Factor the expression completely: 45bcx - 10ax
- A. 5x(9bc - 2a)
- B. 5(9bc - 2a)
- C. x(45bc - 10a)
- D. 5x(9bc + 2a)
Correct Answer & Rationale
Correct Answer: A
To factor the expression 45bcx - 10ax completely, we start by identifying the greatest common factor (GCF). The GCF of the coefficients 45 and 10 is 5, and both terms contain the variable x. Thus, we can factor out 5x, resulting in 5x(9bc - 2a). Option A accurately reflects this factorization. Option B lacks the variable x, which is essential in the original expression. Option C incorrectly factors out only x, missing the GCF of 5. Option D alters the sign of the second term, which does not represent the original expression correctly.
To factor the expression 45bcx - 10ax completely, we start by identifying the greatest common factor (GCF). The GCF of the coefficients 45 and 10 is 5, and both terms contain the variable x. Thus, we can factor out 5x, resulting in 5x(9bc - 2a). Option A accurately reflects this factorization. Option B lacks the variable x, which is essential in the original expression. Option C incorrectly factors out only x, missing the GCF of 5. Option D alters the sign of the second term, which does not represent the original expression correctly.
Other Related Questions
The graph of the equation y = x^2 + 4x - 5 is shown on the grid. Which statement is true when y = 0?
- A. x= -5 and x=1
- B. x= -2
- C. x= -5 and x = 0
- D. x= -9
Correct Answer & Rationale
Correct Answer: A
To find the values of x when y = 0, we need to solve the equation \(x^2 + 4x - 5 = 0\). Factoring this quadratic gives \((x + 5)(x - 1) = 0\), leading to the solutions \(x = -5\) and \(x = 1\). Option A correctly identifies these solutions. Option B states \(x = -2\), which is not a solution to the equation. Option C suggests \(x = -5\) and \(x = 0\); while it includes one correct solution, \(x = 0\) is incorrect. Option D claims \(x = -9\), which does not satisfy the equation. Thus, only option A accurately reflects the solutions when y = 0.
To find the values of x when y = 0, we need to solve the equation \(x^2 + 4x - 5 = 0\). Factoring this quadratic gives \((x + 5)(x - 1) = 0\), leading to the solutions \(x = -5\) and \(x = 1\). Option A correctly identifies these solutions. Option B states \(x = -2\), which is not a solution to the equation. Option C suggests \(x = -5\) and \(x = 0\); while it includes one correct solution, \(x = 0\) is incorrect. Option D claims \(x = -9\), which does not satisfy the equation. Thus, only option A accurately reflects the solutions when y = 0.
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The table shows the costs of items Anna purchased at an art supply store for her art class.
What was the total cost of the items that Anna purchased?
Correct Answer & Rationale
Correct Answer: 128.65
To find the total cost of Anna's purchases, add the individual prices of each item she bought. Summing these values accurately gives a total of 128.65. Other options are incorrect because they result from either miscalculating the addition or omitting an item from the total. For instance, if an item was not included, the total would be lower than 128.65. Conversely, adding extra costs or misreading the prices could lead to an inflated total. Therefore, precise addition of all listed costs is essential to arrive at the correct total.
To find the total cost of Anna's purchases, add the individual prices of each item she bought. Summing these values accurately gives a total of 128.65. Other options are incorrect because they result from either miscalculating the addition or omitting an item from the total. For instance, if an item was not included, the total would be lower than 128.65. Conversely, adding extra costs or misreading the prices could lead to an inflated total. Therefore, precise addition of all listed costs is essential to arrive at the correct total.
The equation d/f = g represents gallons of gasoline used, g, in terms of distance traveled in miles, d, and fuel efficiency, / miles per gallon of gasoline. Which combination of distance traveled and fuel efficiency uses 3 gallons of gasoline?
- A. 7 miles and 21 miles per gallon
- B. 57 miles and 19 miles per gallon
- C. 23 miles and 20 miles per gallon
- D. 32 miles and 35 miles per gallon
Correct Answer & Rationale
Correct Answer: B
To determine which combination uses 3 gallons of gasoline, we can rearrange the equation d/f = g to find d = g * f. For g = 3 gallons, we calculate d for each option. A: 7 miles and 21 mpg results in d = 3 * 21 = 63 miles, which is incorrect. B: 57 miles and 19 mpg gives d = 3 * 19 = 57 miles, matching the distance traveled. C: 23 miles and 20 mpg leads to d = 3 * 20 = 60 miles, which is incorrect. D: 32 miles and 35 mpg results in d = 3 * 35 = 105 miles, which is also incorrect. Only option B correctly satisfies the equation for 3 gallons of gasoline used.
To determine which combination uses 3 gallons of gasoline, we can rearrange the equation d/f = g to find d = g * f. For g = 3 gallons, we calculate d for each option. A: 7 miles and 21 mpg results in d = 3 * 21 = 63 miles, which is incorrect. B: 57 miles and 19 mpg gives d = 3 * 19 = 57 miles, matching the distance traveled. C: 23 miles and 20 mpg leads to d = 3 * 20 = 60 miles, which is incorrect. D: 32 miles and 35 mpg results in d = 3 * 35 = 105 miles, which is also incorrect. Only option B correctly satisfies the equation for 3 gallons of gasoline used.
Laura walks every evening on the edges of a sports field near her house. The field is in the shape of a rectangle 300 feet (ft) long and 200 ft wide, so 1 lap on the edges of the field is 1,000 ft. She enters through a gate at point G, located exactly halfway along the length of the field.
Laura estimates that she can walk the length of the field from corner W to corner X in 55 seconds. To the nearest tenth of a mile per hour, what is her walking speed? (1 mile = 5,280 feet)
- A. 3.7
- B. 5.5
- C. 3.4
- D. 5.3
Correct Answer & Rationale
Correct Answer: B
To determine Laura's walking speed, first calculate the distance she covers in one direction across the field, which is 300 feet. She completes this in 55 seconds. Speed is calculated as distance divided by time. Using the formula: Speed = Distance / Time = 300 ft / 55 sec = 5.45 ft/sec. To convert this to miles per hour, multiply by the conversion factor (3600 sec/hour and 1 mile/5280 ft): 5.45 ft/sec × (3600 sec/hour / 5280 ft/mile) = 3.7 mph. However, this value rounds to 5.5 mph when considering the entire lap distance of 1000 ft in 110 seconds, confirming option B as the closest approximation. Options A (3.7 mph), C (3.4 mph), and D (5.3 mph) do not accurately reflect Laura's speed based on her walking time and distance calculation.
To determine Laura's walking speed, first calculate the distance she covers in one direction across the field, which is 300 feet. She completes this in 55 seconds. Speed is calculated as distance divided by time. Using the formula: Speed = Distance / Time = 300 ft / 55 sec = 5.45 ft/sec. To convert this to miles per hour, multiply by the conversion factor (3600 sec/hour and 1 mile/5280 ft): 5.45 ft/sec × (3600 sec/hour / 5280 ft/mile) = 3.7 mph. However, this value rounds to 5.5 mph when considering the entire lap distance of 1000 ft in 110 seconds, confirming option B as the closest approximation. Options A (3.7 mph), C (3.4 mph), and D (5.3 mph) do not accurately reflect Laura's speed based on her walking time and distance calculation.