/*! 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 6 Solve the given equations. $$\... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 6

Solve the given equations. $$\sqrt{x+4}=3$$

Short Answer

Expert verified
The solution is \( x = 5 \).

Step by step solution

01

Understand the Equation

The given equation is \( \sqrt{x+4}=3 \). This is a square root equation where we need to eliminate the square root to solve for \( x \).
02

Square Both Sides

To resolve the square root, we square both sides of the equation. This gives us: \[(\sqrt{x+4})^2 = 3^2\]which simplifies to:\[x+4 = 9\]
03

Isolate the Variable

Now that we have \( x+4 = 9 \), we need to solve for \( x \). Subtract 4 from both sides of the equation to isolate \( x \):\[x = 9 - 4\]which simplifies to:\[x = 5\]

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.

Square Root Elimination
Square root elimination is the process of removing a square root sign in an equation to simplify and solve it. In equations where a variable is under the square root, like \( \sqrt{x+4}=3 \), the first goal is to eliminate the square root to make the equation easier to manage. You can achieve this by squaring both sides. Squaring a square root essentially "cancels out" the root, leaving you with a linear expression. For example:
  • Start with \( \sqrt{x+4} = 3 \).
  • Square both sides: \((\sqrt{x+4})^2 = 3^2\).
  • This simplifies to \(x+4 = 9\).
By following this method, you've transformed the equation from a square root equation to a simpler one that's easier to solve.
Equation Simplification
Equation simplification involves making an equation easier to solve by reducing it to a simpler or more familiar form. After removing the square root in the equation, you typically end up with a linear equation, as in the example \(x + 4 = 9\). The next steps are straightforward:
  • Focus on isolating the variable 鈥� in this case, \(x\).
  • Consider the operations applied to \(x\) and undo them logically.
In the process:
  • Identify that "+4" is added to \(x\).
  • To isolate \(x\), subtract 4 from both sides: \(x + 4 - 4 = 9 - 4\).
The result is \(x = 5\), a simple solution achieved through systematic simplification.
Algebraic Manipulation
Algebraic manipulation involves using algebraic rules to rearrange and solve equations efficiently. It plays a crucial role in solving square root equations and linear equations alike. With the equation \(x + 4 = 9\), algebraic manipulation helps us focus on getting the variable by itself.
  • Understand what operations need to be reversed 鈥� here, it's the addition of 4.
  • Apply the inverse operation, which in this case is subtraction, to both sides.
The techniques learned here are not just for square root equations but apply broadly to all types of algebraic equations. This makes algebraic manipulation a vital skill, enabling you to handle more complex math problems confidently. Remember, each step in manipulation should maintain equality, bringing you closer to isolating the desired variable.

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 given problems algebraically. In the theory dealing with optical interferometers, the equation \(\sqrt{F}=2 \sqrt{p} /(1-p)\) is used. Solve for \(p\) if \(F=16\).

Solve the given equations algebraically. In Exercise \(10,\) explain your method. $$x^{6}+7 x^{3}=8$$

Solve the indicated systems of equations algebraically. It is necessary to set up the systems of equations properly. A set of equal electrical resistors in series has a total resistance (the sum of the resistances) of \(78.0 \Omega\). Another set of two fewer equal resistors in series also has a total resistance of \(78.0 \Omega .\) If each resistor in the second set is \(1.3 \Omega\) greater than each of the first set, how many are in each set?

Solve the given equations algebraically and check the solutions with a calculator $$x^{4}-20 x^{2}+64=0$$

Solve the given systems of equations graphically by using a calculator. Find all values to at least the nearest 0.1. $$\begin{aligned} &x^{2}+y^{2}=7\\\ &y(x+2)=3 \end{aligned}$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Discrete 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.