/*! 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 12 Find the real solutions, if any,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 12

Find the real solutions, if any, of each equation. $$ \sqrt{5 t+3}=-2 $$

Short Answer

Expert verified
No real solutions.

Step by step solution

01

Understand the Equation

The equation given is \( \sqrt{5t + 3} = -2 \). We first need to recognize the properties of square roots. A square root function, by definition, yields a non-negative result.
02

Analyze the RHS of the Equation

The equation \( \sqrt{5t + 3} = -2 \) suggests that the square root of some expression equals -2. However, since the square root of any real number cannot be negative, this equation has no real solutions.
03

Conclude the Solution

Since the equation implies a mathematical impossibility (a square root equaling a negative number), we conclude that there are no real solutions.

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 a fundamental concept in mathematics. They represent a value that, when multiplied by itself, gives the original number. The symbol for square root is \( \sqrt{...} \). For instance, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \). Note that the principal (main) square root always provides a non-negative number.

It's essential to remember that:\r
  • Square roots of negative numbers are not real鈥攖hey involve imaginary numbers.
    • \r
  • The square root function \( f(x) = \sqrt{x} \) is only defined for non-negative inputs in the realm of real numbers.

Therefore, square roots always produce non-negative results. This is a key point in solving radical equations, as we'll see.
real solutions
Real solutions refer to the values that solve an equation within the set of real numbers. When working with equations, we usually seek solutions that are real, tangible, and can be graphed on a number line.

For example, when solving the equation \( \sqrt{5t + 3} = k \), we look for values of \(t\) that make the equation true. For our solution to be real, a few conditions must be met:\r
  • The expression inside the square root (\( 5t + 3 \) in our case) must be non-negative.
    • \r
  • The right-hand side of the equation, \(k\), must also be non-negative, since the square root of a number is always non-negative.

Understanding that real solutions must fit these criteria helps identify when no real solution exists. In our problem, the equation \( \sqrt{5t + 3} = -2 \) cannot be satisfied by any real number since the square root cannot equal a negative value.
mathematical impossibility
A mathematical impossibility occurs when an equation or problem cannot be satisfied or solved within a given set of rules, such as the real number system. In our case, the equation \( \sqrt{5t + 3} = -2 \) represents such an impossibility.

Let's break down why:\r
  • The square root function results in non-negative outcomes by definition.
    • \r
  • Equating this non-negative result to a negative number, like -2, is logically impossible.

Thus, recognizing mathematical impossibilities, like in this problem, is crucial. It helps save time by directly concluding that no real solutions exist.
Whenever you encounter such conflicts in properties or definitions, it's a strong indication that the problem has no solution under the real number system. This understanding is essential for efficiently solving radical equations and various other mathematical problems.

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

Fat Content Suppose that you order a small McCafeTM chocolate shake ( \(15 \mathrm{~g}\) of fat) and an Artisan Grilled Chicken Sandwich" (7 g of fat) at McDonald's. How many oatmeal raisin cookies can you eat ( \(5 \mathrm{~g}\) of fat) and still keep the total fat content of your meal to no more than \(47 \mathrm{~g} ?\)

Find the real solutions, if any, of each equation. Use any method. $$ \frac{3 x}{x+2}+\frac{1}{x-1}=\frac{4-7 x}{x^{2}+x-2} $$

The sum of the consecutive integers \(1,2,3, \ldots, n\) is given by the formula \(\frac{1}{2} n(n+1) .\) How many consecutive integers, starting with \(1,\) must be added to get a sum of \(703 ?\)

Find all the real solutions of \(12 x^{7 / 5}+3 x^{2 / 5}=13 x^{9 / 10}\).

Find the real solutions, if any, of each equation. $$ 2 x^{2}-13 x+21=0 $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

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.