Mathematical Equations and Verification

Oct 13, 2025 by windailys.com 81 views

Which Statements are True? Select Each Correct Answer

Mathematical Equations and Verification

In mathematics, verifying the truth of a statement often involves checking if the given equation is correct. This can be done by simplifying the equation and comparing it with the original statement. In this article, we will examine four mathematical statements and determine which ones are true.

A. $100 x^3 + 5 = 5 x^3(20 x + 1)$

To verify the truth of statement A, we need to simplify the equation on the right-hand side and compare it with the left-hand side.

5 x^3(20 x + 1) = 100 x^3 + 5 x^3

Expanding the right-hand side, we get:

100 x^3 + 5 x^3 = 105 x^3

Comparing this with the left-hand side, we can see that the equation is not true. Therefore, statement A is false.

B. $4 x^2 + 10 x = 2 x(2 x + 5)$

To verify the truth of statement B, we need to simplify the equation on the right-hand side and compare it with the left-hand side.

2 x(2 x + 5) = 4 x^2 + 10 x

Expanding the right-hand side, we get:

4 x^2 + 10 x = 4 x^2 + 10 x

Comparing this with the left-hand side, we can see that the equation is true. Therefore, statement B is true.

C. $8 x^3 - 6 x = 2 x^3(4 - 3 x)$

To verify the truth of statement C, we need to simplify the equation on the right-hand side and compare it with the left-hand side.

2 x^3(4 - 3 x) = 8 x^3 - 6 x^4

Expanding the right-hand side, we get:

8 x^3 - 6 x^4 = 8 x^3 - 6 x^4

However, the original statement has a negative sign missing in the term $6 x^4$ . Therefore, statement C is false.

D. $30 x^4 - 12 x^3 = 6 x^3(5 x - 2)$

To verify the truth of statement D, we need to simplify the equation on the right-hand side and compare it with the left-hand side.

6 x^3(5 x - 2) = 30 x^4 - 12 x^3

Expanding the right-hand side, we get:

30 x^4 - 12 x^3 = 30 x^4 - 12 x^3

Comparing this with the left-hand side, we can see that the equation is true. Therefore, statement D is true.

Conclusion

In conclusion, out of the four mathematical statements, only two are true. Statement B is true because the equation on the right-hand side simplifies to the same expression as the left-hand side. Statement D is also true because the equation on the right-hand side simplifies to the same expression as the left-hand side. On the other hand, statement A is false because the equation on the right-hand side does not simplify to the same expression as the left-hand side. Statement C is also false because the equation on the right-hand side does not simplify to the same expression as the left-hand side, and there is a missing negative sign in the term $6 x^4$ .

Key Takeaways

Verifying the truth of a mathematical statement involves simplifying the equation and comparing it with the original statement.
The equation on the right-hand side must simplify to the same expression as the left-hand side for the statement to be true.
Missing terms or incorrect signs can make a statement false.

Further Reading

For more information on mathematical equations and verification, you can refer to the following resources:

Mathematical Equations and Verification
Simplifying Algebraic Expressions
Verifying Mathematical Statements
Mathematical Equations and Verification: Q&A

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions related to mathematical equations and verification.

Q: What is the purpose of verifying mathematical equations?

A: The purpose of verifying mathematical equations is to ensure that the equation is correct and that the solution is accurate. Verification involves simplifying the equation and comparing it with the original statement to check for errors.

Q: How do I verify a mathematical equation?

A: To verify a mathematical equation, you need to simplify the equation and compare it with the original statement. This involves expanding and combining like terms, and checking for errors in signs and coefficients.

Q: What are some common mistakes to avoid when verifying mathematical equations?

A: Some common mistakes to avoid when verifying mathematical equations include:

Missing terms or incorrect signs
Incorrectly expanding or combining like terms
Failing to check for errors in coefficients
Not simplifying the equation enough

Q: How do I know if a mathematical equation is true or false?

A: To determine if a mathematical equation is true or false, you need to simplify the equation and compare it with the original statement. If the equation on the right-hand side simplifies to the same expression as the left-hand side, then the equation is true. If not, then the equation is false.

Q: Can I use a calculator to verify mathematical equations?

A: While calculators can be useful for simplifying and solving mathematical equations, they are not always reliable for verifying equations. Calculators can make mistakes, and they may not always be able to simplify complex equations. Therefore, it's always best to verify mathematical equations by hand.

Q: How do I verify a mathematical equation with multiple variables?

A: To verify a mathematical equation with multiple variables, you need to simplify the equation and compare it with the original statement. This involves expanding and combining like terms, and checking for errors in signs and coefficients. You may also need to use algebraic techniques such as factoring and canceling to simplify the equation.

Q: Can I use technology to verify mathematical equations?

A: Yes, there are many software programs and online tools available that can help you verify mathematical equations. These tools can simplify and solve equations, and they can also check for errors in signs and coefficients. Some popular options include Wolfram Alpha, Mathway, and Symbolab.

Q: How do I know if a mathematical equation is a true identity or a false statement?

A: To determine if a mathematical equation is a true identity or a false statement, you need to simplify the equation and compare it with the original statement. If the equation on the right-hand side simplifies to the same expression as the left-hand side, then the equation is a true identity. If not, then the equation is a false statement.

Q: Can I use mathematical equations to solve real-world problems?

A: Yes, mathematical equations can be used to solve real-world problems. Many problems in science, engineering, economics, and other fields can be modeled using mathematical equations. By solving these equations, you can gain insights into the behavior of complex systems and make predictions about future outcomes.

Conclusion

In conclusion, verifying mathematical equations is an essential skill for anyone who works with mathematics. By understanding how to simplify and compare equations, you can ensure that your solutions are accurate and reliable. Whether you're working on a simple algebra problem or a complex calculus equation, the principles of verification remain the same.

Key Takeaways

Verification involves simplifying the equation and comparing it with the original statement.
The equation on the right-hand side must simplify to the same expression as the left-hand side for the equation to be true.
Missing terms or incorrect signs can make a statement false.
Calculators can be useful for simplifying and solving mathematical equations, but they are not always reliable for verifying equations.
Technology can be used to verify mathematical equations, but it's always best to verify equations by hand.

Further Reading

For more information on mathematical equations and verification, you can refer to the following resources: