Solve For B B B . Express Your Answer As An Integer, Integers, Or In Simplest Radical Form. − 72 = − 10 B 2 + 8 -72 = -10b^2 + 8 − 72 = − 10 B 2 + 8 B = □ B = \square B = □

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, $-72 = -10b^2 + 8$, to find the value of $b$. We will break down the solution into manageable steps, using algebraic manipulations and mathematical concepts to arrive at the final answer.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Quadratic equations can be solved using various methods, including factoring, completing the square, and the quadratic formula.

Solving the Quadratic Equation

To solve the quadratic equation $-72 = -10b^2 + 8$, we need to isolate the variable $b$. The first step is to subtract 8 from both sides of the equation, which gives us:

$$-72 - 8 = -10b^2$$

Simplifying the left-hand side, we get:

$$-80 = -10b^2$$

Next, we divide both sides of the equation by -10, which gives us:

$$\frac{-80}{-10} = b^2$$

Simplifying the left-hand side, we get:

$$8 = b^2$$

Finding the Value of $b$

Now that we have the equation $8 = b^2$, we need to find the value of $b$. Since the square of a number is always positive, we can take the square root of both sides of the equation to find the value of $b$. However, we need to consider both the positive and negative square roots.

Taking the square root of both sides, we get:

$$\sqrt{8} = \sqrt{b^2}$$

Simplifying the left-hand side, we get:

$$2\sqrt{2} = b$$

However, we also need to consider the negative square root, which gives us:

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

Conclusion

In this article, we solved the quadratic equation $-72 = -10b^2 + 8$ to find the value of $b$. We broke down the solution into manageable steps, using algebraic manipulations and mathematical concepts to arrive at the final answer. We found that the value of $b$ is either $2\sqrt{2}$ or $-2\sqrt{2}$. This demonstrates the importance of considering both the positive and negative square roots when solving quadratic equations.

Final Answer

Introduction

In our previous article, we solved the quadratic equation $-72 = -10b^2 + 8$ to find the value of $b$. We broke down the solution into manageable steps, using algebraic manipulations and mathematical concepts to arrive at the final answer. In this article, we will answer some frequently asked questions related to solving quadratic equations, including the one we solved earlier.

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. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, completing the square, and the quadratic formula. The method you choose depends 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 gives 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 to solve a quadratic equation?

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

Q: What is the difference between the positive and negative square roots?

A: When solving a quadratic equation, you need to consider both the positive and negative square roots. This is because the square of a number is always positive, but the square root of a number can be either positive or negative.

Q: Why do I need to consider both the positive and negative square roots?

A: You need to consider both the positive and negative square roots because the solutions to a quadratic equation can be either positive or negative. If you only consider the positive square root, you may miss one of the solutions.

Q: Can I use a calculator to solve a quadratic equation?

A: Yes, you can use a calculator to solve a quadratic equation. Many calculators have a built-in quadratic formula function that you can use to find the solutions.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

• Not considering both the positive and negative square roots
• Not simplifying the expression correctly
• Not checking the solutions for extraneous solutions

Conclusion

In this article, we answered some frequently asked questions related to solving quadratic equations, including the one we solved earlier. We covered topics such as the quadratic formula, the difference between the positive and negative square roots, and common mistakes to avoid. By following these tips and techniques, you can become more confident and proficient in solving quadratic equations.

Final Answer

The final answer is $\boxed{2\sqrt{2}}$ or $\boxed{-2\sqrt{2}}$.