What Values Of B B B Satisfy 3 ( 2 B + 3 ) 2 = 36 3(2b+3)^2 = 36 3 ( 2 B + 3 ) 2 = 36 ?A. B = − 3 + 2 3 2 B = \frac{-3 + 2\sqrt{3}}{2} B = 2 − 3 + 2 3 ​ ​ And B = − 3 − 2 3 2 B = \frac{-3 - 2\sqrt{3}}{2} B = 2 − 3 − 2 3 ​ ​ B. B = − 3 + 2 3 3 B = \frac{-3 + 2\sqrt{3}}{3} B = 3 − 3 + 2 3 ​ ​ And B = − 3 − 2 3 3 B = \frac{-3 - 2\sqrt{3}}{3} B = 3 − 3 − 2 3 ​ ​ C. $b =

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Introduction

In this article, we will explore the values of $b$ that satisfy the equation $3(2b+3)^2 = 36$. This equation involves a quadratic expression and requires us to find the values of $b$ that make the equation true. We will use algebraic techniques to solve for $b$ and find the possible values that satisfy the equation.

Understanding the Equation

The given equation is $3(2b+3)^2 = 36$. To solve for $b$, we need to isolate the variable $b$ on one side of the equation. The first step is to expand the squared expression $(2b+3)^2$ using the formula $(a+b)^2 = a^2 + 2ab + b^2$. In this case, $a = 2b$ and $b = 3$, so we have:

$(2b+3)^2 = (2b)^2 + 2(2b)(3) + 3^2$

Expanding the expression, we get:

$(2b+3)^2 = 4b^2 + 12b + 9$

Now, we can rewrite the original equation as:

$3(4b^2 + 12b + 9) = 36$

Expanding and Simplifying

To simplify the equation, we can expand the left-hand side by multiplying $3$ with each term inside the parentheses:

$12b^2 + 36b + 27 = 36$

Subtracting 36 from Both Sides

To isolate the quadratic expression, we can subtract $36$ from both sides of the equation:

$12b^2 + 36b - 9 = 0$

Dividing by 3

To simplify the equation further, we can divide both sides by $3$:

$4b^2 + 12b - 3 = 0$

Using the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form $ax^2 + bx + c = 0$. The formula is given by:

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

In this case, $a = 4$, $b = 12$, and $c = -3$. Plugging these values into the formula, we get:

$b = \frac{-12 \pm \sqrt{12^2 - 4(4)(-3)}}{2(4)}$

Simplifying the Expression

To simplify the expression under the square root, we can calculate the values:

$12^2 = 144$

$-4(4)(-3) = 48$

Now, we can rewrite the expression as:

$b = \frac{-12 \pm \sqrt{144 + 48}}{8}$

Simplifying the Square Root

To simplify the square root, we can add the values inside the square root:

$144 + 48 = 192$

Now, we can rewrite the expression as:

$b = \frac{-12 \pm \sqrt{192}}{8}$

Simplifying the Square Root (continued)

To simplify the square root, we can factor out the largest perfect square from the value inside the square root:

$192 = 16 \times 12$

Now, we can rewrite the expression as:

$b = \frac{-12 \pm \sqrt{16 \times 12}}{8}$

Simplifying the Square Root (continued)

To simplify the square root, we can take the square root of the perfect square:

$\sqrt{16} = 4$

Now, we can rewrite the expression as:

$b = \frac{-12 \pm 4\sqrt{12}}{8}$

Simplifying the Square Root (continued)

To simplify the square root, we can simplify the value inside the square root:

$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$

Now, we can rewrite the expression as:

$b = \frac{-12 \pm 4(2\sqrt{3})}{8}$

Simplifying the Expression

To simplify the expression, we can multiply the values inside the parentheses:

$b = \frac{-12 \pm 8\sqrt{3}}{8}$

Simplifying the Expression (continued)

To simplify the expression, we can divide both the numerator and the denominator by $4$:

$b = \frac{-3 \pm 2\sqrt{3}}{2}$

Conclusion

In this article, we have explored the values of $b$ that satisfy the equation $3(2b+3)^2 = 36$. We have used algebraic techniques to solve for $b$ and find the possible values that satisfy the equation. The values of $b$ that satisfy the equation are:

$b = \frac{-3 + 2\sqrt{3}}{2}$ and $b = \frac{-3 - 2\sqrt{3}}{2}$

These values are the solutions to the equation and represent the possible values of $b$ that make the equation true.

Final Answer

The final answer is:

A. $b = \frac{-3 + 2\sqrt{3}}{2}$ and $b = \frac{-3 - 2\sqrt{3}}{2}$

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Introduction

In our previous article, we explored the values of $b$ that satisfy the equation $3(2b+3)^2 = 36$. We used algebraic techniques to solve for $b$ and find the possible values that satisfy the equation. In this article, we will answer some frequently asked questions related to the equation and its solutions.

Q: What is the equation $3(2b+3)^2 = 36$ trying to solve?

A: The equation $3(2b+3)^2 = 36$ is trying to find the values of $b$ that make the equation true. In other words, it is trying to solve for $b$.

Q: What is the quadratic formula, and how is it used to solve the equation?

A: The quadratic formula is a powerful tool for solving quadratic equations of the form $ax^2 + bx + c = 0$. It is given by:

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

In this case, we used the quadratic formula to solve for $b$.

Q: What are the possible values of $b$ that satisfy the equation?

A: The possible values of $b$ that satisfy the equation are:

$b = \frac{-3 + 2\sqrt{3}}{2}$ and $b = \frac{-3 - 2\sqrt{3}}{2}$

Q: How did we simplify the expression under the square root?

A: We simplified the expression under the square root by factoring out the largest perfect square from the value inside the square root. We then took the square root of the perfect square to simplify the expression.

Q: What is the significance of the square root of 12?

A: The square root of 12 is equal to $2\sqrt{3}$. This is an important simplification that helps us solve for $b$.

Q: Can you explain the concept of the quadratic formula in more detail?

A: The quadratic formula is a powerful tool for solving quadratic equations of the form $ax^2 + bx + c = 0$. It is given by:

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

The formula works by first calculating the value of the expression inside the square root, which is called the discriminant. The discriminant is given by:

$b^2 - 4ac$

If the discriminant is positive, then the equation has two distinct real solutions. If the discriminant is zero, then the equation has one real solution. If the discriminant is negative, then the equation has no real solutions.

Q: How do we know that the values of $b$ we found are the only possible solutions?

A: We know that the values of $b$ we found are the only possible solutions because we used the quadratic formula to solve for $b$. The quadratic formula is a powerful tool for solving quadratic equations, and it guarantees that we will find all possible solutions.

Q: Can you provide more examples of how to use the quadratic formula to solve quadratic equations?

A: Yes, here are a few more examples:

• $x^2 + 5x + 6 = 0$
• $x^2 - 7x + 12 = 0$
• $x^2 + 2x - 15 = 0$

In each of these examples, we can use the quadratic formula to solve for $x$.

Conclusion

In this article, we have answered some frequently asked questions related to the equation $3(2b+3)^2 = 36$ and its solutions. We have used algebraic techniques to solve for $b$ and find the possible values that satisfy the equation. We have also explained the concept of the quadratic formula and provided more examples of how to use it to solve quadratic equations.

Final Answer

The final answer is:

A. $b = \frac{-3 + 2\sqrt{3}}{2}$ and $b = \frac{-3 - 2\sqrt{3}}{2}$