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Chapter 7: Problem 90

Solve each equation. $$3+\sqrt{x^{2}-8 x}=0$$

Short Answer

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Step by step solution

01

Isolate the Square Root

First, isolate the square root on one side of the equation. To do this, subtract 3 from both sides:\[ \sqrt{x^{2} - 8x} = -3 \]
02

Analyze the Square Root

Recall that the square root function \( \sqrt{y} \) where \( y \geq 0 \), always gives a non-negative result. There are no real numbers that can make \( \sqrt{x^{2} - 8x} \) negative. Thus, we can conclude:
03

Explain Non-existent Solution

Since \[ \sqrt{x^{2} - 8x} = -3 \] has no real solution (a square root cannot equal a negative number), the equation has no real solutions.

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Key Concepts

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

Isolating Square Roots
One of the first steps in solving an equation that involves a square root is to isolate the square root term. This means getting the square root by itself on one side of the equation.

In the original exercise, the equation given was: 3 + \( \sqrt{x^{2} - 8x} \)=0 To isolate the square root, we subtract the constant term from both sides. This gives us: \( \sqrt{x^{2} - 8x} \)= -3. By isolating the square root, we set up the equation for further analysis.
Non-Negative Property of Square Roots
A key property of square roots is that they are always non-negative. The square root of any non-negative number gives a non-negative result.

For example, \( \sqrt{9} \)=3 and \( \sqrt{0} \)=0. Similarly, \( \sqrt{x^2 - 8x} \), irrespective of the value inside the square root, must also be non-negative.

This implies that \( \sqrt{x^2 - 8x} \) can never equal a negative number like -3. This understanding helps to identify when a given equation is invalid or has no real solutions.
Non-Existent Solutions in Algebra
In algebra, some equations do not have real solutions. This typically occurs when the math involved contradicts fundamental properties, like the non-negative property of square roots.

In the exercise, after isolating the square root term, we were left with: \( \sqrt{x^2 - 8x} \)= -3. Since a square root can never be negative, this equation has no solution.

Whenever you encounter a square root equaling a negative number in an equation, you can conclude that there are no real solutions to that equation. This highlights the importance of understanding core mathematical properties to identify non-existent solutions correctly.

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Most popular questions from this chapter

Solve each equation. $$(x-3)^{2 / 3}=-4$$

Solve each equation and check for extraneous solutions. $$\sqrt{2 x+5}+\sqrt{x+2}=1$$

Solve each equation. $$\sqrt[3]{2 w+3}=\sqrt[3]{w-2}$$

If we use the product rule to simplify \((-1)^{1 / 2} \cdot(-1)^{1 / 2},\) we get $$(-1)^{1 / 2} \cdot(-1)^{1 / 2}=(-1)^{1}=-1$$ If we use the power of a product rule, we get $$(-1)^{1 / 2} \cdot(-1)^{1 / 2}=(-1 \cdot-1)^{1 / 2}=1^{1 / 2}=1$$ Which of these computations is incorrect? Explain your answer.

Discussion Which of the following equations are identities? Explain your answers. a) \(\sqrt{9 x}=3 \sqrt{x}\) b) \(\sqrt{9+x}=3+\sqrt{x}\) c) \(\sqrt{x-4}=\sqrt{x}-2\) d) \(\sqrt{\frac{x}{4}}=\frac{\sqrt{x}}{2}\)
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