Solving The Equation 3(x - 2)^2 = 147 Find The Values Of X
In this article, we will delve into solving the quadratic equation 3(x - 2)² = 147. Quadratic equations, characterized by the highest power of the variable being 2, are fundamental in mathematics and have widespread applications in various fields, including physics, engineering, and economics. Mastering the techniques to solve them is crucial for anyone pursuing studies or careers in these areas. This guide will provide a step-by-step approach to finding the value(s) of x that satisfy the given equation, ensuring clarity and understanding at every stage.
Understanding Quadratic Equations
Before we dive into the solution, let's briefly discuss quadratic equations. A quadratic equation is generally represented in the form ax² + bx + c = 0, where a, b, and c are constants, and x is the variable we aim to solve for. The solutions to a quadratic equation are also known as the roots or zeros of the equation. These roots represent the x-values where the parabola represented by the quadratic equation intersects the x-axis.
There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. The best method to use often depends on the specific form of the equation. In the case of 3(x - 2)² = 147, we will use a combination of algebraic manipulation and the square root property to arrive at the solution. This approach is particularly effective when the equation is given in a form that is close to a perfect square.
Step-by-Step Solution
Step 1: Isolate the Squared Term
The first step in solving the equation 3(x - 2)² = 147 is to isolate the squared term. This means we want to get the term (x - 2)² by itself on one side of the equation. To do this, we will divide both sides of the equation by 3:
3(x - 2)² / 3 = 147 / 3
This simplifies to:
(x - 2)² = 49
Now we have the squared term isolated, which allows us to proceed to the next step.
Step 2: Apply the Square Root Property
Now that we have (x - 2)² = 49, we can apply the square root property. The square root property states that if x² = a, then x = ±√a. Applying this property to our equation, we take the square root of both sides:
√(x - 2)² = ±√49
This simplifies to:
x - 2 = ±7
It is crucial to remember the ± sign here because both positive and negative 7, when squared, result in 49. This means we have two possible equations to solve:
- x - 2 = 7
- x - 2 = -7
Step 3: Solve for x
We now have two simple linear equations to solve. Let's solve each one separately.
Equation 1: x - 2 = 7
To solve for x, we add 2 to both sides of the equation:
x - 2 + 2 = 7 + 2
This gives us:
x = 9
Equation 2: x - 2 = -7
Similarly, to solve for x, we add 2 to both sides of the equation:
x - 2 + 2 = -7 + 2
This gives us:
x = -5
Step 4: Verify the Solutions
It's always a good practice to verify the solutions we've obtained to ensure they are correct. We do this by substituting each value of x back into the original equation and checking if the equation holds true.
Verification for x = 9
Substitute x = 9 into the original equation 3(x - 2)² = 147:
3(9 - 2)² = 147
3(7)² = 147
3(49) = 147
147 = 147
Since the equation holds true, x = 9 is a valid solution.
Verification for x = -5
Substitute x = -5 into the original equation 3(x - 2)² = 147:
3(-5 - 2)² = 147
3(-7)² = 147
3(49) = 147
147 = 147
Since the equation holds true, x = -5 is also a valid solution.
The Solutions
Therefore, the values of x that satisfy the equation 3(x - 2)² = 147 are x = 9 and x = -5. These are the roots of the quadratic equation.
Alternative Methods for Solving Quadratic Equations
While we've solved the equation using algebraic manipulation and the square root property, it's worth noting other methods available for solving quadratic equations. These methods are particularly useful when the equation is in a different form or when the square root property is not directly applicable.
1. Factoring
Factoring involves expressing the quadratic equation as a product of two binomials. This method is effective when the quadratic expression can be easily factored. However, not all quadratic equations can be factored using integers, so this method has limitations.
To use factoring, we would first need to expand the original equation 3(x - 2)² = 147 and rearrange it into the standard form ax² + bx + c = 0.
3(x² - 4x + 4) = 147
3x² - 12x + 12 = 147
3x² - 12x - 135 = 0
Now, we could try to factor this quadratic expression. However, in this case, factoring might not be the most straightforward method.
2. Completing the Square
Completing the square involves transforming the quadratic equation into a perfect square trinomial. This method is more versatile than factoring and can be used to solve any quadratic equation. However, it can be more complex and time-consuming than other methods, especially when the coefficients are not simple integers.
To use completing the square, we would start with the equation 3x² - 12x - 135 = 0 and follow a series of steps to rewrite it in the form (x - h)² = k, where h and k are constants. Then, we can apply the square root property as we did in our initial solution.
3. The Quadratic Formula
The quadratic formula is a universal method for solving quadratic equations. It can be used to find the roots of any quadratic equation, regardless of whether it can be factored or easily manipulated. The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.
To use the quadratic formula, we would identify the coefficients a, b, and c from the equation 3x² - 12x - 135 = 0, which are a = 3, b = -12, and c = -135. Then, we would substitute these values into the formula and simplify to find the values of x.
Choosing the Best Method
The choice of method for solving a quadratic equation often depends on the specific equation and personal preference. For equations in the form (x - h)² = k, using the square root property is generally the most efficient method. For equations that can be easily factored, factoring may be the quickest approach. Completing the square and the quadratic formula are more versatile and can be used to solve any quadratic equation, but they may involve more complex calculations.
In our case, the original equation 3(x - 2)² = 147 was ideally suited for solving using algebraic manipulation and the square root property. This allowed us to find the solutions x = 9 and x = -5 in a straightforward manner.
Real-World Applications of Quadratic Equations
Quadratic equations are not just abstract mathematical concepts; they have numerous real-world applications. They are used in physics to describe projectile motion, in engineering to design bridges and structures, and in economics to model supply and demand curves. Understanding how to solve quadratic equations is essential for solving problems in these and many other fields.
For example, consider a ball thrown into the air. The height of the ball at any given time can be modeled by a quadratic equation. By solving the equation, we can determine the maximum height the ball reaches and the time it takes for it to hit the ground. Similarly, in engineering, quadratic equations are used to calculate the optimal shape of a bridge arch to ensure its stability and load-bearing capacity.
Conclusion
In this comprehensive guide, we have successfully solved the quadratic equation 3(x - 2)² = 147 using algebraic manipulation and the square root property. We found the values of x to be 9 and -5. We also discussed alternative methods for solving quadratic equations, including factoring, completing the square, and the quadratic formula. Understanding these methods and their applications is crucial for anyone studying mathematics or pursuing careers in related fields. By mastering the techniques for solving quadratic equations, you will be well-equipped to tackle a wide range of problems in both academic and real-world settings.
Remember, the key to success in mathematics is practice. Work through various examples and problems to solidify your understanding of quadratic equations and their solutions. With consistent effort, you will become proficient in solving these important equations and applying them to solve real-world problems.
