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Chapter 6: Problem 140

In Exercises \(137-141,\) factor completely. $$(x+5)^{2}-20(x+5)+100$$

Short Answer

Expert verified
The factored form of the equation \((x+5)^{2}-20(x+5)+100\) is \((x - \sqrt{20} + 5)^2\).

Step by step solution

01

Identify the components

The given equation is \((x+5)^{2}-20(x+5)+100\). We notice this is in the form of \(a^2 - 2ab + b^2\), which is a formula for the square of a binomial difference (i.e., \((a-b)^2\)). Here, \(a = x+5\), \(b = \sqrt{20}\) (Since we know that \(-2ab = -20*(x+5)\), so we can solve for \(b\)).
02

Rewrite the binomial as a square

According to the formula, this binomial can be rewritten as \((a - b)^2\). Replacing \(a\), \(b\) with the identified values, we get \((x + 5 - \sqrt{20})^2\).
03

Simplify the expression

We simplify the obtained expression to give the final answer \((x - \sqrt{20} + 5)^2\).

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Key Concepts

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

Binomial Difference
Recognizing a binomial difference is key in factoring quadratic expressions. This concept involves expressions where one term is subtracted from another. Quadratics often use this in forms like
  • \( a^2 - 2ab + b^2 \)
  • \( (a-b)^2 \)
Understanding these forms helps to rewrite and simplify expressions.
In our exercise, by observing \((x+5)^2 - 20(x+5) + 100\), you can see
  • \( a = x+5 \)
  • \(-2ab = -20(x+5) \)
These are clues that the expression can transform into a squared binomial difference. Recognizing these patterns becomes second nature with practice.
Perfect Square Trinomial
A perfect square trinomial is an expanded form of a binomial squared. Essentially, it's the result of squaring a binomial, like
  • \( (a-b)^2 = a^2 - 2ab + b^2 \)
  • \( (a+b)^2 = a^2 + 2ab + b^2 \)
In our original expression, identifying this form allowed us to condense the quadratic into a neater format.
We were able to rewrite \( (x+5)^2 - 20(x+5) + 100 \) as \( (a-b)^2 \) by finding
  • \( b = \sqrt{20} \)
This transformation simplifies problems significantly, making the expression friendlier to work with. Recognizing perfect square trinomials is crucial for efficient factoring.
Mathematical Simplification
Simplification is the process of making an expression easier to manage. It involves combining like terms and reducing expressions to their simplest form. After identifying a binomial difference and recognizing a perfect square trinomial, the next step is to simplify.
In the example, simplifying \( (a - \sqrt{20})^2 \) involves organizing the expression for clarity, eventually written as
  • \( (x - \sqrt{20} + 5)^2 \)
This step is crucial as it provides the neatest and most efficient way to present a solution. Understanding and applying simplification strengthens problem-solving skills and offers clarity in mathematical communication.

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