Rewriting the Quadratic Expression

Oct 13, 2025 by windailys.com 111 views

Introduction

Quadratic expressions are a fundamental concept in algebra, and they often appear in various mathematical problems. In this article, we will explore how to rewrite and factorize quadratic expressions, focusing on the given expression $30 + x - x^2$ . We will demonstrate how to rewrite this expression as $-(x^2 - x - 30)$ and then fully factorize the quadratic expression.

Rewriting the Quadratic Expression

To rewrite the quadratic expression $30 + x - x^2$ , we need to manipulate the terms to obtain the desired form. Let's start by rearranging the terms:

30 + x - x^2 = -x^2 + x + 30

Now, we can see that the expression can be rewritten as $-(x^2 - x - 30)$ . This is achieved by multiplying the entire expression by $-1$ , which is equivalent to changing the sign of each term.

Justification

To justify this rewriting, let's consider the properties of multiplication. When we multiply an expression by $-1$ , each term is multiplied by $-1$ . Therefore, we can rewrite the expression as:

-1(x^2 - x - 30) = -(x^2 - x - 30)

This demonstrates that the original expression $30 + x - x^2$ can indeed be rewritten as $-(x^2 - x - 30)$ .

Factorizing the Quadratic Expression

Now that we have rewritten the quadratic expression as $-(x^2 - x - 30)$ , we can focus on factorizing the quadratic expression $x^2 - x - 30$ . To factorize this expression, we need to find two numbers whose product is $-30$ and whose sum is $-1$ . These numbers are $-6$ and $5$ , as their product is $-30$ and their sum is $-1$ .

Step 1: Factorize the Quadratic Expression

Using the numbers $-6$ and $5$ , we can factorize the quadratic expression as follows:

x^2 - x - 30 = (x - 6)(x + 5)

This is achieved by splitting the middle term $-x$ into two terms $-6x$ and $5x$ , which are then factored out.

Step 2: Simplify the Factorized Expression

Now that we have factorized the quadratic expression, we can simplify the factorized expression by combining the two factors:

-(x^2 - x - 30) = -(x - 6)(x + 5)

This is achieved by multiplying the two factors together.

Conclusion

In this article, we have demonstrated how to rewrite and factorize the quadratic expression $30 + x - x^2$ . We have shown that the expression can be rewritten as $-(x^2 - x - 30)$ and then fully factorized as $-(x - 6)(x + 5)$ . This process involves manipulating the terms to obtain the desired form and then factorizing the quadratic expression using the numbers $-6$ and $5$ . We hope that this article has provided a clear understanding of how to rewrite and factorize quadratic expressions.

Final Answer

$The final answer is $\boxed{-(x - 6)(x + 5)}$.<br/> **Quadratic Expression Rewrite and Factorization Q&A**$

Introduction

In our previous article, we explored how to rewrite and factorize the quadratic expression $30 + x - x^2$ . We demonstrated how to rewrite the expression as $-(x^2 - x - 30)$ and then fully factorize the quadratic expression as $-(x - 6)(x + 5)$ . In this article, we will address some common questions and provide additional insights on quadratic expression rewrite and factorization.

Q&A

Q: What is the difference between rewriting and factorizing a quadratic expression?

A: Rewriting a quadratic expression involves manipulating the terms to obtain a new form, while factorizing a quadratic expression involves expressing it as a product of two or more linear factors.

Q: How do I determine if a quadratic expression can be rewritten or factorized?

A: To determine if a quadratic expression can be rewritten or factorized, look for common factors or patterns in the expression. If the expression can be rewritten or factorized, it will often involve manipulating the terms to obtain a new form.

Q: What are some common techniques for rewriting and factorizing quadratic expressions?

A: Some common techniques for rewriting and factorizing quadratic expressions include:

Factoring out common factors
Using the distributive property to expand expressions
Rearranging terms to obtain a new form
Using the quadratic formula to solve quadratic equations

Q: How do I factorize a quadratic expression with a negative leading coefficient?

A: To factorize a quadratic expression with a negative leading coefficient, follow these steps:

Rewrite the expression with a positive leading coefficient by multiplying the entire expression by $-1$ .
Factorize the expression using the numbers $-6$ and $5$ .
Simplify the factorized expression by combining the two factors.

Q: What are some common mistakes to avoid when rewriting and factorizing quadratic expressions?

A: Some common mistakes to avoid when rewriting and factorizing quadratic expressions include:

Failing to check for common factors or patterns in the expression
Not using the distributive property to expand expressions
Not rearranging terms to obtain a new form
Not using the quadratic formula to solve quadratic equations

Q: How do I check if a quadratic expression has been correctly rewritten or factorized?

A: To check if a quadratic expression has been correctly rewritten or factorized, follow these steps:

Verify that the rewritten or factorized expression is equivalent to the original expression.
Check that the rewritten or factorized expression has the same solutions as the original expression.
Simplify the rewritten or factorized expression to obtain the final answer.

Conclusion

In this article, we have addressed some common questions and provided additional insights on quadratic expression rewrite and factorization. We hope that this article has provided a clear understanding of how to rewrite and factorize quadratic expressions and has helped to clarify any confusion.

Final Answer

The final answer is $\boxed{-(x - 6)(x + 5)}$ .