/*! 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 100 If a number is decreased by \(3,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 100

If a number is decreased by \(3,\) the principal square root of this difference is 5 less than the number. Find the number(s).

Short Answer

Expert verified
The number that satisfies the given conditions is \(7\).

Step by step solution

01

Formulate the equation

Translate the word problem into an algebraic equation. So, \(\sqrt{x - 3} = x - 5\).
02

Solve the square root equation

Square both sides of the equation to get rid of the square root. So, \((\sqrt{x - 3})^2 = (x - 5)^2\), which simplifies to \(x - 3 = x^2 - 10x + 25\). Standardize the formula to bring it to a quadratic equation form \(x^2 -11x + 28 = 0\).
03

Find the roots

Solve the quadratic equation using the formula for roots \((x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a})\), where \(a = 1,\) \(b = -11,\) and \(c = 28.\) This yields two potential solutions for \(x:\) \(x = 7\) and \(x = 4\).
04

Check the solutions

Substitute the potential solutions back into the original equation to verify. Substituting \(x = 7\) gives us \(\sqrt{7 - 3} = 7 - 5\) => \(\sqrt{4} = 2\), which is true. But substituting \(x = 4\) gives us \(\sqrt{4 - 3} = 4 - 5\) => \(\sqrt{1} = -1\), which is not correct. So, the only number that satisfies the given conditions is \(7\).

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!

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

When 3 times a number is subtracted from 4, the absolute value of the difference is at least 5. Use interval notation to express the set of all numbers that satisfy this condition.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'll win the contest if I can complete the crossword puzzle in 20 minutes plus or minus 5 minutes, so my winning time, \(x\), is modeled by \(|x-20| \leq 5\)

Solve equation by the method of your choice. $$ \frac{x-1}{x-2}+\frac{x}{x-3}=\frac{1}{x^{2}-5 x+6} $$

Solve equation by the method of your choice. $$ \sqrt{3} x^{2}+6 x+7 \sqrt{3}=0 $$

Use the Pythagorean Theorem and the square root property to solve. Express answers in simplified radical form. Then find a decimal approximation to the nearest tenth. A rectangular park is 4 miles long and 2 miles wide. How long is a pedestrian route that runs diagonally across the park?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

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.