What Are The Roots Of The Equation? X 2 ( 6 X − 11 ) ( 7 X − 8 ) = 0 X^2(6x - 11)(7x - 8) = 0 X 2 ( 6 X − 11 ) ( 7 X − 8 ) = 0 Enter Your Answers In The Boxes.$x = $ $\square$ Or $x = $ $\square$ Or $x = $ $\square$

Introduction to Solving Equations

Solving equations is a fundamental concept in mathematics, and it is essential to understand the different methods and techniques used to find the roots of an equation. In this article, we will focus on solving a quadratic equation that involves multiple factors. The given equation is $x^2(6x - 11)(7x - 8) = 0$. Our goal is to find the values of $x$ that satisfy this equation.

Understanding the Concept of Roots

The roots of an equation are the values of the variable that make the equation true. In other words, they are the solutions to the equation. When we solve an equation, we are essentially finding the values of the variable that make the equation equal to zero. In this case, we need to find the values of $x$ that make the equation $x^2(6x - 11)(7x - 8) = 0$ true.

Breaking Down the Equation

To solve the equation, we need to break it down into its individual factors. The equation can be written as $x^2(6x - 11)(7x - 8) = 0$. This means that either $x^2 = 0$, $6x - 11 = 0$, or $7x - 8 = 0$. We will now focus on solving each of these factors individually.

Solving the First Factor: $x^2 = 0$

The first factor is $x^2 = 0$. To solve this equation, we need to find the value of $x$ that makes $x^2$ equal to zero. Since $x^2$ is a squared term, it can only be equal to zero if $x$ is equal to zero. Therefore, the solution to this factor is $x = 0$.

Solving the Second Factor: $6x - 11 = 0$

The second factor is $6x - 11 = 0$. To solve this equation, we need to isolate the variable $x$. We can do this by adding 11 to both sides of the equation, which gives us $6x = 11$. Then, we can divide both sides of the equation by 6, which gives us $x = \frac{11}{6}$.

Solving the Third Factor: $7x - 8 = 0$

The third factor is $7x - 8 = 0$. To solve this equation, we need to isolate the variable $x$. We can do this by adding 8 to both sides of the equation, which gives us $7x = 8$. Then, we can divide both sides of the equation by 7, which gives us $x = \frac{8}{7}$.

Conclusion

In conclusion, the roots of the equation $x^2(6x - 11)(7x - 8) = 0$ are $x = 0$, $x = \frac{11}{6}$, and $x = \frac{8}{7}$. These values of $x$ make the equation true, and they are the solutions to the equation.

Final Answer

The final answer is:

$x = $ 0 or $x = $ $\frac{11}{6}$ or $x = $ $\frac{8}{7}$

Discussion

The equation $x^2(6x - 11)(7x - 8) = 0$ is a quadratic equation that involves multiple factors. To solve this equation, we need to break it down into its individual factors and solve each of them individually. The roots of the equation are the values of $x$ that make the equation true, and they are the solutions to the equation. In this case, the roots of the equation are $x = 0$, $x = \frac{11}{6}$, and $x = \frac{8}{7}$.

Introduction to FAQs

In the previous article, we discussed how to solve the equation $x^2(6x - 11)(7x - 8) = 0$. We broke down the equation into its individual factors and solved each of them individually. In this article, we will answer some frequently asked questions (FAQs) about solving equations.

Q: What is the difference between a root and a solution?

A: A root and a solution are often used interchangeably, but technically, a root is a value that makes the equation true, while a solution is the process of finding that value.

Q: How do I know if an equation has multiple roots?

A: If an equation has multiple factors, it is likely to have multiple roots. For example, the equation $x^2(6x - 11)(7x - 8) = 0$ has three factors, so it has three roots.

Q: Can I use the same method to solve all types of equations?

A: No, different types of equations require different methods to solve them. For example, linear equations can be solved using basic algebra, while quadratic equations require more advanced techniques.

Q: What is the significance of the order of operations in solving equations?

A: The order of operations is crucial in solving equations. It ensures that the correct operations are performed in the correct order, which can affect the accuracy of the solution.

Q: Can I use a calculator to solve equations?

A: Yes, calculators can be a useful tool in solving equations, but they should be used with caution. It's essential to understand the underlying math and not rely solely on the calculator.

Q: How do I know if an equation has no solution?

A: If an equation has no factors or if the factors do not intersect, it is likely to have no solution. For example, the equation $x^2 + 1 = 0$ has no real solution because there is no real number that can satisfy the equation.

Q: Can I use the same method to solve equations with different variables?

A: No, different variables require different methods to solve them. For example, equations with multiple variables require more advanced techniques, such as substitution or elimination.

Q: What is the importance of checking the solution?

A: Checking the solution is crucial to ensure that the solution is accurate. It's essential to plug the solution back into the original equation to verify that it satisfies the equation.

Q: Can I use the same method to solve equations with different coefficients?

A: No, different coefficients require different methods to solve them. For example, equations with large coefficients require more advanced techniques, such as approximation or numerical methods.

Conclusion

In conclusion, solving equations is a complex process that requires a deep understanding of the underlying math. By understanding the different methods and techniques used to solve equations, you can become proficient in solving a wide range of equations.

Final Answer

The final answer is:

• Q: What is the difference between a root and a solution? A: A root and a solution are often used interchangeably, but technically, a root is a value that makes the equation true, while a solution is the process of finding that value.
• Q: How do I know if an equation has multiple roots? A: If an equation has multiple factors, it is likely to have multiple roots.
• Q: Can I use the same method to solve all types of equations? A: No, different types of equations require different methods to solve them.
• Q: What is the significance of the order of operations in solving equations? A: The order of operations is crucial in solving equations.
• Q: Can I use a calculator to solve equations? A: Yes, calculators can be a useful tool in solving equations, but they should be used with caution.
• Q: How do I know if an equation has no solution? A: If an equation has no factors or if the factors do not intersect, it is likely to have no solution.
• Q: Can I use the same method to solve equations with different variables? A: No, different variables require different methods to solve them.
• Q: What is the importance of checking the solution? A: Checking the solution is crucial to ensure that the solution is accurate.
• Q: Can I use the same method to solve equations with different coefficients? A: No, different coefficients require different methods to solve them.