/*! 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 29 Solve by using the square root p... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 29

Solve by using the square root property. \((k+2)^{2}=28\)

Short Answer

Expert verified
k = -2 \pm 2\sqrt{7}

Step by step solution

01

- Understand the equation

We start with the equation \( (k+2)^2 = 28 \)
02

- Apply the square root property

To solve for k, apply the square root property by taking the square root of both sides of the equation. This gives us: \( \sqrt{(k+2)^2} = \pm \sqrt{28} \). Simplifying the left side, we get: \( k+2 = \pm \sqrt{28} \)
03

- Simplify the square root

Simplify \sqrt{28} \ to \sqrt{4 \cdot 7} \ which equals \sqrt{4}\cdot\sqrt{7} \ thus giving us: \( k+2 = \pm 2\sqrt{7} \)
04

- Isolate the variable k

To isolate k, subtract 2 from both sides of the equation: \( k = -2 \pm 2\sqrt{7} \)

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.

Solving Equations
Solving equations means finding the value(s) of the variable that make the equation true. In the given problem, we begin with the equation \((k+2)^2 = 28\). Here, we need to determine the value of \(k\) that satisfies this equation.

To solve an equation, we typically follow these steps:
  • Identify the equation
  • Simplify and rearrange the equation as needed
  • Apply mathematical properties (like square root property) to solve for the variable
  • Check the solution to ensure it satisfies the original equation
In this problem, it's crucial to understand how to use the square root property to solve for \(k\). This approach helps in breaking down complicated expressions involving squares of variables into simpler forms.
Simplifying Square Roots
Simplifying square roots is an important step when solving equations involving squared terms. In the exercise, we encountered \(\sqrt{28}\). To simplify, we look for the largest square factor of 28, which is 4. This leads us to rewrite:
\[ \sqrt{28} = \sqrt{4 \cdot 7} \]
Using the property of square roots (\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)), we get:
\[ \sqrt{28} = \sqrt{4} \cdot \sqrt{7} = 2\sqrt{7} \]
This simplifies our equation to \( k+2 = \pm 2\sqrt{7} \). Simplifying square roots helps in making the equation easier to solve and understand.
Isolation of Variables
To solve for a specific variable, we must isolate it on one side of the equation. In this problem, after simplifying the square root, our equation was transformed to \(k+2 = \pm 2\sqrt{7}\). To isolate \(k\), we perform the following steps:
  • Subtract 2 from both sides: \(k + 2 - 2 = \pm 2\sqrt{7} - 2\)
  • This simplifies to: \(k = -2 \pm 2\sqrt{7}\)
Here, \(\pm\) indicates two possible solutions:
  • \(k = -2 + 2\sqrt{7}\)
  • \(k = -2 - 2\sqrt{7}\)
Thus, isolating variables involves rearranging the equation to solve for the target variable, ensuring we perform the same operation on both sides.

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

Solve the inequality. Write the solution set in interval notation. $$-1<\frac{4-x}{-2} \leq 3$$

A die is a six-sided cube with sides labeled with \(1,2,3,4,5,\) or 6 dots. The die is a "fair" die if when rolled, each outcome is equally likely. Therefore, the probability that it lands on " \(1 "\) is \(\frac{1}{6}\). If a fair die is rolled 360 times, we would expect it to land as a "1" roughly 60 times. Let \(x\) represent the number of times a "1" is rolled. The inequality \(\left|\frac{x-60}{\sqrt{50}}\right|<1.96\) gives the "reasonable" range for the number of times that a "1" comes up in 360 rolls. a. Solve the inequality and interpret the answer in the context of this problem. b. If the die is rolled 360 times, and a "1" comes up 30 times, does it appear that the die is a fair die?

Solve the inequality, and write the solution set in interval notation. \(\left|\frac{m-4}{2}\right|<14\)

Solve the equations. \(|k-3|=|k+3|\)

Solve the inequality. Write the solution set in interval notation. $$-11<6 y+7 \text { and } 6 y+7<-5$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

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.