What are the solutions to (x-2)(x+4) = 0?
- A. -4 and 2
- B. -3 and 1
- C. -2 and 4
- D. -1 and 1
- E. -1 and 3
Correct Answer & Rationale
Correct Answer: A
To solve the equation (x-2)(x+4) = 0, we apply the zero product property, which states that if a product of factors equals zero, at least one of the factors must equal zero. Setting each factor to zero gives us the equations x - 2 = 0 and x + 4 = 0. Solving these yields x = 2 and x = -4, confirming that the solutions are -4 and 2. Options B, C, D, and E provide incorrect pairs of solutions that do not satisfy the original equation when substituted back in. Each of these pairs results in non-zero products for the factors, thus failing to meet the requirement of the equation.
To solve the equation (x-2)(x+4) = 0, we apply the zero product property, which states that if a product of factors equals zero, at least one of the factors must equal zero. Setting each factor to zero gives us the equations x - 2 = 0 and x + 4 = 0. Solving these yields x = 2 and x = -4, confirming that the solutions are -4 and 2. Options B, C, D, and E provide incorrect pairs of solutions that do not satisfy the original equation when substituted back in. Each of these pairs results in non-zero products for the factors, thus failing to meet the requirement of the equation.
Other Related Questions
Quadrilateral ABCD satisfies the following conditions: Side AB is parallel to side CD, Side BC is not parallel to side AD. Which term is the best classification for quadrilateral ABCD?
- A. Parallelogram
- B. Rectangle
- C. Rhombus
- D. Square
- E. Trapezoid
Correct Answer & Rationale
Correct Answer: E
Quadrilateral ABCD has one pair of parallel sides (AB and CD), which defines it as a trapezoid. Option A, parallelogram, is incorrect because both pairs of opposite sides must be parallel. Option B, rectangle, is a specific type of parallelogram with right angles, so it also requires two pairs of parallel sides. Option C, rhombus, similarly demands both pairs of opposite sides to be parallel, along with equal side lengths. Option D, square, is a special type of rectangle and rhombus, necessitating both pairs of parallel sides and equal side lengths. Thus, the only classification that fits is trapezoid.
Quadrilateral ABCD has one pair of parallel sides (AB and CD), which defines it as a trapezoid. Option A, parallelogram, is incorrect because both pairs of opposite sides must be parallel. Option B, rectangle, is a specific type of parallelogram with right angles, so it also requires two pairs of parallel sides. Option C, rhombus, similarly demands both pairs of opposite sides to be parallel, along with equal side lengths. Option D, square, is a special type of rectangle and rhombus, necessitating both pairs of parallel sides and equal side lengths. Thus, the only classification that fits is trapezoid.
A temperature of F degrees Fahrenheit will be converted to C degrees Celsius. Given F = 9/5C + 32, which of the following expressions represents that temperature in degrees Celsius?
- A. 5/9(F-32)
- B. 5/9F-32
- C. 9/5(F-32)
- D. 9/5(F+32)
- E. 9/5F+32
Correct Answer & Rationale
Correct Answer: A
To convert Fahrenheit (F) to Celsius (C), the formula is rearranged from F = 9/5C + 32 to isolate C. Starting with F = 9/5C + 32, subtracting 32 from both sides gives F - 32 = 9/5C. Multiplying both sides by 5/9 yields C = 5/9(F - 32), which matches option A. Option B (5/9F - 32) incorrectly places 32 outside the parentheses, misrepresenting the conversion. Option C (9/5(F - 32)) incorrectly applies the conversion factor, while D (9/5(F + 32)) and E (9/5F + 32) misapply the formula entirely by not correctly isolating C.
To convert Fahrenheit (F) to Celsius (C), the formula is rearranged from F = 9/5C + 32 to isolate C. Starting with F = 9/5C + 32, subtracting 32 from both sides gives F - 32 = 9/5C. Multiplying both sides by 5/9 yields C = 5/9(F - 32), which matches option A. Option B (5/9F - 32) incorrectly places 32 outside the parentheses, misrepresenting the conversion. Option C (9/5(F - 32)) incorrectly applies the conversion factor, while D (9/5(F + 32)) and E (9/5F + 32) misapply the formula entirely by not correctly isolating C.
Josh takes 6 hours to paint a room. Margaret can paint the same room in 4 hours. Assuming their individual rates do not change, how long will it take them to paint the room together?
- A. 1.5 hours
- B. 2.4 hours
- C. 4.8 hours
- D. 5 hours
- E. 10 hours
Correct Answer & Rationale
Correct Answer: B
To determine how long it takes Josh and Margaret to paint the room together, we first calculate their individual rates. Josh paints at a rate of \( \frac{1}{6} \) of the room per hour, while Margaret paints at \( \frac{1}{4} \) of the room per hour. Combined, their rates are: \[ \frac{1}{6} + \frac{1}{4} = \frac{2}{12} + \frac{3}{12} = \frac{5}{12} \] This means together they paint \( \frac{5}{12} \) of the room per hour. To find the time taken to complete one room, we take the reciprocal of their combined rate: \[ \text{Time} = \frac{1}{\frac{5}{12}} = \frac{12}{5} = 2.4 \text{ hours} \] Option A (1.5 hours) is too short, as it implies a higher combined rate than possible. Option C (4.8 hours) suggests they are slower than working alone, which is incorrect. Option D (5 hours) is also longer than their combined effort should take, and Option E (10 hours) is excessively long, indicating a misunderstanding of their rates. Thus, 2.4 hours accurately reflects their collaborative efficiency.
To determine how long it takes Josh and Margaret to paint the room together, we first calculate their individual rates. Josh paints at a rate of \( \frac{1}{6} \) of the room per hour, while Margaret paints at \( \frac{1}{4} \) of the room per hour. Combined, their rates are: \[ \frac{1}{6} + \frac{1}{4} = \frac{2}{12} + \frac{3}{12} = \frac{5}{12} \] This means together they paint \( \frac{5}{12} \) of the room per hour. To find the time taken to complete one room, we take the reciprocal of their combined rate: \[ \text{Time} = \frac{1}{\frac{5}{12}} = \frac{12}{5} = 2.4 \text{ hours} \] Option A (1.5 hours) is too short, as it implies a higher combined rate than possible. Option C (4.8 hours) suggests they are slower than working alone, which is incorrect. Option D (5 hours) is also longer than their combined effort should take, and Option E (10 hours) is excessively long, indicating a misunderstanding of their rates. Thus, 2.4 hours accurately reflects their collaborative efficiency.
Connor sprinted 55 yards in 6.25 seconds. What was Connor's average speed in miles per hour?
- A. 6
- B. 9
- C. 15
- D. 18
- E. 26
Correct Answer & Rationale
Correct Answer: D
To find Connor's average speed in miles per hour, we first convert 55 yards to miles. There are 1,760 yards in a mile, so 55 yards is approximately 0.0312 miles. Next, we convert 6.25 seconds to hours by dividing by 3,600 (the number of seconds in an hour), resulting in about 0.001736 hours. Average speed is calculated by dividing distance by time: 0.0312 miles / 0.001736 hours ≈ 18 mph. Option A (6 mph) and B (9 mph) underestimate Connor's speed, while C (15 mph) is also too low. E (26 mph) overestimates it. Thus, 18 mph is the accurate average speed.
To find Connor's average speed in miles per hour, we first convert 55 yards to miles. There are 1,760 yards in a mile, so 55 yards is approximately 0.0312 miles. Next, we convert 6.25 seconds to hours by dividing by 3,600 (the number of seconds in an hour), resulting in about 0.001736 hours. Average speed is calculated by dividing distance by time: 0.0312 miles / 0.001736 hours ≈ 18 mph. Option A (6 mph) and B (9 mph) underestimate Connor's speed, while C (15 mph) is also too low. E (26 mph) overestimates it. Thus, 18 mph is the accurate average speed.