What are the solutions to the equation: x² - 10?
- A. ±5
- B. ±√10
- C. ±10
- D. ±10²
- E. ±20
Correct Answer & Rationale
Correct Answer: B
To solve the equation \( x^2 - 10 = 0 \), we first isolate \( x^2 \) by adding 10 to both sides, resulting in \( x^2 = 10 \). Taking the square root of both sides gives us \( x = \pm\sqrt{10} \), which corresponds to option B. Option A, \( \pm5 \), is incorrect as \( 5^2 = 25 \), not 10. Option C, \( \pm10 \), is also wrong because \( 10^2 = 100 \). Option D, \( \pm10^2 \), misinterprets the operation, yielding \( \pm100 \), which is not relevant here. Lastly, option E, \( \pm20 \), is incorrect since \( 20^2 = 400 \). Thus, only option B accurately represents the solutions to the equation.
To solve the equation \( x^2 - 10 = 0 \), we first isolate \( x^2 \) by adding 10 to both sides, resulting in \( x^2 = 10 \). Taking the square root of both sides gives us \( x = \pm\sqrt{10} \), which corresponds to option B. Option A, \( \pm5 \), is incorrect as \( 5^2 = 25 \), not 10. Option C, \( \pm10 \), is also wrong because \( 10^2 = 100 \). Option D, \( \pm10^2 \), misinterprets the operation, yielding \( \pm100 \), which is not relevant here. Lastly, option E, \( \pm20 \), is incorrect since \( 20^2 = 400 \). Thus, only option B accurately represents the solutions to the equation.
Other Related Questions
One online movie-streaming service costs $8 per month and charges $1.50 per movie. A second service costs $2 per month and charges $2 per movie. For what number of movies per month is the monthly cost of both services the same?
- A. 3
- B. 6
- C. 5
- D. 12
- E. 20
Correct Answer & Rationale
Correct Answer: D
To determine when the costs of both services are equal, we can set up equations based on the monthly fees and per-movie charges. For the first service: Cost = $8 + $1.50 * number of movies (m) Cost = $8 + 1.5m For the second service: Cost = $2 + $2 * number of movies (m) Cost = $2 + 2m Setting the two equations equal gives us: $8 + 1.5m = $2 + 2m Rearranging leads to: $6 = 0.5m m = 12 Thus, when 12 movies are rented, the costs are equal. Options A (3), B (6), and C (5) yield different costs, as they do not satisfy the equation. Option E (20) results in a higher cost for the second service, confirming that 12 is the only solution where both services cost the same.
To determine when the costs of both services are equal, we can set up equations based on the monthly fees and per-movie charges. For the first service: Cost = $8 + $1.50 * number of movies (m) Cost = $8 + 1.5m For the second service: Cost = $2 + $2 * number of movies (m) Cost = $2 + 2m Setting the two equations equal gives us: $8 + 1.5m = $2 + 2m Rearranging leads to: $6 = 0.5m m = 12 Thus, when 12 movies are rented, the costs are equal. Options A (3), B (6), and C (5) yield different costs, as they do not satisfy the equation. Option E (20) results in a higher cost for the second service, confirming that 12 is the only solution where both services cost the same.
Isabel earns $15.80 per hour for the first 8 hours she works each day. She earns 1.5 times her hourly rate for all time after the first 8 hours. How much does Isabel earn on a day when she works 8.5 hours?
- A. 126.4
- B. 138.25
- C. 189.6
- D. 201.45
- E. 237
Correct Answer & Rationale
Correct Answer: B
To determine Isabel's earnings for an 8.5-hour workday, first calculate her earnings for the first 8 hours at $15.80 per hour, which totals $126.40 (8 hours × $15.80/hour). For the additional 0.5 hours, she earns 1.5 times her hourly rate, which is $23.70 (1.5 × $15.80). Therefore, for the extra half hour, she earns $11.85 (0.5 hours × $23.70/hour). Adding these amounts together gives $138.25 ($126.40 + $11.85). Option A ($126.40) only accounts for the first 8 hours. Option C ($189.60) incorrectly assumes full-time pay without considering the overtime rate. Option D ($201.45) miscalculates the overtime pay, while Option E ($237) overestimates by not applying the correct hourly rates.
To determine Isabel's earnings for an 8.5-hour workday, first calculate her earnings for the first 8 hours at $15.80 per hour, which totals $126.40 (8 hours × $15.80/hour). For the additional 0.5 hours, she earns 1.5 times her hourly rate, which is $23.70 (1.5 × $15.80). Therefore, for the extra half hour, she earns $11.85 (0.5 hours × $23.70/hour). Adding these amounts together gives $138.25 ($126.40 + $11.85). Option A ($126.40) only accounts for the first 8 hours. Option C ($189.60) incorrectly assumes full-time pay without considering the overtime rate. Option D ($201.45) miscalculates the overtime pay, while Option E ($237) overestimates by not applying the correct hourly rates.
sqrt(45) is between what two consecutive whole numbers?
- A. 4 and 5
- B. 5 and 6
- C. 6 and 7
- D. 14 and 15
- E. 22 and 23
Correct Answer & Rationale
Correct Answer: C
To determine between which two consecutive whole numbers \(\sqrt{45}\) lies, we can evaluate the squares of whole numbers around it. Calculating, \(6^2 = 36\) and \(7^2 = 49\). Since \(36 < 45 < 49\), it follows that \(6 < \sqrt{45} < 7\). Therefore, \(\sqrt{45}\) is between 6 and 7. Option A (4 and 5) is incorrect as \(4^2 = 16\) and \(5^2 = 25\), which are both less than 45. Option B (5 and 6) is also wrong since \(5^2 = 25\) and \(6^2 = 36\) are still below 45. Option D (14 and 15) and Option E (22 and 23) are far too high, as \(14^2 = 196\) and \(22^2 = 484\) exceed 45.
To determine between which two consecutive whole numbers \(\sqrt{45}\) lies, we can evaluate the squares of whole numbers around it. Calculating, \(6^2 = 36\) and \(7^2 = 49\). Since \(36 < 45 < 49\), it follows that \(6 < \sqrt{45} < 7\). Therefore, \(\sqrt{45}\) is between 6 and 7. Option A (4 and 5) is incorrect as \(4^2 = 16\) and \(5^2 = 25\), which are both less than 45. Option B (5 and 6) is also wrong since \(5^2 = 25\) and \(6^2 = 36\) are still below 45. Option D (14 and 15) and Option E (22 and 23) are far too high, as \(14^2 = 196\) and \(22^2 = 484\) exceed 45.
What is the product of the two polynomials: (x - 5)(x² - 3x + 6)?
- A. x³ - 8x² + 21x - 30
- B. x³ - 8x² - 21x - 30
- C. x³ - 8x² - 9x - 30
- D. x³ + 8x² + 21x + 30
- E. x³ + 8x² - 9x + 30
Correct Answer & Rationale
Correct Answer: A
To find the product of the polynomials (x - 5)(x² - 3x + 6), we apply the distributive property (FOIL method). 1. Multiply x by each term in the second polynomial: - x * x² = x³ - x * (-3x) = -3x² - x * 6 = 6x 2. Multiply -5 by each term in the second polynomial: - -5 * x² = -5x² - -5 * (-3x) = 15x - -5 * 6 = -30 Combining these results yields: x³ + (-3x² - 5x²) + (6x + 15x) - 30 = x³ - 8x² + 21x - 30. Option A matches this result. Options B and C have incorrect signs for the x terms. Option D has incorrect signs for all terms, and option E has incorrect signs for the x² and x terms. Thus, only option A accurately represents the product of the polynomials.
To find the product of the polynomials (x - 5)(x² - 3x + 6), we apply the distributive property (FOIL method). 1. Multiply x by each term in the second polynomial: - x * x² = x³ - x * (-3x) = -3x² - x * 6 = 6x 2. Multiply -5 by each term in the second polynomial: - -5 * x² = -5x² - -5 * (-3x) = 15x - -5 * 6 = -30 Combining these results yields: x³ + (-3x² - 5x²) + (6x + 15x) - 30 = x³ - 8x² + 21x - 30. Option A matches this result. Options B and C have incorrect signs for the x terms. Option D has incorrect signs for all terms, and option E has incorrect signs for the x² and x terms. Thus, only option A accurately represents the product of the polynomials.