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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.

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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.

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Most popular questions from this chapter

What Is Wrong? One step in the following list contains an error. Identify it and explain what is wrong. $$ \begin{aligned} x &=2 \\ 3 x-2 x &=2 \\ 3 x &=2 x+2 \\ x^{2}+3 x &=x^{2}+2 x+2 \\ x^{2}+3 x-10 &=x^{2}+2 x-8 \\ (x-2)(x+5) &=(x-2)(x+4) \\ x+5 &=x+4 \\ 1 &=0 \end{aligned} $$

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