What Is The Missing Coefficient? ( 15 X 2 + 11 Y 2 + 8 X ) − ( 7 X 2 + 5 Y 2 + 2 X ) = □ X 2 + 6 Y 2 + 6 X \left(15 X^2 + 11 Y^2 + 8 X\right) - \left(7 X^2 + 5 Y^2 + 2 X\right) = \square X^2 + 6 Y^2 + 6 X ( 15 X 2 + 11 Y 2 + 8 X ) − ( 7 X 2 + 5 Y 2 + 2 X ) = □ X 2 + 6 Y 2 + 6 X A. 4 B. 8 C. 10 D. 22

Mar 11, 2025 by ADMIN 304 views

**What is the Missing Coefficient?**

Understanding the Problem

The given problem involves simplifying an algebraic expression by combining like terms. We are presented with two quadratic expressions, and we need to find the missing coefficient that makes the equation true. The equation is:

$\left(15 x^2 + 11 y^2 + 8 x\right) - \left(7 x^2 + 5 y^2 + 2 x\right) = \square x^2 + 6 y^2 + 6 x$

Breaking Down the Problem

To solve this problem, we need to simplify the left-hand side of the equation by combining like terms. We can start by distributing the negative sign to the terms inside the second set of parentheses:

$\left(15 x^2 + 11 y^2 + 8 x\right) - \left(7 x^2 + 5 y^2 + 2 x\right)$

$= 15 x^2 - 7 x^2 + 11 y^2 - 5 y^2 + 8 x - 2 x$

Combining Like Terms

Now, we can combine the like terms on the left-hand side of the equation:

$= (15 - 7) x^2 + (11 - 5) y^2 + (8 - 2) x$

$= 8 x^2 + 6 y^2 + 6 x$

Finding the Missing Coefficient

We are given that the simplified expression is equal to $\square x^2 + 6 y^2 + 6 x$ . We can see that the coefficients of the $y^2$ and $x$ terms are the same on both sides of the equation. Therefore, we can focus on finding the missing coefficient of the $x^2$ term.

Comparing Coefficients

We can compare the coefficients of the $x^2$ term on both sides of the equation:

$8 x^2 = \square x^2$

This implies that the missing coefficient is equal to 8.

Conclusion

In conclusion, the missing coefficient is 8. This is the value that makes the equation true.

Answer

The correct answer is B. 8.

Additional Examples

Here are a few additional examples of simplifying algebraic expressions by combining like terms:

$\left(2 x^2 + 3 y^2 + 4 x\right) - \left(x^2 + 2 y^2 + 3 x\right) = \square x^2 + y^2 + x$
$\left(5 x^2 + 2 y^2 + 3 x\right) - \left(2 x^2 + 3 y^2 + 4 x\right) = \square x^2 + 5 y^2 - x$

These examples demonstrate the importance of combining like terms when simplifying algebraic expressions.

Tips and Tricks

Here are a few tips and tricks for simplifying algebraic expressions by combining like terms:

Make sure to distribute the negative sign to all the terms inside the second set of parentheses.
Combine the like terms on the left-hand side of the equation.
Compare the coefficients of the like terms on both sides of the equation.
Focus on finding the missing coefficient of the like term.

Q: What is the missing coefficient in the given equation?

A: The missing coefficient is 8. This is the value that makes the equation true.

Q: How do I simplify an algebraic expression by combining like terms?

A: To simplify an algebraic expression by combining like terms, you need to follow these steps:

Distribute the negative sign to all the terms inside the second set of parentheses.
Combine the like terms on the left-hand side of the equation.
Compare the coefficients of the like terms on both sides of the equation.
Focus on finding the missing coefficient of the like term.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, $x^2$ and $x^2$ are like terms, while $x^2$ and $x$ are not like terms.

Q: How do I identify like terms in an algebraic expression?

A: To identify like terms in an algebraic expression, you need to look for terms that have the same variable raised to the same power. For example, in the expression $2x^2 + 3y^2 + 4x$ , the like terms are $2x^2$ and $4x$ .

Q: What is the importance of combining like terms in algebra?

A: Combining like terms is an important step in simplifying algebraic expressions. It helps to eliminate unnecessary terms and make the expression easier to work with. It also helps to identify the missing coefficient of a like term.

Q: Can you provide more examples of simplifying algebraic expressions by combining like terms?

A: Here are a few more examples:

$\left(3x^2 + 2y^2 + 5x\right) - \left(2x^2 + 3y^2 + 4x\right) = \square x^2 + 5y^2 + x$
$\left(4x^2 + 3y^2 + 2x\right) - \left(2x^2 + 4y^2 + 3x\right) = \square x^2 + 5y^2 - x$

Q: How do I know which terms to combine in an algebraic expression?

A: To know which terms to combine in an algebraic expression, you need to look for terms that have the same variable raised to the same power. You can then combine these terms by adding or subtracting their coefficients.

Q: Can you provide tips and tricks for simplifying algebraic expressions by combining like terms?

A: Here are a few tips and tricks:

Make sure to distribute the negative sign to all the terms inside the second set of parentheses.
Combine the like terms on the left-hand side of the equation.
Compare the coefficients of the like terms on both sides of the equation.
Focus on finding the missing coefficient of the like term.

By following these tips and tricks, you can simplify algebraic expressions by combining like terms and find the missing coefficient.

Q: What are some common mistakes to avoid when simplifying algebraic expressions by combining like terms?

A: Here are a few common mistakes to avoid:

Not distributing the negative sign to all the terms inside the second set of parentheses.
Not combining the like terms on the left-hand side of the equation.
Not comparing the coefficients of the like terms on both sides of the equation.
Not focusing on finding the missing coefficient of the like term.

By avoiding these mistakes, you can simplify algebraic expressions by combining like terms and find the missing coefficient.