Solving For B Values That Satisfy 3(2b+3)^2=36

Solving quadratic equations is a fundamental skill in algebra, and this problem presents an excellent opportunity to demonstrate that skill. The given equation, 3(2b+3)^2=36, involves a squared term, which means we're dealing with a quadratic equation in disguise. To find the values of b that satisfy the equation, we need to carefully follow the steps of simplification, expansion, and ultimately, solving for b. Let's embark on this algebraic journey step by step.

Our main objective in this exploration is to determine the precise values of b that render the equation 3(2b+3)^2=36 valid. This involves a series of algebraic manipulations, each crucial in isolating b and unveiling its possible values. We begin by simplifying the equation, aiming to reduce its complexity and make it more manageable. The first step is to divide both sides of the equation by 3. This simple division is a pivotal move, effectively reducing the coefficients and setting the stage for further simplification. Now our equation looks like this: (2b+3)^2=12. This is a much more approachable form, allowing us to move forward with expanding the squared term. This expansion is a critical step, as it transforms the equation from a compact form to a more detailed expression, revealing the underlying quadratic structure.

Expanding (2b+3)^2 is a key step in solving this equation. When we expand the left side of the equation, (2b+3)^2, we apply the FOIL method or the binomial square formula, resulting in (2b+3)(2b+3)=4b^2+12b+9. This expansion is a crucial transformation, converting the squared term into a standard quadratic expression. Now, our equation takes the form 4b^2+12b+9=12. This equation is now a clearer representation of a quadratic equation, with terms involving b^2, b, and constant terms. To proceed further, we need to bring all terms to one side, setting the equation to zero. This is a standard practice in solving quadratic equations, as it allows us to apply techniques like factoring or the quadratic formula. Subtracting 12 from both sides, we get 4b^2+12b+9-12=0, which simplifies to 4b^2+12b-3=0. This simplified quadratic equation is now ready for the next stage of our solving process. The stage is where we find the values of b that make this equation true.

With our equation now in the standard quadratic form, 4b^2+12b-3=0, we face the task of finding the values of b that satisfy it. There are several methods to solve quadratic equations, but given the coefficients, the quadratic formula appears to be the most straightforward approach in this case. The quadratic formula is a powerful tool that provides the solutions for any quadratic equation of the form ax^2+bx+c=0, and it is given by: x = [-b ± √(b^2 - 4ac)] / (2a). In our equation, 4b^2+12b-3=0, we can identify the coefficients as follows: a = 4, b = 12, and c = -3. These values are crucial for plugging into the quadratic formula. We substitute these values into the formula carefully, ensuring that we handle the signs and arithmetic operations correctly. The substitution is a critical step, and any error here can lead to incorrect solutions. Once we have substituted the values, we proceed with simplifying the expression, which involves evaluating the square root and performing the arithmetic operations. This process requires careful attention to detail, ensuring that we follow the order of operations and simplify the expression correctly. The final result will give us the two possible values of b that satisfy the original equation. Let's substitute and simplify to find these values.

Substituting the values of a, b, and c into the quadratic formula, we get: b = [-12 ± √(12^2 - 4 * 4 * (-3))] / (2 * 4). This substitution is a crucial step, as it sets up the equation for solving. Now, we simplify the expression step by step. First, we evaluate the expression under the square root: 12^2 - 4 * 4 * (-3) = 144 + 48 = 192. So, the equation becomes: b = [-12 ± √192] / 8. Next, we simplify the square root. We can express 192 as a product of its prime factors: 192 = 2^6 * 3. Therefore, √192 = √(2^6 * 3) = 2^3√3 = 8√3. Substituting this back into the equation, we have: b = [-12 ± 8√3] / 8. Finally, we simplify the entire expression by dividing both the numerator and the denominator by their greatest common divisor, which is 4. This gives us: b = [-3 ± 2√3] / 2. This final simplification reveals the two distinct values of b that satisfy the equation. These values represent the solutions to the quadratic equation, and they are the values we were seeking to find. Therefore, we have successfully navigated through the algebraic manipulations and arrived at the solutions.

We have arrived at two potential solutions for b: b = (-3 + 2√3) / 2 and b = (-3 - 2√3) / 2. It is crucial to verify these solutions by substituting them back into the original equation, 3(2b+3)^2=36, to ensure that they indeed satisfy the equation. This verification process is a fundamental step in solving any equation, as it confirms the accuracy of our calculations and algebraic manipulations. Substituting each value of b back into the original equation allows us to check if the equation holds true. If the left-hand side of the equation equals the right-hand side after the substitution, then the solution is valid. This verification step is particularly important when dealing with quadratic equations, as they can sometimes have extraneous solutions that do not satisfy the original equation. By verifying our solutions, we can confidently confirm that we have found the correct values of b. This process involves careful attention to detail and accurate arithmetic calculations. Let's proceed with the verification to ensure the correctness of our solutions.

Let's substitute b = (-3 + 2√3) / 2 into the original equation: 3[2((-3 + 2√3) / 2) + 3]^2 = 36. Simplifying the expression inside the brackets, we get: 2((-3 + 2√3) / 2) + 3 = -3 + 2√3 + 3 = 2√3. Now, substituting this back into the equation, we have: 3(2√3)^2 = 3(4 * 3) = 3(12) = 36. This confirms that b = (-3 + 2√3) / 2 is a valid solution. Next, let's substitute b = (-3 - 2√3) / 2 into the original equation: 3[2((-3 - 2√3) / 2) + 3]^2 = 36. Simplifying the expression inside the brackets, we get: 2((-3 - 2√3) / 2) + 3 = -3 - 2√3 + 3 = -2√3. Now, substituting this back into the equation, we have: 3(-2√3)^2 = 3(4 * 3) = 3(12) = 36. This also confirms that b = (-3 - 2√3) / 2 is a valid solution. Therefore, both values of b satisfy the original equation, and we have successfully verified our solutions. This verification step adds a layer of confidence to our results and ensures that we have accurately solved the problem.

In conclusion, the values of b that satisfy the equation 3(2b+3)^2=36 are b = (-3 + 2√3) / 2 and b = (-3 - 2√3) / 2. We arrived at these solutions by meticulously simplifying the equation, expanding the squared term, applying the quadratic formula, and verifying our results. This problem showcases the importance of mastering algebraic techniques, particularly those related to solving quadratic equations. The quadratic formula, in particular, is a powerful tool that enables us to find solutions for a wide range of quadratic equations. Furthermore, the process of verifying solutions is a crucial step in ensuring the accuracy of our work and avoiding extraneous solutions. By following a systematic approach and paying careful attention to detail, we can confidently tackle such algebraic challenges and arrive at the correct answers. This exploration not only provides the specific solutions to the given equation but also reinforces the broader skills and concepts necessary for success in algebra and mathematics in general. Therefore, understanding and practicing these techniques are essential for anyone seeking to excel in mathematics and related fields.

The final answer is A. b=(-3+2√3)/2 and (-3-2√3)/2