Choose the best answer. If necessary, use the paper you were given.
Which of the following is equivalent to 12x +8?
- A. 4(3x+2)
- B. 4(3x+8)
- C. 4(3x+2x)
- D. 20x
Correct Answer & Rationale
Correct Answer: A
To determine the equivalent expression for \(12x + 8\), we can factor out the greatest common factor, which is 4. Option A, \(4(3x + 2)\), simplifies to \(12x + 8\) when distributed, making it equivalent to the original expression. Option B, \(4(3x + 8)\), simplifies to \(12x + 32\), which is not equivalent. Option C, \(4(3x + 2x)\), simplifies to \(4(5x)\) or \(20x\), which is also not equivalent. Option D, \(20x\), does not match the original expression either. Thus, only option A is correct.
To determine the equivalent expression for \(12x + 8\), we can factor out the greatest common factor, which is 4. Option A, \(4(3x + 2)\), simplifies to \(12x + 8\) when distributed, making it equivalent to the original expression. Option B, \(4(3x + 8)\), simplifies to \(12x + 32\), which is not equivalent. Option C, \(4(3x + 2x)\), simplifies to \(4(5x)\) or \(20x\), which is also not equivalent. Option D, \(20x\), does not match the original expression either. Thus, only option A is correct.
Other Related Questions
For what values of x does 5x ^ 2 + 4x - 4 = 0 ?
- A. x = 1/5 and x = - 1
- B. x = - 4/5 and x = 1
- C. x = (- 2±6 * √(2))/5
- D. x = (- 2±2 * √(6))/5
Correct Answer & Rationale
Correct Answer: D
To solve the quadratic equation \(5x^2 + 4x - 4 = 0\), one can apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 5\), \(b = 4\), and \(c = -4\). Calculating the discriminant gives \(b^2 - 4ac = 16 + 80 = 96\), leading to \(x = \frac{-4 \pm \sqrt{96}}{10} = \frac{-4 \pm 4\sqrt{6}}{10} = \frac{-2 \pm 2\sqrt{6}}{5}\), which matches option D. Option A provides incorrect roots not derived from the quadratic formula. Option B also presents incorrect values, failing to satisfy the equation. Option C miscalculates the discriminant, leading to an incorrect expression. Thus, D accurately reflects the solution to the equation.
To solve the quadratic equation \(5x^2 + 4x - 4 = 0\), one can apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 5\), \(b = 4\), and \(c = -4\). Calculating the discriminant gives \(b^2 - 4ac = 16 + 80 = 96\), leading to \(x = \frac{-4 \pm \sqrt{96}}{10} = \frac{-4 \pm 4\sqrt{6}}{10} = \frac{-2 \pm 2\sqrt{6}}{5}\), which matches option D. Option A provides incorrect roots not derived from the quadratic formula. Option B also presents incorrect values, failing to satisfy the equation. Option C miscalculates the discriminant, leading to an incorrect expression. Thus, D accurately reflects the solution to the equation.
Which of the following is a factor of u²+uv-2v²?
- A. (u-v)
- B. (2u-v)
- C. (u-2v)
- D. (u+v)
Correct Answer & Rationale
Correct Answer: C
To determine the factors of \( u^2 + uv - 2v^2 \), we can factor the expression. Option C, \( (u - 2v) \), is a valid factor. When we perform polynomial long division or synthetic division using \( (u - 2v) \), we find that it divides evenly, confirming it as a factor. Option A, \( (u - v) \), does not satisfy the factorization, as substituting \( v \) does not yield a zero remainder. Option B, \( (2u - v) \), also fails to factor the expression correctly, leading to a non-zero remainder upon division. Option D, \( (u + v) \), similarly does not yield a zero remainder, confirming it is not a factor. Thus, only \( (u - 2v) \) is a valid factor of the expression.
To determine the factors of \( u^2 + uv - 2v^2 \), we can factor the expression. Option C, \( (u - 2v) \), is a valid factor. When we perform polynomial long division or synthetic division using \( (u - 2v) \), we find that it divides evenly, confirming it as a factor. Option A, \( (u - v) \), does not satisfy the factorization, as substituting \( v \) does not yield a zero remainder. Option B, \( (2u - v) \), also fails to factor the expression correctly, leading to a non-zero remainder upon division. Option D, \( (u + v) \), similarly does not yield a zero remainder, confirming it is not a factor. Thus, only \( (u - 2v) \) is a valid factor of the expression.
The average of 4 numbers is 9. If one of the numbers is 7, what is the sum of the other 3 numbers?
- A. 2
- B. 12
- C. 29
- D. 36
Correct Answer & Rationale
Correct Answer: C
To find the sum of the other three numbers, start by calculating the total sum of all four numbers. Since the average is 9, multiply this by 4, yielding a total of 36. Given that one of the numbers is 7, subtract this from the total: 36 - 7 = 29. Therefore, the sum of the other three numbers is 29. Option A (2) is too low, as it does not account for the total sum needed. Option B (12) underestimates the remaining numbers. Option D (36) mistakenly includes the known number, rather than calculating the sum of the others.
To find the sum of the other three numbers, start by calculating the total sum of all four numbers. Since the average is 9, multiply this by 4, yielding a total of 36. Given that one of the numbers is 7, subtract this from the total: 36 - 7 = 29. Therefore, the sum of the other three numbers is 29. Option A (2) is too low, as it does not account for the total sum needed. Option B (12) underestimates the remaining numbers. Option D (36) mistakenly includes the known number, rather than calculating the sum of the others.
If (2w + 7)(3w - 1) = 0 which of the following is a possible value of w?
- A. -3
- B. -0.28571
- C. 01-Mar
- D. 07-Feb
Correct Answer & Rationale
Correct Answer: D
To solve the equation (2w + 7)(3w - 1) = 0, we set each factor to zero. 1. For 2w + 7 = 0, solving gives w = -3. This corresponds to option A, which is a valid solution. 2. For 3w - 1 = 0, solving gives w = 1/3, approximately 0.333. Option B, -0.28571, does not match this value. 3. Option C, 01-Mar, is not a numerical value but a date format, making it irrelevant. 4. Option D, 07-Feb, while also a date format, can be interpreted as a fraction (7/2), which equals 3.5, not a solution to the equation. Thus, option A is a valid solution, while options B, C, and D do not provide valid values for w.
To solve the equation (2w + 7)(3w - 1) = 0, we set each factor to zero. 1. For 2w + 7 = 0, solving gives w = -3. This corresponds to option A, which is a valid solution. 2. For 3w - 1 = 0, solving gives w = 1/3, approximately 0.333. Option B, -0.28571, does not match this value. 3. Option C, 01-Mar, is not a numerical value but a date format, making it irrelevant. 4. Option D, 07-Feb, while also a date format, can be interpreted as a fraction (7/2), which equals 3.5, not a solution to the equation. Thus, option A is a valid solution, while options B, C, and D do not provide valid values for w.