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

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific quadratic equation, $3(2b+3)^2=36$, and explore the values of $b$ that satisfy this 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 term using the formula $(a+b)^2 = a^2 + 2ab + b^2$.

Expanding the Squared Term

Using the formula, we can expand the squared term as follows:

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

Simplifying the expression, we get:

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

Now, we can substitute this expression back into the original equation:

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

Simplifying the Equation

To simplify the equation, we can start by distributing the coefficient $3$ to the terms inside the parentheses:

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

Next, we can subtract $36$ from both sides of the equation to get:

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

Solving the Quadratic Equation

Now that we have a quadratic equation in the form $ax^2 + bx + c = 0$, we can use the quadratic formula to solve for $b$. The quadratic formula is given by:

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

In this case, $a = 12$, $b = 36$, and $c = -9$. Plugging these values into the quadratic formula, we get:

$$b = \frac{-36 \pm \sqrt{36^2 - 4(12)(-9)}}{2(12)}$$

Simplifying the expression under the square root, we get:

$$b = \frac{-36 \pm \sqrt{1296 + 432}}{24}$$

$$b = \frac{-36 \pm \sqrt{1728}}{24}$$

$$b = \frac{-36 \pm 41.569}{24}$$

Finding the Values of $b$

Now that we have the solutions to the quadratic equation, we can find the values of $b$ that satisfy the equation. We have two possible solutions:

$$b = \frac{-36 + 41.569}{24}$$

$$b = \frac{-36 - 41.569}{24}$$

Simplifying the expressions, we get:

$$b = \frac{5.569}{24}$$

$$b = \frac{-77.569}{24}$$

However, we can simplify these expressions further by factoring out the common factor of $-1$:

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

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

Conclusion

In this article, we solved the quadratic equation $3(2b+3)^2=36$ and found the values of $b$ that satisfy the equation. We used the quadratic formula to solve for $b$ and simplified the expressions to find the final solutions. The values of $b$ that satisfy the equation are $b=\frac{-3+2 \sqrt{3}}{2}$ and $b=\frac{-3-2 \sqrt{3}}{2}$.

Answer

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Introduction

In our previous article, we solved the quadratic equation $3(2b+3)^2=36$ and found the values of $b$ that satisfy the equation. In this article, we will provide a comprehensive Q&A guide to help you understand the concepts and techniques used to solve quadratic equations.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It is typically written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and completing the square. The method you choose will depend on the specific equation and the values of the coefficients.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by:

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

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the quadratic equation. Then, you can plug these values into the formula and simplify to find the solutions.

Q: What is the difference between the two solutions of a quadratic equation?

A: The two solutions of a quadratic equation are given by the quadratic formula. The difference between the two solutions is the value of the expression under the square root, which is $b^2 - 4ac$.

Q: Can I have a negative value under the square root?

A: Yes, it is possible to have a negative value under the square root. In this case, the solutions will be complex numbers, which are numbers that involve the imaginary unit $i$.

Q: How do I simplify complex solutions?

A: To simplify complex solutions, you need to multiply the numerator and denominator by the conjugate of the denominator. This will eliminate the imaginary unit and give you a simplified solution.

Q: Can I have a repeated root in a quadratic equation?

A: Yes, it is possible to have a repeated root in a quadratic equation. This occurs when the discriminant $b^2 - 4ac$ is equal to zero.

Q: How do I find the repeated root?

A: To find the repeated root, you can set the discriminant equal to zero and solve for the variable. This will give you the repeated root.

Q: What is the significance of the discriminant?

A: The discriminant is a value that determines the nature of the solutions to a quadratic equation. If the discriminant is positive, the solutions are real and distinct. If the discriminant is zero, the solutions are real and repeated. If the discriminant is negative, the solutions are complex.

Conclusion

In this article, we provided a comprehensive Q&A guide to help you understand the concepts and techniques used to solve quadratic equations. We covered topics such as the quadratic formula, complex solutions, and repeated roots. We hope this guide has been helpful in your understanding of quadratic equations.

Frequently Asked Questions

• Q: What is the quadratic formula?
• A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation.
• Q: How do I use the quadratic formula?
• A: To use the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the quadratic equation and plug these values into the formula.
• Q: What is the difference between the two solutions of a quadratic equation?
• A: The two solutions of a quadratic equation are given by the quadratic formula, and the difference between the two solutions is the value of the expression under the square root.
• Q: Can I have a negative value under the square root?
• A: Yes, it is possible to have a negative value under the square root, which will result in complex solutions.
• Q: How do I simplify complex solutions?
• A: To simplify complex solutions, you need to multiply the numerator and denominator by the conjugate of the denominator.
• Q: Can I have a repeated root in a quadratic equation?
• A: Yes, it is possible to have a repeated root in a quadratic equation, which occurs when the discriminant is equal to zero.
• Q: How do I find the repeated root?
• A: To find the repeated root, you can set the discriminant equal to zero and solve for the variable.

Additional Resources

• For more information on quadratic equations, please refer to our previous article on solving quadratic equations.
• For a comprehensive guide to algebra, please refer to our article on algebra basics.
• For a list of common algebra formulas, please refer to our article on algebra formulas.