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Chapter 5: Problem 107

Factor. $$ (x+3)^{2}-2(x+3)-35 $$

Short Answer

Expert verified
(x - 4)(x + 8)

Step by step solution

01

Substitute a Variable

Let \(u = x + 3\). This simplifies the expression to \(u^2 - 2u - 35\).
02

Factor the Quadratic Expression

Identify two numbers that multiply to -35 and add up to -2. These numbers are -7 and 5. Rewrite the expression as \((u - 7)(u + 5)\).
03

Substitute Back the Original Expression

Replace \(u\) with \(x + 3\) to get \((x + 3 - 7)(x + 3 + 5)\), which simplifies to \((x - 4)(x + 8)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

substitution method
The substitution method is a strategy used in algebra to make complex expressions easier to work with. In this exercise, we replaced the expression \(x + 3\) with a new variable \(u\). This transformation can help to simplify and solve problems more efficiently.

Here are some steps to understand the substitution method better:
  • Choose a complex expression to substitute with a simple variable.
  • Rewrite the entire expression using the new variable.
  • Solve the simplified problem using standard algebraic techniques.
  • Substitute back the original expression for the variable.
In this problem, by setting \(u = x + 3\), we transformed \((x+3)^{2} - 2(x+3) - 35\) into a simpler form \(u^{2} - 2u - 35\), making the factorization process easier.
quadratic factoring
Quadratic factoring is a technique used to break down quadratic expressions into the product of simpler binomial expressions.

The general form of a quadratic expression is \(ax^{2} + bx + c\). To factorize this, we look for two numbers that both:
  • Multiply to the constant term \(ac\).
  • Add up to the coefficient of the middle term \(b\).

In this exercise, with our expression \(u^{2} - 2u - 35\), we identify -7 and 5 as two numbers that multiply to -35 and add up to -2.
So, we rewrite \(u^{2} - 2u - 35\) as \((u - 7)(u + 5)\). This is the factorized form of the quadratic expression.
algebraic expressions
Algebraic expressions are combinations of variables, numbers, and functions that represent a value.

They can take various forms, such as constants (e.g., 3), variables (e.g., \(x\)), and terms that combine both (e.g., \(2x\)). In this problem, we started with \((x + 3)^{2} - 2(x + 3) - 35\).

Understanding how to manipulate these kinds of expressions is a fundamental skill in algebra.

Specifically, in this exercise:
  • We used substitution to simplify a complex expression.
  • We factored a quadratic expression to find its roots.
  • We then substituted back to get the final factored form of the original expression.

Algebraic expressions like these are at the heart of solving many types of mathematical problems.

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Most popular questions from this chapter

Solve. Ignacio is planning a rectangular garden that is \(25 \mathrm{m}\) longer than it is wide. The garden will have an area of \(7500 \mathrm{m}^{2} .\) What will its dimensions be?

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