Solve For { B $}$ In The Equation:${ 14b^2 - 2 = -3b }$

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 quadratic equation of the form $14b^2 - 2 = -3b$. Our goal is to isolate the variable $b$ and find its value. We will use algebraic techniques to simplify the equation and solve for $b$.

Understanding the Equation

The given equation is $14b^2 - 2 = -3b$. To solve for $b$, we need to isolate the variable on one side of the equation. The first step is to move all the terms involving $b$ to one side of the equation and the constant terms to the other side.

Step 1: Rearrange the Equation

We can start by adding $3b$ to both sides of the equation to get:

$$14b^2 - 2 + 3b = -3b + 3b$$

This simplifies to:

$$14b^2 + 3b - 2 = 0$$

Step 2: Factor the Quadratic Expression

The next step is to factor the quadratic expression on the left-hand side of the equation. We can try to find two numbers whose product is $14 \times -2 = -28$ and whose sum is $3$. These numbers are $7$ and $-4$, so we can write the quadratic expression as:

$$(7b - 4)(2b + 1) = 0$$

Step 3: Solve for b

Now that we have factored the quadratic expression, we can set each factor equal to zero and solve for $b$. We have two possible solutions:

$$7b - 4 = 0 \quad \text{or} \quad 2b + 1 = 0$$

Solving the first equation for $b$, we get:

$$7b = 4 \quad \Rightarrow \quad b = \frac{4}{7}$$

Solving the second equation for $b$, we get:

$$2b = -1 \quad \Rightarrow \quad b = -\frac{1}{2}$$

Step 4: Check the Solutions

Before we can be confident that we have found the correct solutions, we need to check them by plugging them back into the original equation. Let's start with the first solution, $b = \frac{4}{7}$.

Substituting this value into the original equation, we get:

$$14\left(\frac{4}{7}\right)^2 - 2 = -3\left(\frac{4}{7}\right)$$

Simplifying this expression, we get:

$$14\left(\frac{16}{49}\right) - 2 = -\frac{12}{7}$$

This simplifies to:

$$\frac{224}{49} - \frac{98}{49} = -\frac{12}{7}$$

Which is equal to:

$$\frac{126}{49} = -\frac{12}{7}$$

This is not true, so $b = \frac{4}{7}$ is not a valid solution.

Now let's try the second solution, $b = -\frac{1}{2}$.

Substituting this value into the original equation, we get:

$$14\left(-\frac{1}{2}\right)^2 - 2 = -3\left(-\frac{1}{2}\right)$$

Simplifying this expression, we get:

$$14\left(\frac{1}{4}\right) - 2 = \frac{3}{2}$$

This simplifies to:

$$\frac{14}{4} - \frac{8}{4} = \frac{3}{2}$$

Which is equal to:

$$\frac{6}{4} = \frac{3}{2}$$

This is true, so $b = -\frac{1}{2}$ is a valid solution.

Conclusion

In this article, we solved a quadratic equation of the form $14b^2 - 2 = -3b$. We used algebraic techniques to simplify the equation and solve for $b$. We found that the only valid solution is $b = -\frac{1}{2}$. This solution satisfies the original equation and is therefore the correct value of $b$.

Final Answer

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Introduction

In our previous article, we solved a quadratic equation of the form $14b^2 - 2 = -3b$. We used algebraic techniques to simplify the equation and solve for $b$. In this article, we will answer some common questions related to solving quadratic equations.

Q: What is a quadratic equation?

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?

To solve a quadratic equation, you can use the following steps:

1. Rearrange the equation: Move all the terms involving the variable to one side of the equation and the constant terms to the other side.
2. Factor the quadratic expression: Try to factor the quadratic expression on the left-hand side of the equation.
3. Solve for the variable: Set each factor equal to zero and solve for the variable.
4. Check the solutions: Plug the solutions back into the original equation to check if they are valid.

Q: What are the different methods for solving quadratic equations?

There are several methods for solving quadratic equations, including:

1. Factoring: This method involves factoring the quadratic expression on the left-hand side of the equation.
2. Quadratic formula: This method involves using the quadratic formula to find the solutions.
3. Graphing: This method involves graphing the quadratic function and finding the x-intercepts.
4. Completing the square: This method involves completing the square to find the solutions.

Q: What is the quadratic formula?

The quadratic formula is a formula that can be used to find the solutions of a quadratic equation. It is given by:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula?

To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. You will then get two solutions, which are given by:

$$x = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \quad \text{and} \quad x = \frac{-b - \sqrt{b^2 - 4ac}}{2a}$$

Q: What are the advantages and disadvantages of using the quadratic formula?

The advantages of using the quadratic formula are:

• It can be used to find the solutions of a quadratic equation even if it cannot be factored.
• It is a general method that can be used for any quadratic equation.

The disadvantages of using the quadratic formula are:

• It can be complex and difficult to use.
• It may not be as efficient as other methods, such as factoring.

Q: Can I use the quadratic formula to solve a quadratic equation with complex solutions?

Yes, you can use the quadratic formula to solve a quadratic equation with complex solutions. The quadratic formula will give you two complex solutions, which are given by:

$$x = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \quad \text{and} \quad x = \frac{-b - \sqrt{b^2 - 4ac}}{2a}$$

Conclusion

In this article, we answered some common questions related to solving quadratic equations. We discussed the different methods for solving quadratic equations, including factoring, the quadratic formula, graphing, and completing the square. We also discussed the advantages and disadvantages of using the quadratic formula. We hope that this article has been helpful in answering your questions about solving quadratic equations.

Final Answer

The final answer is $\boxed{0}$.