Rewrite The Expression: $9x^2 - Y^2 + 10y - 25$

Introduction

In mathematics, rewriting an expression is an essential skill that helps in simplifying complex equations and making them easier to solve. It involves rearranging the terms in an expression to make it more manageable and understandable. In this article, we will focus on rewriting the given expression $9x^2 - y^2 + 10y - 25$.

Step 1: Factor Out the Greatest Common Factor (GCF)

The first step in rewriting the expression is to factor out the greatest common factor (GCF) from the terms. In this case, the GCF of the terms is 1, which means that we cannot factor out any common factor from the terms.

However, we can rewrite the expression by grouping the terms that have a common factor. We can group the terms $9x^2$ and $-25$ as they have a common factor of 1, and the terms $-y^2$ and $10y$ as they have a common factor of 1.

9x^2 - y^2 + 10y - 25 = (9x^2 - 25) + (-y^2 + 10y)

Step 2: Factor the Quadratic Terms

The next step is to factor the quadratic terms in the expression. We can factor the term $9x^2 - 25$ as $(3x)^2 - 5^2$ using the difference of squares formula.

(9x^2 - 25) = (3x)^2 - 5^2

Similarly, we can factor the term $-y^2 + 10y$ as $-(y^2 - 10y)$ using the difference of squares formula.

(-y^2 + 10y) = -(y^2 - 10y)

Step 3: Simplify the Expression

Now that we have factored the quadratic terms, we can simplify the expression by combining the like terms.

(9x^2 - 25) + (-y^2 + 10y) = (3x)^2 - 5^2 - (y^2 - 10y)

We can further simplify the expression by combining the like terms.

(3x)^2 - 5^2 - (y^2 - 10y) = (3x)^2 - 5^2 - y^2 + 10y

Step 4: Rewrite the Expression in Standard Form

The final step is to rewrite the expression in standard form. We can rewrite the expression as follows:

(3x)^2 - 5^2 - y^2 + 10y = 9x^2 - 25 - y^2 + 10y

Conclusion

In this article, we have rewritten the expression $9x^2 - y^2 + 10y - 25$ by factoring out the greatest common factor, factoring the quadratic terms, and simplifying the expression. We have also rewritten the expression in standard form. The rewritten expression is $9x^2 - 25 - y^2 + 10y$.

Final Answer

Introduction

In our previous article, we rewrote the expression $9x^2 - y^2 + 10y - 25$ by factoring out the greatest common factor, factoring the quadratic terms, and simplifying the expression. In this article, we will answer some frequently asked questions related to rewriting the expression.

Q: What is the greatest common factor (GCF) of the terms in the expression?

A: The greatest common factor (GCF) of the terms in the expression is 1, which means that we cannot factor out any common factor from the terms.

Q: How do I factor the quadratic terms in the expression?

A: To factor the quadratic terms in the expression, we can use the difference of squares formula. For example, we can factor the term $9x^2 - 25$ as $(3x)^2 - 5^2$ using the difference of squares formula.

Q: What is the difference of squares formula?

A: The difference of squares formula is a mathematical formula that states that $a^2 - b^2 = (a + b)(a - b)$, where $a$ and $b$ are any two numbers.

Q: How do I simplify the expression after factoring the quadratic terms?

A: To simplify the expression after factoring the quadratic terms, we can combine the like terms. For example, we can combine the terms $9x^2$ and $-25$ as $(3x)^2 - 5^2$, and the terms $-y^2$ and $10y$ as $-(y^2 - 10y)$.

Q: What is the final answer to the expression?

A: The final answer to the expression is $9x^2 - 25 - y^2 + 10y$.

Q: Can I use any other method to rewrite the expression?

A: Yes, there are other methods to rewrite the expression. For example, we can use the method of completing the square to rewrite the expression.

Q: What is the method of completing the square?

A: The method of completing the square is a mathematical technique that involves rewriting a quadratic expression in the form $(x + a)^2 + b$, where $a$ and $b$ are any two numbers.

Q: How do I use the method of completing the square to rewrite the expression?

A: To use the method of completing the square to rewrite the expression, we can start by adding and subtracting a constant term to the expression. For example, we can add and subtract $25$ to the expression $9x^2 - y^2 + 10y - 25$.

Q: What is the final answer to the expression using the method of completing the square?

A: The final answer to the expression using the method of completing the square is $9x^2 - (y - 5)^2 + 25$.

Conclusion

In this article, we have answered some frequently asked questions related to rewriting the expression $9x^2 - y^2 + 10y - 25$. We have also provided some additional information on the method of completing the square.

Final Answer

The final answer is: $\boxed{9x^2 - 25 - y^2 + 10y}$