Questions and Answers
What is lesson 25 about?
Solving radical equations.
Why did Joey use the exponent of 1/4 in line 2?
Joey needed to rewrite the radical as an exponent to solve the equation, and an exponent of 1/4 is equivalent to the fourth root of a number.
What is the value of x in the given equation √7x-12 = x?
x = 3 and x = 4.
What is the solution to the radical equation 2(4)√x^3 - 4 = 12?
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Which of the solutions are extraneous in the equation √-2x+12 = x-6?
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Which of the solutions are extraneous in the equation √x = 2 - x?
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What are the valid solutions to the equation x = √x+6?
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What is the solution to the equation √7x-10 = x?
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What is the value of x in the equation √2x+15 - x = 0?
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What is the solution to the equation 3√x+1 = 15?
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Why did Phoebe use the exponent of 1/3 in line 2?
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What mistake did Anna make during the solution process?
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Which of the solutions are extraneous in the equation √4x-8 = x-5?
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What is lesson 26 about?
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Which statements about f(x) are true?
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Which statements are true about the function represented by the graph of f(x)?
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Which statements about f(x) are true?
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Which statements are true about the function represented by this graph?
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For the function f(x) = 3√x+4, as x approaches infinity, f(x) approaches ________
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Which statements about the function f(x) = 3√x-3 + 1 are true?
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Which statements about g(x) = (3)√x+4 are true?
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Which statement about g(x) = (3)√x+2 + 2 is true?
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Study Notes
Radical Equations Overview
- Lesson 25 focuses on solving radical equations.
- Exponents can be used to rewrite radicals, such as 1/4 for the fourth root.
- Solutions to various radical equations can yield numerical results such as x = 3 and x = 4 from √7x-12 = x.
Solving Exemplified Equations
- From the equation 2(4)√x^3 - 4 = 12, the solution is x = 16.
- In the case of (3)√x^2 = 4, the steps include rewriting the expression and using reciprocal exponents during simplification.
Extraneous Solutions
- In the equation √-2x+12 = x-6, only x = 4 is an extraneous solution (valid solution: x = 6).
- For √x = 2 - x, similarly only x = 4 is extraneous (valid solution: x = 1).
Additional Example Solutions
- The valid solution for x = √x+6 is x = 3.
- Solving √7x-10 = x gives solutions x = 2 and x = 5.
- For √2x+15 - x = 0, the solution is x = 5.
- The equation 3√x+1 = 15 results in x = 24.
Errors and Corrections
- Phoebe utilized the exponent of 1/3 for cube roots, similar to how Joey used 1/4 for fourth roots.
- Anna's mistake involved failing to raise both sides of her equation to the power of 2, which is necessary to maintain equality.
Graphing Radical Functions
- Lesson 26 discusses graphing techniques for radical functions.
- For the graph of f(x), key features include:
- A range of [−3, ∞) and a domain of [−4, ∞).
- The x-intercept at (5, 0) and y-intercept at (0, −1).
Characteristics of Graphs
- Functions can often lack maximum or minimum values; for instance, one function has no maximum and a minimum of -3.
- Another function shows that as x approaches negative infinity, f(x) approaches infinity, while it may approach negative infinity as x approaches positive infinity.
Function Behavior at Infinity
- For f(x) = 3√x+4, f(x) tends towards infinity as x approaches infinity.
- The function f(x) = 3√x-3 + 1 has a minimum of 1, with its domain starting at 3 and extending to infinity.
General Properties of Radical Functions
- Functions like g(x) = (3)√x+4 have a domain and range consisting of all real numbers, indicating they are unbounded.
- As x approaches negative or positive infinity, g(x) behaves accordingly, approaching negative infinity for negative x and positive infinity for positive x.
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Description
This quiz covers the key concepts and problem-solving techniques related to radical equations. It includes examples, solutions, and discussions on extraneous solutions. Test your understanding of rewriting radicals and finding valid solutions through various exercises.
