Solving (4x + 3)^2 = 18 A Step By Step Guide

Understanding the Equation

Before diving into the solution, it's crucial to understand the structure of the equation (4x + 3)^2 = 18. This equation is a quadratic equation because it involves a squared term. However, it is presented in a slightly different form than the standard quadratic equation form (ax^2 + bx + c = 0). The left-hand side of the equation is a binomial squared, which means we are squaring the expression (4x + 3). The right-hand side is a constant, 18. The goal is to find the values of x that, when substituted into the equation, make the left-hand side equal to the right-hand side. To solve this, we need to undo the operations applied to x. The first step is to take the square root of both sides of the equation. This will eliminate the square on the left-hand side and give us two possible equations to solve. We will then isolate x in each equation to find the solutions. The solutions will be the values of x that satisfy the original equation. Understanding the structure of the equation and the operations involved is crucial for solving it correctly.

Step 1: Taking the Square Root of Both Sides

The first step in solving the equation (4x + 3)^2 = 18 is to take the square root of both sides. This operation is crucial because it undoes the square on the left-hand side, allowing us to work with a simpler equation. When taking the square root of both sides, it is essential to remember that we must consider both the positive and negative square roots. This is because both the positive and negative square roots, when squared, will result in the same positive number. For example, both 3 and -3, when squared, equal 9. Applying this to our equation, taking the square root of both sides gives us: √(4x + 3)^2 = ±√18. This simplifies to 4x + 3 = ±√18. Now we have two separate equations to solve: 4x + 3 = √18 and 4x + 3 = -√18. These two equations represent the two possible solutions for x. Before proceeding, it's often helpful to simplify the square root term if possible. In this case, √18 can be simplified to 3√2, as 18 = 9 * 2, and √9 = 3. This simplification makes the subsequent steps easier to manage. By taking the square root of both sides and considering both positive and negative roots, we have set the stage for solving for x. This step is a fundamental technique in solving quadratic equations and similar algebraic problems.

Step 2: Simplifying the Square Root

As mentioned in the previous step, simplifying the square root is a crucial step in solving the equation (4x + 3)^2 = 18. We arrived at the equation 4x + 3 = ±√18 after taking the square root of both sides. The term √18 can be simplified because 18 has a perfect square factor. We can express 18 as the product of 9 and 2 (18 = 9 * 2), where 9 is a perfect square (3^2 = 9). Using the property of square roots that √(a * b) = √a * √b, we can rewrite √18 as √(9 * 2) = √9 * √2. Since √9 = 3, we have √18 = 3√2. This simplification makes the equation easier to work with. Substituting this back into our equation, we now have 4x + 3 = ±3√2. This means we have two equations: 4x + 3 = 3√2 and 4x + 3 = -3√2. Simplifying the square root not only makes the numbers more manageable but also allows us to express the solutions in their simplest form. This step is particularly important in mathematics as it ensures that the final answer is presented in the most concise and understandable way. By simplifying the square root, we are one step closer to isolating x and finding the solutions to the equation.

Step 3: Isolating x

After simplifying the square root, the next step is to isolate x in the equations 4x + 3 = 3√2 and 4x + 3 = -3√2. Isolating x means getting x by itself on one side of the equation. To do this, we need to undo the operations that are being performed on x. In both equations, x is being multiplied by 4 and then 3 is being added. To isolate x, we will first subtract 3 from both sides of each equation. For the first equation, 4x + 3 = 3√2, subtracting 3 from both sides gives us 4x = 3√2 - 3. For the second equation, 4x + 3 = -3√2, subtracting 3 from both sides gives us 4x = -3√2 - 3. Now, in both equations, x is being multiplied by 4. To undo this multiplication, we will divide both sides of each equation by 4. For the first equation, 4x = 3√2 - 3, dividing both sides by 4 gives us x = (3√2 - 3) / 4. For the second equation, 4x = -3√2 - 3, dividing both sides by 4 gives us x = (-3√2 - 3) / 4. These two values of x are the solutions to the original equation. By systematically undoing the operations performed on x, we have successfully isolated x and found the values that satisfy the equation. This process of isolating the variable is a fundamental skill in algebra and is used extensively in solving various types of equations.

Step 4: Presenting the Solutions

Having isolated x in the previous step, we have found the two solutions to the equation (4x + 3)^2 = 18. These solutions are x = (3√2 - 3) / 4 and x = (-3√2 - 3) / 4. It is important to present these solutions clearly and in a standard format. We can write the solutions as a set, or we can list them individually. In this case, we can write the solutions as: x = (3√2 - 3) / 4 and x = (-3√2 - 3) / 4. These solutions represent the two values of x that, when substituted into the original equation, will make the equation true. It is good practice to check the solutions by substituting them back into the original equation to ensure they are correct. However, in this case, we have followed the steps carefully, so we can be confident in our solutions. The solutions can also be written in a slightly different form by factoring out a -1 from the numerator of the second solution, which gives us x = (-3√2 - 3) / 4 = -(3√2 + 3) / 4. However, both forms are equally correct. Presenting the solutions clearly and accurately is the final step in solving the equation. It is important to show the solutions in their simplified form and to ensure that they are easily understandable.

Conclusion

In conclusion, solving the equation (4x + 3)^2 = 18 involves a series of algebraic steps that, when followed carefully, lead to the solutions. The process begins with understanding the structure of the equation and recognizing it as a quadratic equation in disguise. The first key step is to take the square root of both sides, remembering to consider both positive and negative roots. This leads to two separate equations. The next crucial step is to simplify the square root term, if possible, to make the subsequent calculations easier. In this case, √18 simplifies to 3√2. Then, the goal is to isolate x in each equation by undoing the operations performed on it. This involves subtracting 3 from both sides and then dividing by 4. Finally, the solutions are presented clearly and accurately, showing the two values of x that satisfy the original equation: x = (3√2 - 3) / 4 and x = (-3√2 - 3) / 4. By understanding each step and the underlying principles, you can confidently solve similar equations. This step-by-step approach not only provides the solutions but also enhances your understanding of algebraic problem-solving techniques. Remember to always check your work and present your solutions in a clear and concise manner.

Therefore, the solution to the equation (4x + 3)^2 = 18 is C. x = (-3 + 3√2) / 4 and x = (-3 - 3√2) / 4.