/*! 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 18 Solve each equation. $$ \sqr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 18

Solve each equation. $$ \sqrt{4 x-2}=\sqrt{3 x+5} $$

Short Answer

Expert verified
x = 7

Step by step solution

01

Square both sides to eliminate the square roots

The equation has square roots on both sides, so start by squaring both sides to eliminate the square roots.\[ (\text{sqrt}(4x - 2))^2 = (\text{sqrt}(3x + 5))^2 \]This simplifies to:\[ 4x - 2 = 3x + 5 \]
02

Solve for x

Now, isolate the variable x. First, subtract 3x from both sides:\[ 4x - 3x - 2 = 5 \]This simplifies to:\[ x - 2 = 5 \]Next, add 2 to both sides to solve for x:\[ x = 7 \]
03

Verify the solution

It's important to verify the solution by substituting it back into the original equation to ensure both sides are equal. Substitute \( x = 7 \) back into the original equation:\[ \text{sqrt}(4(7) - 2) = \text{sqrt}(3(7) + 5) \]This simplifies to:\[ \text{sqrt}(28 - 2) = \text{sqrt}(21 + 5) \]\[ \text{sqrt}(26) = \text{sqrt}(26) \]Since both sides are equal, \( x = 7 \) is a valid solution.

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 roots
Square roots are fundamental in many areas of mathematics. The square root of a number is a value which, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because 3 multiplied by 3 equals 9. We represent the square root of a number using the radical symbol, 鈭�. When dealing with square root equations, our goal is often to eliminate the square root so we can solve for the unknown variable. This usually involves squaring both sides of the equation, as we'll demonstrate in our example.
equation solving
Equation solving is the process of finding the value of the unknown variable that satisfies the equation. Let's break down our example:

  • Step 1: To solve the given equation \[ \text{sqrt}(4x - 2) = \text{sqrt}(3x + 5) \], we first square both sides to eliminate the square roots. This gives us: \[ (\text{sqrt}(4x - 2))^2 = (\text{sqrt}(3x + 5))^2 \] which simplifies to \[ 4x - 2 = 3x + 5 \].
  • Step 2: Next, we isolate the variable x. Subtract 3x from both sides: \[ 4x - 3x - 2 = 5 \], simplifying to \[ x - 2 = 5 \]. Then, add 2 to both sides to solve for x: \[ x = 7 \].
Now, we can proceed to verification to ensure our solution is correct.
verification
Verification is an essential step to confirm that our solution is accurate. We substitute the found value of x back into the original equation to check if both sides are indeed equal. For our example:

  • Substitute x = 7 into the original equation: \[ \text{sqrt}(4(7) - 2) = \text{sqrt}(3(7) + 5) \], which simplifies to \[ \text{sqrt}(28 - 2) = \text{sqrt}(21 + 5) \].
  • This further simplifies to \[ \text{sqrt}(26) = \text{sqrt}(26) \].
Since both sides are equal, our solution x = 7 is verified to be correct. Always remember to verify your solution, as it helps avoid mistakes and confirms your answer.

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

Find each quotient. $$ \frac{5+i}{-i} $$

Multiply or divide as indicated. $$ \sqrt{-10} \cdot \sqrt{2} $$

If the sides of a triangle are \(\sqrt{65}\) in. \(\sqrt{35}\) in. , and \(\sqrt{26}\) in., which one of the following is the best estimate of its perimeter? A. 20 in. B. 26 in. C. 19 in. D. 24 in.

Solve each formula for the indicated variable. $$ r=\sqrt{\frac{M m}{F}} \text { for } M $$

Fill in the blank with the correct response: Because \(\frac{-5}{2-i}=-2-i,\) using the definition of division, we can check this to find that \((-2-i)(2-i)=\) _____.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

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.