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Chapter 1: Problem 120

Make up a radical equation that has no solution.

Short Answer

Expert verified
The equation \( \sqrt{x+1} = -x \) has no solution because the square root is always non-negative while the right-hand side can be negative.

Step by step solution

01

- Understand Radical Equations

A radical equation is an equation in which a variable is inside a radical, usually a square root. The general form of a radical equation is \( \sqrt{f(x)} = g(x) \).
02

- Create Example Equation

To create a radical equation that has no solution, we need to ensure the contents under the radical cannot equal the other side of the equation.\ For example, consider the equation \( \sqrt{x+1} = -x \).
03

- Analyze the Equation

Recall that the square root of a number is always non-negative, so \( \sqrt{x+1} \geq 0 \). However, the right-hand side of our equation is \( -x \), which can be negative or positive depending on the value of \( x \).
04

- Identify No Solution

For the equation \( \sqrt{x+1} = -x \) to have any solution, both sides of the equation must be non-negative. But since \( \sqrt{x+1} \geq 0 \) and \(-x \leq 0\), they can never be equal for any real value of \( x \). Therefore, the equation has no solution.

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Key Concepts

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

square root
The square root is a fundamental concept in mathematics. It deals with finding a number which, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because 3 * 3 = 9.
When working with square roots, especially in equations, it is important to remember that the square root of a non-negative number is also non-negative. This means if we have a square root, like \(\sqrt{a}\), it can never be negative as long as 'a' is a non-negative number. This property plays a critical role in solving radical equations.
no solution
An equation has no solution when no value for the variable can satisfy it. In the case of radical equations, this can often happen due to the properties of square roots.
For example, consider the equation \(\sqrt{x+1} = -x\). The left side is a square root and hence always non-negative. But the right side is \(-x\) which can be negative if 'x' is positive.
Since a non-negative number cannot equal a negative number, there is no real number 'x' that can satisfy this equation, so it has no solution.
variable under radical
In a radical equation, the variable is often inside a radical, such as within a square root. This makes the equation different and sometimes more challenging to solve compared to regular equations.
When solving radical equations, it is crucial to isolate the radical expression first and then square both sides of the equation.
However, care must be taken since squaring both sides can introduce extraneous solutions. Always check your solutions in the original equation. In our example, \(\sqrt{x+1} = -x\), even after isolating the radical and squaring both sides, we still find that no real 'x' satisfies this equation due to the properties of square roots and negative numbers.

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Most popular questions from this chapter

An air rescue plane averages 300 miles per hour in still air. It carries enough fuel for 5 hours of flying time. If, upon takeoff, it encounters a head wind of \(30 \mathrm{mi} / \mathrm{h}\), how far can it fly and return safely? (Assume that the wind remains constant.)

An inequality is given. Write the inequality obtained by: (a) Adding 3 to both sides of the given inequality. (b) Subtracting 5 from both sides of the given inequality. (c) Multiplying both sides of the given inequality by 3 . (d) Multiplying both sides of the given inequality by -2. 1-2 x>5

Multiple Choice A chemist mixes 10 liters of a \(20 \%\) solution with \(x\) liters of a \(35 \%\) solution. Which of the following expressions represents the total number of liters in the mixture? (a) \(x\) (b) \(20-x\) (c) \(\frac{35}{x}\) (d) \(10+x\)

Critical Thinking You are the manager of a clothing store and have just purchased 100 dress shirts for $20.00 each. After 1 month of selling the shirts at the regular price, you plan to have a sale giving 40% off the original selling price. However, you still want to make a profit of 4 on each shirt at the sale price. What should you price the shirts at initially to ensure this? If, instead of 40% off at the sale, you give 50% off, by how much is your profit reduced?

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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