Multiply (5x - 1)(5x - 1)
- A. 25x^2 + 1
- B. 25x^2 - 1
- C. 25x^2 - 2x + 1
- D. 25x^2 - 10x + 1
Correct Answer & Rationale
Correct Answer: D
To find the product of (5x - 1)(5x - 1), we can use the formula for squaring a binomial, which states that (a - b)² = a² - 2ab + b². Here, a = 5x and b = 1. Calculating this gives: - a² = (5x)² = 25x² - 2ab = 2(5x)(1) = 10x - b² = 1² = 1 Thus, the expanded form is 25x² - 10x + 1, matching option D. Option A (25x² + 1) incorrectly omits the linear term. Option B (25x² - 1) miscalculates the constant term. Option C (25x² - 2x + 1) incorrectly computes the coefficient of the x term. Each of these options fails to accurately reflect the multiplication of the binomials.
To find the product of (5x - 1)(5x - 1), we can use the formula for squaring a binomial, which states that (a - b)² = a² - 2ab + b². Here, a = 5x and b = 1. Calculating this gives: - a² = (5x)² = 25x² - 2ab = 2(5x)(1) = 10x - b² = 1² = 1 Thus, the expanded form is 25x² - 10x + 1, matching option D. Option A (25x² + 1) incorrectly omits the linear term. Option B (25x² - 1) miscalculates the constant term. Option C (25x² - 2x + 1) incorrectly computes the coefficient of the x term. Each of these options fails to accurately reflect the multiplication of the binomials.
Other Related Questions
The world's highest suspension bridge spans the Arkansas River at a height of 1,053 feet above the water. If a ball is dropped from the bridge. The height of the ball, In feet, after t seconds can be modeled by the equation f(t)= -16(t)^2 + 1053. How many feet above the water is the ball 7 seconds after being dropped?
Correct Answer & Rationale
Correct Answer: A
To determine the height of the ball 7 seconds after being dropped, substitute \( t = 7 \) into the equation \( f(t) = -16(t)^2 + 1053 \). Calculating this gives \( f(7) = -16(7)^2 + 1053 = -16(49) + 1053 = -784 + 1053 = 269 \) feet. Option A provides this correct height of 269 feet. Other options are incorrect because they result from miscalculations or incorrect substitutions into the equation. For example, using an incorrect value for \( t \) or failing to properly apply the formula leads to heights that do not reflect the physics of the scenario.
To determine the height of the ball 7 seconds after being dropped, substitute \( t = 7 \) into the equation \( f(t) = -16(t)^2 + 1053 \). Calculating this gives \( f(7) = -16(7)^2 + 1053 = -16(49) + 1053 = -784 + 1053 = 269 \) feet. Option A provides this correct height of 269 feet. Other options are incorrect because they result from miscalculations or incorrect substitutions into the equation. For example, using an incorrect value for \( t \) or failing to properly apply the formula leads to heights that do not reflect the physics of the scenario.
Which expression is equivalent to (3a + 4ab - 7b) - (a + 2ab - 4b)?
- A. 2a + 2ab - 11b
- B. 2a + 6ab - 11b
- C. 2a + 2ab - 3b
- D. 2a + 6ab - 35
Correct Answer & Rationale
Correct Answer: C
To simplify the expression \((3a + 4ab - 7b) - (a + 2ab - 4b)\), start by distributing the negative sign across the second set of parentheses: \[ 3a + 4ab - 7b - a - 2ab + 4b \] Next, combine like terms: - For \(a\): \(3a - a = 2a\) - For \(ab\): \(4ab - 2ab = 2ab\) - For \(b\): \(-7b + 4b = -3b\) This results in the expression \(2a + 2ab - 3b\), matching option C. Option A introduces an incorrect coefficient for \(b\), while option B miscalculates the \(ab\) term. Option D incorrectly combines terms, leading to an erroneous constant. Thus, option C is the only accurate simplification.
To simplify the expression \((3a + 4ab - 7b) - (a + 2ab - 4b)\), start by distributing the negative sign across the second set of parentheses: \[ 3a + 4ab - 7b - a - 2ab + 4b \] Next, combine like terms: - For \(a\): \(3a - a = 2a\) - For \(ab\): \(4ab - 2ab = 2ab\) - For \(b\): \(-7b + 4b = -3b\) This results in the expression \(2a + 2ab - 3b\), matching option C. Option A introduces an incorrect coefficient for \(b\), while option B miscalculates the \(ab\) term. Option D incorrectly combines terms, leading to an erroneous constant. Thus, option C is the only accurate simplification.
The daily cost, C(x), for a company to produce x microscopes is given by the equation C(x) = 300 + 10.5x. What is the cost of producing 50 microscopes?
- A. $41,250
- B. $360.50
- C. $15,525
- D. $825
Correct Answer & Rationale
Correct Answer: D
To determine the cost of producing 50 microscopes, substitute x = 50 into the equation C(x) = 300 + 10.5x. This gives C(50) = 300 + 10.5(50) = 300 + 525 = 825. Thus, the total cost is $825. Option A ($41,250) is incorrect as it miscalculates the cost by multiplying incorrectly. Option B ($360.50) results from a misunderstanding of the equation, possibly neglecting the fixed cost. Option C ($15,525) likely arises from an error in multiplying the variable cost without adding the fixed cost. Each incorrect option fails to follow the proper calculation method outlined in the cost equation.
To determine the cost of producing 50 microscopes, substitute x = 50 into the equation C(x) = 300 + 10.5x. This gives C(50) = 300 + 10.5(50) = 300 + 525 = 825. Thus, the total cost is $825. Option A ($41,250) is incorrect as it miscalculates the cost by multiplying incorrectly. Option B ($360.50) results from a misunderstanding of the equation, possibly neglecting the fixed cost. Option C ($15,525) likely arises from an error in multiplying the variable cost without adding the fixed cost. Each incorrect option fails to follow the proper calculation method outlined in the cost equation.
Compare the zeros of function P and function Q. Which statement about the zeros of the functions is true?
- A. Function P has the greater zero, which is 9.
- B. Function P has the greater zero, which is 1.
- C. Function Q has the greater zero, which is 5.
- D. Function Q has the greater zero, which is 4.
Correct Answer & Rationale
Correct Answer: C
To determine which statement is true regarding the zeros of functions P and Q, it's essential to analyze the values given. Option A claims that function P's greater zero is 9; however, this contradicts the provided information, as 9 is not a zero for P. Option B asserts that function P's greater zero is 1, which is also incorrect if 1 is not the highest zero of P. Option D states that function Q's greater zero is 4, but if Q's zeros are higher, this option cannot be true. In contrast, option C correctly identifies that function Q has a greater zero, specifically 5, which aligns with the provided data about the functions' zeros.
To determine which statement is true regarding the zeros of functions P and Q, it's essential to analyze the values given. Option A claims that function P's greater zero is 9; however, this contradicts the provided information, as 9 is not a zero for P. Option B asserts that function P's greater zero is 1, which is also incorrect if 1 is not the highest zero of P. Option D states that function Q's greater zero is 4, but if Q's zeros are higher, this option cannot be true. In contrast, option C correctly identifies that function Q has a greater zero, specifically 5, which aligns with the provided data about the functions' zeros.