/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 72 \(16-(x-11)^{2}\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 72

\(16-(x-11)^{2}\)

Short Answer

Expert verified
The simplified form of the equation is \( -x^2 + 22x - 105 \).

Step by step solution

01

Expand the Square of the Binomial

Expand the square of the binomial \( (x-11)^{2} \). This can be redone as \( (x - 11)*(x - 11) \) . When expanded, it gives \( x^{2} - 22x + 121 \).
02

Insert Binomial Expansion into the Given Equation

Now we substitute the expanded form of \( (x - 11)^{2} \) into the given equation. The equation becomes \( 16 - (x^{2} - 22x + 121) \).
03

Simplify the Equation

To simplify the equation, we combine like terms and simplify the constant. The equation now looks like \( -x^2 + 22x - 105 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Understanding Polynomial Expressions
Polynomial expressions are mathematical expressions that consist of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication. The expression in our original exercise, \((x-11)^2\), is a binomial鈥攁 polynomial with two terms: \(x\) and \(-11\).
To expand this binomial, we apply the distributive property, also known as the FOIL method for binomials, where we multiply each term in the first binomial by each term in the second. Expanding \((x-11)*(x-11)\) results in \(x^2 - 22x + 121\). Understanding how to recognize and manipulate polynomial expressions is crucial as they form the foundation of more complex algebraic operations.
Steps in Equation Simplification
Equation simplification is the process of reducing an expression to its simplest form. It involves combining like terms and simplifying coefficients, which are essential skills for solving equations efficiently.
In our problem, after binomial expansion, the equation becomes \(16 - (x^2 - 22x + 121)\). To simplify, distribute the negative sign through the parentheses, ensuring each term is correctly adjusted:
  • Remove brackets and change the sign of each term inside: \(-x^2 + 22x - 121\).
  • Combine with the constant 16 to get \(-x^2 + 22x - 105\).
Each step brings you closer to the simplest possible expression, making it easier to solve or evaluate.
Mastering Algebraic Substitution
Algebraic substitution involves replacing a variable or expression in an equation with another equivalent expression. This technique helps to simplify or transform complex equations into forms that are easier to manage.
In the original problem, the substitution step involved inserting the expanded form of the binomial, \(x^2 - 22x + 121\), into the equation \(16 - (x - 11)^2\). Substituting correctly ensures each subsequent step of simplification retains accuracy.
Execute substitution carefully to avoid mistakes, such as forgetting to change signs when subtracting terms. Mastering this skill allows you to handle more complex algebraic tasks with confidence.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

\(\frac{4}{x}-\frac{x}{6}=\frac{5}{3}\)

\(\frac{\left(\frac{6 x^{2}-17 x+5}{3 x^{2}+3 x}\right)}{\left(\frac{3 x-1}{3 x+1}\right)}\)

Alternate Methods How can you simplify a complex fraction without inverting the divisor?

\(2 y=\frac{y+6}{y+1}\)

\(\frac{-9}{2 x}+\frac{9-2 x}{4 x}\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.