Each Expression Is Given In Standard Form. Rewrite It In Factored Form.Enter The Negative Factor First.A. X 2 − 900 = ( X − □ ) ( X + □ X^2 - 900 = (x - \square)(x + \square X 2 − 900 = ( X − □ ) ( X + □ ]B. X 2 + 18 X − 40 = ( X + □ ) ( X + □ X^2 + 18x - 40 = (x + \square)(x + \square X 2 + 18 X − 40 = ( X + □ ) ( X + □ ]C. $x^2 - 3x - 10 = (x + \square)(x

Introduction

In algebra, quadratic expressions are a fundamental concept that plays a crucial role in solving equations and inequalities. A quadratic expression is a polynomial of degree two, which means the highest power of the variable is two. These expressions can be written in two forms: standard form and factored form. In this article, we will focus on rewriting quadratic expressions in factored form, with a negative factor first.

What is Factored Form?

Factored form is a way of writing a quadratic expression as a product of two binomials. This form is useful because it allows us to easily identify the roots of the quadratic equation, which are the values of the variable that make the expression equal to zero. The factored form of a quadratic expression is typically written as:

(a + b)(a - b)

where a and b are constants.

Rewriting Quadratic Expressions in Factored Form

A. $x^2 - 900 = (x - \square)(x + \square)$

To rewrite this expression in factored form, we need to find two numbers whose product is -900 and whose sum is 0. These numbers are -30 and 30, because (-30)(30) = -900 and -30 + 30 = 0. Therefore, we can rewrite the expression as:

$x^2 - 900 = (x - 30)(x + 30)$

B. $x^2 + 18x - 40 = (x + \square)(x + \square)$

To rewrite this expression in factored form, we need to find two numbers whose product is -40 and whose sum is 18. These numbers are 20 and -2, because (20)(-2) = -40 and 20 + (-2) = 18. Therefore, we can rewrite the expression as:

$x^2 + 18x - 40 = (x + 20)(x - 2)$

C. $x^2 - 3x - 10 = (x + \square)(x - \square)$

To rewrite this expression in factored form, we need to find two numbers whose product is -10 and whose sum is -3. These numbers are -5 and 2, because (-5)(2) = -10 and -5 + 2 = -3. Therefore, we can rewrite the expression as:

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

Tips and Tricks

When rewriting a quadratic expression in factored form, it's essential to remember the following tips and tricks:

• The product of the two binomials must be equal to the original expression.
• The sum of the two binomials must be equal to the coefficient of the middle term.
• The two binomials must have the same variable.

Conclusion

Rewriting quadratic expressions in factored form is a crucial skill in algebra. By following the steps outlined in this article, you can easily rewrite expressions in factored form, with a negative factor first. Remember to identify the two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the middle term. With practice, you'll become proficient in rewriting quadratic expressions in factored form.

Common Mistakes to Avoid

When rewriting quadratic expressions in factored form, it's easy to make mistakes. Here are some common mistakes to avoid:

• Not identifying the two numbers whose product is equal to the constant term.
• Not identifying the two numbers whose sum is equal to the coefficient of the middle term.
• Not ensuring that the two binomials have the same variable.

Real-World Applications

Rewriting quadratic expressions in factored form has numerous real-world applications. For example:

• In physics, factored form is used to describe the motion of objects under the influence of gravity.
• In engineering, factored form is used to design and optimize systems.
• In economics, factored form is used to model and analyze economic systems.

Practice Problems

To practice rewriting quadratic expressions in factored form, try the following problems:

• $x^2 + 14x + 44 = (x + \square)(x + \square)$
• $x^2 - 12x - 45 = (x + \square)(x - \square)$
• $x^2 + 2x - 15 = (x + \square)(x - \square)$

Conclusion

Introduction

In our previous article, we discussed how to rewrite quadratic expressions in factored form. In this article, we will answer some frequently asked questions about quadratic expressions in factored form.

Q: What is the difference between standard form and factored form?

A: Standard form is a way of writing a quadratic expression as a polynomial, while factored form is a way of writing a quadratic expression as a product of two binomials.

Q: How do I know if a quadratic expression can be rewritten in factored form?

A: A quadratic expression can be rewritten in factored form if it can be expressed as a product of two binomials. This means that the expression must have a constant term and a middle term.

Q: What are the steps to rewrite a quadratic expression in factored form?

A: The steps to rewrite a quadratic expression in factored form are:

1. Identify the two numbers whose product is equal to the constant term.
2. Identify the two numbers whose sum is equal to the coefficient of the middle term.
3. Write the expression as a product of two binomials.

Q: How do I identify the two numbers whose product is equal to the constant term?

A: To identify the two numbers whose product is equal to the constant term, you can use the following methods:

• Factorization: Factor the constant term into two numbers.
• Prime factorization: Find the prime factors of the constant term and multiply them together.
• Trial and error: Try different combinations of numbers until you find two numbers whose product is equal to the constant term.

Q: How do I identify the two numbers whose sum is equal to the coefficient of the middle term?

A: To identify the two numbers whose sum is equal to the coefficient of the middle term, you can use the following methods:

• Factorization: Factor the coefficient of the middle term into two numbers.
• Prime factorization: Find the prime factors of the coefficient of the middle term and multiply them together.
• Trial and error: Try different combinations of numbers until you find two numbers whose sum is equal to the coefficient of the middle term.

Q: What are some common mistakes to avoid when rewriting quadratic expressions in factored form?

A: Some common mistakes to avoid when rewriting quadratic expressions in factored form are:

• Not identifying the two numbers whose product is equal to the constant term.
• Not identifying the two numbers whose sum is equal to the coefficient of the middle term.
• Not ensuring that the two binomials have the same variable.

Q: How do I check if my factored form is correct?

A: To check if your factored form is correct, you can multiply the two binomials together and see if you get the original expression.

Q: What are some real-world applications of quadratic expressions in factored form?

A: Some real-world applications of quadratic expressions in factored form include:

• Physics: Factored form is used to describe the motion of objects under the influence of gravity.
• Engineering: Factored form is used to design and optimize systems.
• Economics: Factored form is used to model and analyze economic systems.

Q: How do I practice rewriting quadratic expressions in factored form?

A: To practice rewriting quadratic expressions in factored form, you can try the following:

• Use online resources such as Khan Academy or Mathway to practice rewriting quadratic expressions in factored form.
• Work with a tutor or teacher to practice rewriting quadratic expressions in factored form.
• Try rewriting quadratic expressions in factored form on your own and check your work with a calculator or online resource.

Conclusion

Rewriting quadratic expressions in factored form is a fundamental skill in algebra. By following the steps outlined in this article, you can easily rewrite expressions in factored form, with a negative factor first. Remember to identify the two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the middle term. With practice, you'll become proficient in rewriting quadratic expressions in factored form.