Problem 6 Solve each radical equation. $... [FREE SOLUTION]

Chapter 10: Problem 6

Solve each radical equation. $$\sqrt{2 x+5}+11=6$$

Short Answer

Expert verified

The equation has no solution due to the negative square root.

Step by step solution

Isolate the square root term

Subtract 11 from both sides to isolate the square root term on one side of the equation: $ \sqrt{2x+5} = 6 - 11 $ which simplifies down to $ \sqrt{2x+5} = -5 $

Check if the equation is valid

From the previous step, it鈥檚 found that the square root term equals to -5. This situation isn鈥檛 possible since the square root of any real number is always non-negative. Hence, the equation has no solution.

Declare the conclusion

There are no solutions because the square root of a number can't be negative. The original equation is therefore impossible to satisfy.

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Key Concepts

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

Solving Radical Equations

Radical equations include variables within a radical, often a square root. Solving them involves isolating the radical and then eliminating it by squaring both sides. This transforms the equation into a polynomial equation, making it easier to solve.

Here鈥檚 the general approach:

Isolate the Radical: Move other terms to the other side to focus on the radical expression.
Square Both Sides: By squaring, the radical is eliminated, leaving a simpler equation.
Solve the Resulting Equation: This can involve basic solving methods or factoring, depending on the complexity.
Verify Solutions: Some solutions may not work due to the squaring process. It鈥檚 essential to check if they satisfy the original equation.

For the given problem, isolation leads to $ \sqrt{2x+5} = -5 $. Since a square root can鈥檛 be negative, there鈥檚 no solution in this case.

Square Root Properties

Square roots have unique properties that are crucial in equations. They are key to understanding why certain equations may have no solutions. One vital property is that the square root of a non-negative number is always non-negative.

This property gives rise to important implications:

Positive Results: No real number's square root can ever be negative.
Non-Negative Input: You can only take square roots of non-negative numbers in the real number system.

A negative result when isolating a square root, as seen in $ \sqrt{2x+5} = -5 $, immediately means no real solutions exist, confirming why some problems yield no possible answers.

Algebraic Concepts

Algebra relies on rules that allow simplification and manipulation of expressions and equations. A solid understanding of algebraic concepts is crucial for solving radical equations.

Important concepts include:

Equality Manipulation: Adding, subtracting, multiplying, or dividing both sides of an equation by the same number keeps it balanced.
Inverse Operations: Operations that undo each other, like squaring and square-rooting, are often exploited to solve equations.
Checking Solutions: Especially for transformed equations, verifying is essential to ensure solutions satisfy the initial problem.

When working with radical equations, using these concepts helps in transforming and solving them effectively. In cases like $ \sqrt{2x+5} = -5 $, recognizing the impossibility through these principles saves effort and guides the correct conclusion of no solution.

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91影视

Solve each radical equation. $$\sqrt{2 x+5}+11=6$$

Short Answer

Step by step solution

Isolate the square root term

Check if the equation is valid

Declare the conclusion

Key Concepts

Solving Radical Equations

Square Root Properties

Algebraic Concepts

One App. One Place for Learning.

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