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Questions and Answers
What are the solutions for the equation $x²-9x+18 = 0$?
What are the solutions for the equation $x²-9x+18 = 0$?
- x = -3, 6
- x = 3, 9
- x = 3, 6 (correct)
- x = -3, -6
- x = 3, -6
What is the average speed for a round trip between two cities if the speeds are 600 miles/hour and 300 miles/hour?
What is the average speed for a round trip between two cities if the speeds are 600 miles/hour and 300 miles/hour?
- 350 miles/hour
- 500 miles/hour
- 300 miles/hour
- 450 miles/hour
- 400 miles/hour (correct)
What is $g(f(x))$ if $f(x) = 7x² - 3$ and $g(y) = 2y + 9$?
What is $g(f(x))$ if $f(x) = 7x² - 3$ and $g(y) = 2y + 9$?
- 14x² - 3
- 14x² + 3 (correct)
- 7y² - 3
- 2x + 9
- 14y² + 3
Comparing Quantity A: x and Quantity B: 2x, which statement is true?
Comparing Quantity A: x and Quantity B: 2x, which statement is true?
What is the value of n if one less than three-eighths of n is the third prime integer?
What is the value of n if one less than three-eighths of n is the third prime integer?
If $(x - 5)² = 900$, what is one possible value for x?
If $(x - 5)² = 900$, what is one possible value for x?
What is the value of x if $3x + y = 13$ and $x - 2y = -12$?
What is the value of x if $3x + y = 13$ and $x - 2y = -12$?
For the equation $(b * b⁴ * b⁷)^{1/2}/(b³ * b^{x}) = b⁵$, if b is not negative, then what is x?
For the equation $(b * b⁴ * b⁷)^{1/2}/(b³ * b^{x}) = b⁵$, if b is not negative, then what is x?
If $6h - 2g = 4g + 3h$, what is h in terms of g?
If $6h - 2g = 4g + 3h$, what is h in terms of g?
What is the value of x in the equations $4x + 3y = 6$ and $2x + 2y = 4$?
What is the value of x in the equations $4x + 3y = 6$ and $2x + 2y = 4$?
If Sally is 2 years younger than Abby, Daisy is 5 years older than Tracy, and Abby is 6 years older than Tracy, which quantity is greater?
If Sally is 2 years younger than Abby, Daisy is 5 years older than Tracy, and Abby is 6 years older than Tracy, which quantity is greater?
Which is greater: Quantity A: $4^3$ or Quantity B: $3^4$?
Which is greater: Quantity A: $4^3$ or Quantity B: $3^4$?
If $a = 1/3b$ and $b = 4c$, then what is $a - b + c$ in terms of c?
If $a = 1/3b$ and $b = 4c$, then what is $a - b + c$ in terms of c?
What is the value of $2y - 1$ if $x = 4$ and $y = 3x + 5$?
What is the value of $2y - 1$ if $x = 4$ and $y = 3x + 5$?
How can $25x²-36y²$ be factored?
How can $25x²-36y²$ be factored?
How much greater is the degree measure of the sector representing workers in the construction industry compared to the financial industry if the values are 18° and 25°?
How much greater is the degree measure of the sector representing workers in the construction industry compared to the financial industry if the values are 18° and 25°?
If $[(7/8)]^n = \sqrt{([(7/8)]^5)},$ what is the value of n?
If $[(7/8)]^n = \sqrt{([(7/8)]^5)},$ what is the value of n?
Which is greater, Quantity A: |x| or Quantity B: $x^3$?
Which is greater, Quantity A: |x| or Quantity B: $x^3$?
If y is a real number such that $y ≥ 1$, compare the quantities: Quantity A: $(y²)(y⁴)$ and Quantity B: $y⁸$.
If y is a real number such that $y ≥ 1$, compare the quantities: Quantity A: $(y²)(y⁴)$ and Quantity B: $y⁸$.
Bill can build k toys in 6 hours and Bob can build k toys in 3 hours. How long would it take them to build 4k toys together?
Bill can build k toys in 6 hours and Bob can build k toys in 3 hours. How long would it take them to build 4k toys together?
If Kim is twice as old as Claire, Nick is 3 years older than Claire, and their combined age equals 81, how old is Nick?
If Kim is twice as old as Claire, Nick is 3 years older than Claire, and their combined age equals 81, how old is Nick?
What is the next consecutive odd integer after the sum of two consecutive odd integers that is equal to 32?
What is the next consecutive odd integer after the sum of two consecutive odd integers that is equal to 32?
In a comparison, Quantity A: $x²$ and Quantity B: $x³$, which is greater?
In a comparison, Quantity A: $x²$ and Quantity B: $x³$, which is greater?
Given one root of the equation $x² + kx - 12 = 0$ is 3, compare Quantity A: the value of k and Quantity B: -1.
Given one root of the equation $x² + kx - 12 = 0$ is 3, compare Quantity A: the value of k and Quantity B: -1.
What digit appears in the units place when $2^{102}$ is multiplied out?
What digit appears in the units place when $2^{102}$ is multiplied out?
Which best describes the relationship between $(x+y)³$ and $x³+y³$ if $x, y
eq 0$?
Which best describes the relationship between $(x+y)³$ and $x³+y³$ if $x, y eq 0$?
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Study Notes
Algebra and Equations
- Quadratic equation: x² - 9x + 18 = 0, roots are x = 3 and x = 6.
- Average speed calculated using the formula for two speeds: (2 * Speed1 * Speed2) / (Speed1 + Speed2) results in an average speed of 400 miles/hour for a jet returning from City 2 to City 1.
- Composition of functions: g(f(x)) results in 14x² + 3 for f(x) = 7x² - 3 and g(y) = 2y + 9.
Quantitative Comparisons
- Comparison of x (Quantity A) and 2x (Quantity B) shows that the relationship cannot be determined as x could be positive or negative.
- Problem involving prime numbers states that if one less than three-eighths of a positive integer n equals the third prime (5), then n is 16.
Solving Equations
- Equation (x - 5)² = 900 indicates potential solutions: x = 35 or x = -25.
- System of equations: Solving 3x + y = 13 and x - 2y = -12 yields value x = 2.
- Expression manipulation reveals if 6h - 2g = 4g + 3h, then h = 2g.
Factorization and Age Problems
- The factorization of 25x² - 36y² results in (5x - 6y)(5x + 6y).
- Age relationships indicate Sally (younger than Abby) and Daisy (older than Tracy) leads to a conclusion that Daisy's age is greater than Sally's.
Exponential and Quantitative Comparisons
- Comparing powers: Quantity A (4³) is less than Quantity B (3⁴).
- Given relationships a = 1/3b and b = 4c, the expression a - b + c simplifies to -5/3c.
Toys Production Problem
- Work rate problem: Bob and Bill together can build 4k toys in 8 hours based on their respective speeds (k toys in 6 hours for Bill and k in 3 hours for Bob).
Additional Challenges
- Finding the next consecutive odd integer after the sum of two consecutive odd integers (sum = 32) leads to identifying the next integer as 19.
- The equation x² + kx - 12 = 0 having one root as 3 determines that k > -1.
Exponents and Digits
- The units digit of 2^102 parallels observing the cycle of units digits in powers of 2, confirming it is 4.
- The relationship between (x + y)³ and x³ + y³ indicates expansion to include terms related to both x and y if neither is equal to 0.
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