Simplify The Expression:$\[ 9x^2 - Y^2 + 10y - 25 \\]

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Introduction

In mathematics, simplifying expressions is a crucial step in solving equations and inequalities. It involves rewriting the expression in a more manageable form, often by combining like terms or factoring out common factors. In this article, we will simplify the given expression: $9x^2 - y^2 + 10y - 25$. We will use various techniques such as factoring, combining like terms, and completing the square to simplify the expression.

Step 1: Factor Out Common Factors

The first step in simplifying the expression is to factor out any common factors. In this case, we can factor out a negative sign from the first three terms:

${ - (9x^2 + y^2 - 10y + 25) }$

However, there are no common factors among the terms, so we cannot factor out any further.

Step 2: Combine Like Terms

The next step is to combine like terms. We can combine the $x^2$ terms and the $y$ terms separately:

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

However, there are no like terms that can be combined.

Step 3: Factor the Quadratic Expression

We can try to factor the quadratic expression $9x^2 - y^2$ as a difference of squares:

${ 9x^2 - y^2 = (3x)^2 - y^2 }$

This can be factored as:

${ (3x + y)(3x - y) }$

However, this does not simplify the expression further.

Step 4: Complete the Square

We can try to complete the square for the $y$ terms. We can rewrite the expression as:

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

This can be rewritten as:

${ 9x^2 - (y - 5)^2 + 25 }$

This is the simplified form of the expression.

Conclusion

In this article, we simplified the given expression $9x^2 - y^2 + 10y - 25$ using various techniques such as factoring, combining like terms, and completing the square. We found that the simplified form of the expression is $9x^2 - (y - 5)^2 + 25$. This form is more manageable and can be used to solve equations and inequalities.

Final Answer

The final answer is: $\boxed{9x^2 - (y - 5)^2 + 25}$

Discussion

The expression $9x^2 - y^2 + 10y - 25$ can be simplified using various techniques. The simplified form of the expression is $9x^2 - (y - 5)^2 + 25$. This form is more manageable and can be used to solve equations and inequalities.

Related Topics

• Factoring quadratic expressions
• Completing the square
• Simplifying expressions

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak

Tags

• Simplifying expressions
• Factoring quadratic expressions
• Completing the square
• Algebra
• Calculus
  

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Introduction

In our previous article, we simplified the expression $9x^2 - y^2 + 10y - 25$ using various techniques such as factoring, combining like terms, and completing the square. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q&A

Q: What is the simplified form of the expression $9x^2 - y^2 + 10y - 25$?

A: The simplified form of the expression is $9x^2 - (y - 5)^2 + 25$.

Q: How do I factor the quadratic expression $9x^2 - y^2$?

A: The quadratic expression $9x^2 - y^2$ can be factored as a difference of squares: $(3x)^2 - y^2 = (3x + y)(3x - y)$.

Q: Can I combine like terms in the expression $9x^2 - y^2 + 10y - 25$?

A: No, there are no like terms that can be combined in the expression $9x^2 - y^2 + 10y - 25$.

Q: What is completing the square?

A: Completing the square is a technique used to rewrite a quadratic expression in the form $(x - a)^2 + b$. It involves adding and subtracting a constant term to create a perfect square trinomial.

Q: How do I complete the square for the $y$ terms in the expression $9x^2 - y^2 + 10y - 25$?

A: To complete the square for the $y$ terms, we can rewrite the expression as $9x^2 - (y^2 - 10y + 25) + 25$. This can be rewritten as $9x^2 - (y - 5)^2 + 25$.

Q: What is the final answer to the expression $9x^2 - y^2 + 10y - 25$?

A: The final answer to the expression $9x^2 - y^2 + 10y - 25$ is $\boxed{9x^2 - (y - 5)^2 + 25}$.

Discussion

The expression $9x^2 - y^2 + 10y - 25$ can be simplified using various techniques such as factoring, combining like terms, and completing the square. The simplified form of the expression is $9x^2 - (y - 5)^2 + 25$. This form is more manageable and can be used to solve equations and inequalities.

Related Topics

• Factoring quadratic expressions
• Completing the square
• Simplifying expressions
• Algebra
• Calculus

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak

Tags

• Simplifying expressions
• Factoring quadratic expressions
• Completing the square
• Algebra
• Calculus

Additional Resources

• [1] Khan Academy: Simplifying Expressions
• [2] Mathway: Simplifying Expressions
• [3] Wolfram Alpha: Simplifying Expressions

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression $9x^2 - y^2 + 10y - 25$. We provided step-by-step solutions to the questions and discussed the related topics. We hope that this article has been helpful in understanding the simplification of the expression.