Subtract { -2x^2 + 2$}$ From { -3x^2 + 10$}$.

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Introduction

In algebra, quadratic expressions are a fundamental concept that plays a crucial role in solving various mathematical problems. When working with quadratic expressions, it's often necessary to perform operations such as addition and subtraction. In this article, we will focus on subtracting quadratic expressions, specifically subtracting −2x2+2{-2x^2 + 2} from −3x2+10{-3x^2 + 10}.

Understanding Quadratic Expressions

A quadratic expression is a polynomial expression of degree two, which means the highest power of the variable (in this case, x) is two. Quadratic expressions can be written in the general form of ax2+bx+c{ax^2 + bx + c}, where a, b, and c are constants.

Example of a Quadratic Expression

The expression −3x2+10{-3x^2 + 10} is a quadratic expression because the highest power of x is two. Similarly, the expression −2x2+2{-2x^2 + 2} is also a quadratic expression.

Subtracting Quadratic Expressions

To subtract one quadratic expression from another, we need to follow the same rules as subtracting like terms. When subtracting quadratic expressions, we need to subtract the corresponding terms.

Step-by-Step Guide to Subtracting Quadratic Expressions

  1. Identify the like terms: The first step is to identify the like terms in both expressions. In this case, the like terms are the terms with the same power of x.
  2. Subtract the coefficients: Once we have identified the like terms, we need to subtract the coefficients of the corresponding terms. The coefficient of a term is the number in front of the variable.
  3. Combine the constant terms: After subtracting the coefficients of the like terms, we need to combine the constant terms.

Subtracting −2x2+2{-2x^2 + 2} from −3x2+10{-3x^2 + 10}

Let's apply the steps outlined above to subtract −2x2+2{-2x^2 + 2} from −3x2+10{-3x^2 + 10}.

  1. Identify the like terms: The like terms in both expressions are the terms with the same power of x, which are −2x2{-2x^2} and −3x2{-3x^2}.
  2. Subtract the coefficients: The coefficient of −2x2{-2x^2} is -2, and the coefficient of −3x2{-3x^2} is -3. To subtract these coefficients, we need to subtract -2 from -3, which gives us -1.
  3. Combine the constant terms: The constant terms in both expressions are 2 and 10. To combine these terms, we need to subtract 2 from 10, which gives us 8.

Conclusion

Subtracting quadratic expressions involves following the same rules as subtracting like terms. By identifying the like terms, subtracting the coefficients, and combining the constant terms, we can perform the subtraction. In this article, we have demonstrated how to subtract −2x2+2{-2x^2 + 2} from −3x2+10{-3x^2 + 10} using a step-by-step guide.

Example Problems

Problem 1

Subtract 4x2+6{4x^2 + 6} from 2x2+8{2x^2 + 8}.

Solution

To solve this problem, we need to follow the same steps outlined above.

  1. Identify the like terms: The like terms in both expressions are the terms with the same power of x, which are 4x2{4x^2} and 2x2{2x^2}.
  2. Subtract the coefficients: The coefficient of 4x2{4x^2} is 4, and the coefficient of 2x2{2x^2} is 2. To subtract these coefficients, we need to subtract 4 from 2, which gives us -2.
  3. Combine the constant terms: The constant terms in both expressions are 6 and 8. To combine these terms, we need to subtract 6 from 8, which gives us 2.

The final answer is −2x2+2{-2x^2 + 2}.

Problem 2

Subtract 3x2+9{3x^2 + 9} from 5x2+2{5x^2 + 2}.

Solution

To solve this problem, we need to follow the same steps outlined above.

  1. Identify the like terms: The like terms in both expressions are the terms with the same power of x, which are 3x2{3x^2} and 5x2{5x^2}.
  2. Subtract the coefficients: The coefficient of 3x2{3x^2} is 3, and the coefficient of 5x2{5x^2} is 5. To subtract these coefficients, we need to subtract 3 from 5, which gives us 2.
  3. Combine the constant terms: The constant terms in both expressions are 9 and 2. To combine these terms, we need to subtract 9 from 2, which gives us -7.

The final answer is 2x2−7{2x^2 - 7}.

Final Thoughts

Subtracting quadratic expressions is an essential skill in algebra that can be applied to a wide range of mathematical problems. By following the steps outlined in this article, you can perform the subtraction of quadratic expressions with ease. Remember to identify the like terms, subtract the coefficients, and combine the constant terms to get the final answer. With practice and patience, you can master the art of subtracting quadratic expressions.

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Frequently Asked Questions

Q: What is the difference between subtracting quadratic expressions and subtracting linear expressions?

A: The main difference between subtracting quadratic expressions and subtracting linear expressions is that quadratic expressions have a squared variable (x^2), while linear expressions do not. When subtracting quadratic expressions, we need to follow the same rules as subtracting like terms, but we also need to consider the squared variable.

Q: How do I identify like terms when subtracting quadratic expressions?

A: To identify like terms when subtracting quadratic expressions, we need to look for terms with the same power of x. In this case, the like terms are the terms with the same power of x, such as −2x2{-2x^2} and −3x2{-3x^2}.

Q: What is the coefficient of a term in a quadratic expression?

A: The coefficient of a term in a quadratic expression is the number in front of the variable. For example, in the expression −2x2{-2x^2}, the coefficient is -2.

Q: How do I subtract the coefficients of like terms when subtracting quadratic expressions?

A: To subtract the coefficients of like terms when subtracting quadratic expressions, we need to subtract the coefficients of the corresponding terms. For example, in the expression −2x2+2{-2x^2 + 2} subtracted from −3x2+10{-3x^2 + 10}, we need to subtract -2 from -3, which gives us -1.

Q: What is the constant term in a quadratic expression?

A: The constant term in a quadratic expression is the term that does not have a variable. For example, in the expression −3x2+10{-3x^2 + 10}, the constant term is 10.

Q: How do I combine the constant terms when subtracting quadratic expressions?

A: To combine the constant terms when subtracting quadratic expressions, we need to subtract the constant terms of the two expressions. For example, in the expression −2x2+2{-2x^2 + 2} subtracted from −3x2+10{-3x^2 + 10}, we need to subtract 2 from 10, which gives us 8.

Q: Can I subtract a quadratic expression from a linear expression?

A: Yes, you can subtract a quadratic expression from a linear expression. However, you need to follow the same rules as subtracting like terms, but you also need to consider the squared variable in the quadratic expression.

Q: What is the final answer when subtracting −2x2+2{-2x^2 + 2} from −3x2+10{-3x^2 + 10}?

A: The final answer when subtracting −2x2+2{-2x^2 + 2} from −3x2+10{-3x^2 + 10} is −x2+8{-x^2 + 8}.

Q: Can I use a calculator to subtract quadratic expressions?

A: Yes, you can use a calculator to subtract quadratic expressions. However, it's always a good idea to double-check your work by following the steps outlined in this article.

Q: What are some common mistakes to avoid when subtracting quadratic expressions?

A: Some common mistakes to avoid when subtracting quadratic expressions include:

  • Not identifying like terms correctly
  • Not subtracting the coefficients of like terms correctly
  • Not combining the constant terms correctly
  • Not following the order of operations (PEMDAS)

Conclusion

Subtracting quadratic expressions can be a challenging task, but with practice and patience, you can master the art of subtracting quadratic expressions. By following the steps outlined in this article and avoiding common mistakes, you can ensure that you get the correct answer. Remember to identify like terms, subtract the coefficients, and combine the constant terms to get the final answer.