/*! 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 1 Is the equation \(3 x-\sqrt{2}=\... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 1

Is the equation \(3 x-\sqrt{2}=\sqrt{6}\) a radical equation? Explain your reasoning.

Short Answer

Expert verified
No, \(3 x -\sqrt{2} =\sqrt{6}\) is not a radical equation because the variable \(x\) is not under a radical.

Step by step solution

01

Identify any radicals in the equation

Look at each term and determine if any of them include the variable under a radical. In the equation \(3 x -\sqrt{2} =\sqrt{6}\), neither of the radical terms \(\sqrt{2}\) or \(\sqrt{6}\) contain the variable \(x\).
02

Identify if the variable is under the radical

A radical equation has the variable under a radical. The variable in this equation, \(x\), is not under a radical but is multiplied by 3.
03

Conclusion

Since the variable \(x\) is not under a radical, we can conclude that \(3 x -\sqrt{2} =\sqrt{6}\) is not a radical equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Radical Equations
Solving radical equations involves isolating the radical expression and then eliminating the radical by raising both sides of the equation to the power of the index of the radical. The result is an algebraic equation that can usually be solved through standard means.

Consider the following typical steps in solving a radical equation:
  • Isolate the radical expression on one side of the equation.
  • Raise both sides of the equation to the power that matches the index of the radical. For square roots, this would be the second power.
  • Solve the resulting algebraic equation.
  • Check your solutions by substituting them back into the original equation to ensure they do not result in extraneous solutions - solutions that are mathematically correct but do not satisfy the original radical equation.
Despite the simplicity of the process, careful attention is necessary to avoid common pitfalls, such as forgetting to check for extraneous solutions or incorrectly handling equations with multiple radicals.
Radicals in Algebra
Radicals in algebra, often represented by the root symbol, play a significant role in algebraic expressions and equations. A radical can indicate a square root when no index is shown, or a higher root when an index is present, such as the cube root represented by \( \sqrt[3]{...} \).

Understanding radicals is key for:
  • Manipulating and simplifying algebraic expressions that contain roots.
  • Using properties of exponents, since radicals can be expressed as exponents. For example, \( \sqrt{x} = x^{\frac{1}{2}} \).
  • Approaching complex problems that involve both polynomial and non-polynomial terms.
Let's remember that a radical alone does not make an equation a 'radical equation.' It's the presence of the variable under the radical that qualifies it as such.
Algebraic Equations
Algebraic equations are mathematical statements that show the equality of two expressions. They involve constants, variables, and various operations such as addition, subtraction, multiplication, and division, as well as potentially more complex operations like taking roots or raising to powers.

Key features in understanding algebraic equations include:
  • Knowing the order of operations to correctly solve equations.
  • Being able to simplify expressions before solving.
  • Recognizing different types of equations, such as linear, quadratic, or higher-degree polynomial equations, and applying the appropriate solving techniques.
  • Interpreting solutions in the appropriate context, whether they are real, complex, or no solution at all.
Equations like \(3 x - \sqrt{2} = \sqrt{6}\), although they contain radicals, are more straightforward because the variable is not under a radical, thus making them algebraic rather than radical equations.

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Most popular questions from this chapter

\(f(x)=\sqrt[3]{x^2+10 x}, g(x)=\frac{1}{4} f(-x)+6\)

In Exercises 35鈥�46, determine whether the inverse of \(f\) is a function. Then find the inverse. $$ f(x)=-\sqrt[3]{\frac{2 x+4}{3}} $$

Consider the function \(f(x)=-x\). a. Graph \(f(x)=-x\) and explain why it is its own inverse. Also, verify that \(f(x)=-x\) is its own inverse algebraically. b. Graph other linear functions that are their own inverses. Write equations of the lines you graphed. c. Use your results from part (b) to write a general equation describing the family of linear functions that are their own inverses.

\(f(x)=\sqrt{2 x^2+x+1}\)

MULTIPLE REPRESENTATIONS The terminal velocity \(v_t\) (in feet per second) of a skydiver who weighs 140 pounds is given by $$ v_t=33.7 \sqrt{\frac{140}{A}} $$ where \(A\) is the cross-sectional surface area (in square feet) of the skydiver. The table shows the terminal velocities (in feet per second) for various surface areas (in square feet) of a skydiver who weighs 165 pounds. \begin{tabular}{|c|c|} \hline Cross-sectional surface area, \(\boldsymbol{A}\) & Terminal velocity, \(\boldsymbol{v}_{\boldsymbol{t}}\) \\ \hline 1 & \(432.9\) \\ 3 & \(249.9\) \\ 5 & \(193.6\) \\ 7 & \(163.6\) \\ \hline \end{tabular} a. Which skydiver has a greater terminal velocity for each value of \(A\) ? b. Describe how the different values of \(A\) given in the table relate to the possible positions of the falling skydiver.
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