This step is critical because it unveils the two possible solutions for $x$. The square root operation undoes the squaring, but it introduces the possibility of both positive and negative values. This is because both $(\sqrt{13})^2$ and $(-\sqrt{13})^2$ equal 13. Failing to consider both roots is a common mistake in solving quadratic equations, and it can lead to an incomplete solution. Therefore, it's essential to always remember the $\pm$ sign when taking the square root in an equation.

Step 3: Solve for $x$

Finally, to solve for $x$, we need to isolate $x$ by subtracting 4 from both sides of the equation. This gives us two possible solutions:

$x = -4 + \sqrt{13}$ and $x = -4 - \sqrt{13}$

These are the two values of $x$ that satisfy the original equation. The solutions are expressed in terms of $\sqrt{13}$, which is an irrational number. This means that the solutions are not whole numbers or simple fractions, but they are still valid and precise. In some cases, you might need to approximate these values as decimals, but expressing them in terms of the square root is often the most accurate representation. The two solutions indicate that the quadratic equation has two points where the parabola intersects the x-axis, which is a characteristic feature of quadratic equations.

Presenting the Solutions

The solutions to the quadratic equation $3(x+4)^2=39$ are:

$x = -4 + \sqrt{13}$

and

$x = -4 - \sqrt{13}$

These values represent the exact solutions to the equation. If needed, we can approximate these values to decimal places. $\sqrt{13}$ is approximately 3.606. Thus,

$x ≈ -4 + 3.606 ≈ -0.394$

and

$x ≈ -4 - 3.606 ≈ -7.606$

These decimal approximations can be useful in real-world applications where an exact value might not be necessary. However, it's crucial to remember that these are approximations, and the exact solutions expressed in terms of $\sqrt{13}$ are more precise. Presenting both the exact solutions and the decimal approximations provides a comprehensive understanding of the solution set.

Verification of Solutions

To ensure that our solutions are correct, we can substitute them back into the original equation $3(x+4)^2=39$. Let's start with $x = -4 + \sqrt{13}$:

$3((-4 + \sqrt{13})+4)^2 = 3(\sqrt{13})^2 = 3(13) = 39$

This confirms that $x = -4 + \sqrt{13}$ is indeed a solution. Now, let's verify the second solution, $x = -4 - \sqrt{13}$:

$3((-4 - \sqrt{13})+4)^2 = 3(-\sqrt{13})^2 = 3(13) = 39$

This also confirms that $x = -4 - \sqrt{13}$ is a valid solution. The verification process is a critical step in solving any equation. By substituting the solutions back into the original equation, we can catch any errors made during the solving process. This not only ensures the accuracy of our solutions but also reinforces our understanding of the equation and the steps involved in solving it. In this case, the successful verification of both solutions gives us confidence in our final answer.

Alternative Methods for Solving Quadratic Equations

While we solved the equation $3(x+4)^2=39$ by simplifying and taking the square root, it's important to know that there are other methods to solve quadratic equations. Two common methods are factoring and using the quadratic formula. Let's briefly discuss these methods.

Factoring

Factoring involves rewriting the quadratic equation in the form $(ax + b)(cx + d) = 0$. This method is effective when the quadratic expression can be easily factored. However, not all quadratic equations can be factored easily, especially when the roots are irrational or complex. In such cases, other methods might be more appropriate. Factoring relies on finding two binomial expressions that, when multiplied, give the original quadratic expression. This method is particularly useful when the coefficients are integers, and the roots are rational numbers. However, for equations with irrational or complex roots, factoring can be challenging or even impossible.

Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations of the form $ax^2 + bx + c = 0$. The formula is:

$x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}$

This formula can be used to solve any quadratic equation, regardless of whether it can be factored or not. The quadratic formula is derived by completing the square on the general form of a quadratic equation. It provides a direct way to find the solutions, even when the roots are irrational or complex. The term $b^2 - 4ac$ under the square root is called the discriminant, and it provides information about the nature of the roots. If the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root (a repeated root); and if it is negative, there are two complex roots. The quadratic formula is a powerful tool in solving quadratic equations and is a fundamental concept in algebra.

Conclusion

In this article, we have thoroughly explored how to solve the quadratic equation $3(x+4)^2=39$. We started by simplifying the equation, then took the square root, and finally solved for $x$. We obtained two solutions: $x = -4 + \sqrt{13}$ and $x = -4 - \sqrt{13}$. We also verified these solutions by substituting them back into the original equation. Furthermore, we discussed alternative methods for solving quadratic equations, such as factoring and using the quadratic formula. Understanding these different methods equips you with a versatile toolkit for tackling various quadratic equations. The ability to solve quadratic equations is a fundamental skill in mathematics and has wide-ranging applications in various fields. By mastering these techniques, you can confidently approach and solve complex problems in mathematics and beyond. Remember to always verify your solutions and consider different methods to ensure accuracy and efficiency.