If a +√x= b then x =
- A. √b-√a
- B. √(b-1)
- C. (b-a)²
- D. b²-a²
Correct Answer & Rationale
Correct Answer: C
To solve for \( x \) in the equation \( a + \sqrt{x} = b \), we first isolate \( \sqrt{x} \) by rearranging the equation to \( \sqrt{x} = b - a \). Squaring both sides gives \( x = (b - a)^2 \), which corresponds to option C. Option A, \( \sqrt{b} - \sqrt{a} \), does not account for squaring the expression and thus cannot represent \( x \). Option B, \( \sqrt{(b-1)} \), is unrelated to the original equation and lacks the necessary operations. Option D, \( b^2 - a^2 \), applies the difference of squares incorrectly and does not solve for \( x \) directly.
To solve for \( x \) in the equation \( a + \sqrt{x} = b \), we first isolate \( \sqrt{x} \) by rearranging the equation to \( \sqrt{x} = b - a \). Squaring both sides gives \( x = (b - a)^2 \), which corresponds to option C. Option A, \( \sqrt{b} - \sqrt{a} \), does not account for squaring the expression and thus cannot represent \( x \). Option B, \( \sqrt{(b-1)} \), is unrelated to the original equation and lacks the necessary operations. Option D, \( b^2 - a^2 \), applies the difference of squares incorrectly and does not solve for \( x \) directly.
Other Related Questions
Which of the following represents the cost, in dollars, of renting a car for d days and driving m miles?
- A. 45+.25+ d+m
- B. 45.25+ dm
- C. 45d +.25m
- D. 45/d +25/m
Correct Answer & Rationale
Correct Answer: C
Option C accurately represents the cost of renting a car, where $45 is a fixed daily rental fee multiplied by the number of days (d) and $0.25 is the cost per mile multiplied by the number of miles driven (m). Option A incorrectly adds the fixed cost and variable costs without proper multiplication, leading to an illogical expression. Option B misrepresents the relationship by multiplying the daily rate by the miles driven, which does not reflect the cost structure. Option D divides the fixed cost by days and the cost per mile by miles, which does not align with standard cost calculations for renting a car.
Option C accurately represents the cost of renting a car, where $45 is a fixed daily rental fee multiplied by the number of days (d) and $0.25 is the cost per mile multiplied by the number of miles driven (m). Option A incorrectly adds the fixed cost and variable costs without proper multiplication, leading to an illogical expression. Option B misrepresents the relationship by multiplying the daily rate by the miles driven, which does not reflect the cost structure. Option D divides the fixed cost by days and the cost per mile by miles, which does not align with standard cost calculations for renting a car.
If the combined amount of donations collected by Kevin, Fran, and Brooke exceeded the amount Lamar collected by $250, what was the total amount of donations collected by all five club members?
- A. $500
- B. $1,200
- C. $2,500
- D. $3,200
Correct Answer & Rationale
Correct Answer: C
To determine the total amount of donations collected by all five club members, we start with the information that the combined donations of Kevin, Fran, and Brooke exceeded Lamar's by $250. If we denote Lamar's donations as \( L \), then the amount collected by Kevin, Fran, and Brooke is \( L + 250 \). Thus, the total donations from all five members can be expressed as \( L + (L + 250) = 2L + 250 \). To find a plausible total, we consider the options. - A: $500 is too low, as it doesn't allow for both \( L \) and the excess amount. - B: $1,200 also falls short since it would imply \( L \) is negative. - D: $3,200 would require \( L \) to be too high, exceeding reasonable donation limits. C: $2,500 fits perfectly, allowing \( L \) to be $1,125, which is a feasible figure. Therefore, the total amount is logically $2,500.
To determine the total amount of donations collected by all five club members, we start with the information that the combined donations of Kevin, Fran, and Brooke exceeded Lamar's by $250. If we denote Lamar's donations as \( L \), then the amount collected by Kevin, Fran, and Brooke is \( L + 250 \). Thus, the total donations from all five members can be expressed as \( L + (L + 250) = 2L + 250 \). To find a plausible total, we consider the options. - A: $500 is too low, as it doesn't allow for both \( L \) and the excess amount. - B: $1,200 also falls short since it would imply \( L \) is negative. - D: $3,200 would require \( L \) to be too high, exceeding reasonable donation limits. C: $2,500 fits perfectly, allowing \( L \) to be $1,125, which is a feasible figure. Therefore, the total amount is logically $2,500.
Which equation is a correct way to calculate x?
- A. sin x=5,000 /7,000
- B. sin x= 7,000 /5,000
- C. tan x= 5,000/7,000
- D. tan x=7,000/5,000
Correct Answer & Rationale
Correct Answer: C
To solve for \( x \), the correct relationship involves the tangent function, as \( \tan \) is defined as the ratio of the opposite side to the adjacent side in a right triangle. Option C, \( \tan x = \frac{5,000}{7,000} \), accurately represents this ratio. Option A misapplies the sine function, which should represent the ratio of the opposite side to the hypotenuse, not the adjacent side. Similarly, option B incorrectly uses sine but with the sides reversed, leading to an inaccurate representation. Option D misuses tangent, suggesting the opposite and adjacent sides are swapped, which does not align with the definition of tangent. Thus, only option C correctly applies the tangent function to find \( x \).
To solve for \( x \), the correct relationship involves the tangent function, as \( \tan \) is defined as the ratio of the opposite side to the adjacent side in a right triangle. Option C, \( \tan x = \frac{5,000}{7,000} \), accurately represents this ratio. Option A misapplies the sine function, which should represent the ratio of the opposite side to the hypotenuse, not the adjacent side. Similarly, option B incorrectly uses sine but with the sides reversed, leading to an inaccurate representation. Option D misuses tangent, suggesting the opposite and adjacent sides are swapped, which does not align with the definition of tangent. Thus, only option C correctly applies the tangent function to find \( x \).
The system of equations above has how many solutions? x+4y=3, 2x+8y=4
- A. None
- B. One
- C. Two
- D. Infinitely many
Correct Answer & Rationale
Correct Answer: A
To determine the number of solutions for the system of equations, we first analyze the equations: \(x + 4y = 3\) and \(2x + 8y = 4\). The second equation can be simplified by dividing all terms by 2, resulting in \(x + 4y = 2\). Now, we have two equations: \(x + 4y = 3\) and \(x + 4y = 2\). Since both equations represent parallel lines (same slope, different y-intercepts), they will never intersect, indicating there are no solutions. Option B suggests one solution, which is incorrect as parallel lines do not meet. Option C suggests two solutions, which is also incorrect for the same reason. Option D proposes infinitely many solutions, which applies only to identical lines, not parallel ones. Thus, the system has no solutions.
To determine the number of solutions for the system of equations, we first analyze the equations: \(x + 4y = 3\) and \(2x + 8y = 4\). The second equation can be simplified by dividing all terms by 2, resulting in \(x + 4y = 2\). Now, we have two equations: \(x + 4y = 3\) and \(x + 4y = 2\). Since both equations represent parallel lines (same slope, different y-intercepts), they will never intersect, indicating there are no solutions. Option B suggests one solution, which is incorrect as parallel lines do not meet. Option C suggests two solutions, which is also incorrect for the same reason. Option D proposes infinitely many solutions, which applies only to identical lines, not parallel ones. Thus, the system has no solutions.