/*! 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 40 Find all real numbers \(x\) that... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 40

Find all real numbers \(x\) that satisfy the indicated equation. $$ x-7 \sqrt{x}+12=0 $$

Short Answer

Expert verified
The real numbers \(x\) that satisfy the equation \(x-7\sqrt{x}+12=0\) are \(x=8\) and \(x=9\).

Step by step solution

01

Isolate the square root term

First, we'll isolate the square root term by moving the remaining terms to the other side of the equation. \(7\sqrt{x}=x-12\)
02

Square both sides

Now, we will square both sides to eliminate the square root. \((7\sqrt{x})^2=(x-12)^2\)
03

Simplify the equation

After squaring both sides, simplify the equation: \(49x=(x^2-24x+144)\)
04

Rearrange into a quadratic equation

Move all terms to one side to form a quadratic equation: \(x^2-73x+144=0\)
05

Solve the quadratic equation

Now, we can solve the quadratic equation by factoring it. Observe that \(144=8\times 9\), and \(8+9=17\). We can rewrite the equation as: \(x^2-8x-9x+144=0\) Now factor by grouping: \(x(x-8)-9(x-8)=0\) \((x-8)(x-9)=0\)
06

Find the solutions

This gives us two possible solutions: \(x-8=0\) implies \(x=8\) or \(x-9=0\) implies \(x=9\)
07

Check for extraneous solutions

Plug both solutions back into the original equation to ensure they are valid: For \(x=8\): \[8-7\sqrt{8}+12=8-14+12=0\] For \(x=9\): \[9-7\sqrt{9}+12=9-21+12=0\] Both solutions satisfy the original equation. No extraneous solutions were encountered in this case.
08

Write down the final answer

So we have found two valid real number solutions for the given equation: The real numbers \(x\) that satisfy the equation \(x-7\sqrt{x}+12=0\) are \(x=8\) and \(x=9\).

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!

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

Suppose a savings account pays \(5 \%\) interest per year, compounded four times per year. If the savings account starts with \(\$ 600\), how many years would it take for the savings account to exceed \(\$ 1400 ?\)

Find all numbers \(x\) that satisfy the given equation. $$ \frac{\log _{9}(13 x)}{\log _{9}(4 x)}=2 $$

Suppose your vacuum cleaner makes a noise of 80 decibels and you normally speak at 60 decibels. (a) What is the ratio of the sound intensity of your vacuum cleaner to the sound intensity of your normal speech? (b) How many times louder does your vacuum cleaner seem as compared to your normal speech?

Explain why $$ 1+\log x=\log (10 x) $$ for every positive number \(x\)

Show that if \(0
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

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.