At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Our platform connects you with professionals ready to provide precise answers to all your questions in various areas of expertise. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
To determine which expression evaluates to 3 when \( p = -2 \) and \( q = 3 \), let's evaluate each of the given expressions step-by-step.
1. Expression 1: \( p^2 + q - 4 q^2 \)
1. Calculate \( p^2 \):
[tex]\[ p^2 = (-2)^2 = 4 \][/tex]
2. Calculate \( q \):
[tex]\[ q = 3 \][/tex]
3. Calculate \( 4 q^2 \):
[tex]\[ 4 q^2 = 4 \times (3)^2 = 4 \times 9 = 36 \][/tex]
4. Evaluate the entire expression:
[tex]\[ p^2 + q - 4 q^2 = 4 + 3 - 36 = 7 - 36 = -29 \][/tex]
The result is \(-29\), not 3.
2. Expression 2: \( 3 p^2 + 2 p q - 6 q + 2 \)
1. Calculate \( 3 p^2 \):
[tex]\[ 3 p^2 = 3 \times (-2)^2 = 3 \times 4 = 12 \][/tex]
2. Calculate \( 2 p q \):
[tex]\[ 2 p q = 2 \times (-2) \times 3 = 2 \times -6 = -12 \][/tex]
3. Calculate \( -6 q \):
[tex]\[ -6 q = -6 \times 3 = -18 \][/tex]
4. Sum up with the constant term:
[tex]\[ 12 - 12 - 18 + 2 = 12 - 30 + 2 = -16 \][/tex]
The result is \(-16\), not 3.
3. Expression 3: \( p^3 + 2 p^2 q - p^2 + 2 p q + q \)
1. Calculate \( p^3 \):
[tex]\[ p^3 = (-2)^3 = -8 \][/tex]
2. Calculate \( 2 p^2 q \):
[tex]\[ 2 p^2 q = 2 \times (-2)^2 \times 3 = 2 \times 4 \times 3 = 24 \][/tex]
3. Calculate \( - p^2 \):
[tex]\[ - p^2 = -(-2)^2 = -4 \][/tex]
4. Calculate \( 2 p q \):
[tex]\[ 2 p q = 2 \times (-2) \times 3 = -12 \][/tex]
5. Sum up with \( q \):
[tex]\[ p^3 + 2 p^2 q - p^2 + 2 p q + q = -8 + 24 - 4 - 12 + 3 = -8 + 24 - 4 - 12 + 3 = 3 \][/tex]
The result is \( 3 \), which matches the student's result.
4. Expression 4: \( p^2 + 3 q^2 - q^2 + p \)
1. Calculate \( p^2 \):
[tex]\[ p^2 = (-2)^2 = 4 \][/tex]
2. Calculate \( 3 q^2 \):
[tex]\[ 3 q^2 = 3 \times (3)^2 = 3 \times 9 = 27 \][/tex]
3. Calculate \( - q^2 \):
[tex]\[ - q^2 = - (3)^2 = -9 \][/tex]
4. Evaluate \( p \):
[tex]\[ p = -2 \][/tex]
5. Sum up:
[tex]\[ p^2 + 3 q^2 - q^2 + p = 4 + 27 - 9 - 2 = 4 + 18 - 2 = 20 \][/tex]
The result is \( 20 \), not 3.
Therefore, the expression the student evaluated to get a result of 3 is:
[tex]\[ p^3 + 2 p^2 q - p^2 + 2 p q + q \][/tex]
1. Expression 1: \( p^2 + q - 4 q^2 \)
1. Calculate \( p^2 \):
[tex]\[ p^2 = (-2)^2 = 4 \][/tex]
2. Calculate \( q \):
[tex]\[ q = 3 \][/tex]
3. Calculate \( 4 q^2 \):
[tex]\[ 4 q^2 = 4 \times (3)^2 = 4 \times 9 = 36 \][/tex]
4. Evaluate the entire expression:
[tex]\[ p^2 + q - 4 q^2 = 4 + 3 - 36 = 7 - 36 = -29 \][/tex]
The result is \(-29\), not 3.
2. Expression 2: \( 3 p^2 + 2 p q - 6 q + 2 \)
1. Calculate \( 3 p^2 \):
[tex]\[ 3 p^2 = 3 \times (-2)^2 = 3 \times 4 = 12 \][/tex]
2. Calculate \( 2 p q \):
[tex]\[ 2 p q = 2 \times (-2) \times 3 = 2 \times -6 = -12 \][/tex]
3. Calculate \( -6 q \):
[tex]\[ -6 q = -6 \times 3 = -18 \][/tex]
4. Sum up with the constant term:
[tex]\[ 12 - 12 - 18 + 2 = 12 - 30 + 2 = -16 \][/tex]
The result is \(-16\), not 3.
3. Expression 3: \( p^3 + 2 p^2 q - p^2 + 2 p q + q \)
1. Calculate \( p^3 \):
[tex]\[ p^3 = (-2)^3 = -8 \][/tex]
2. Calculate \( 2 p^2 q \):
[tex]\[ 2 p^2 q = 2 \times (-2)^2 \times 3 = 2 \times 4 \times 3 = 24 \][/tex]
3. Calculate \( - p^2 \):
[tex]\[ - p^2 = -(-2)^2 = -4 \][/tex]
4. Calculate \( 2 p q \):
[tex]\[ 2 p q = 2 \times (-2) \times 3 = -12 \][/tex]
5. Sum up with \( q \):
[tex]\[ p^3 + 2 p^2 q - p^2 + 2 p q + q = -8 + 24 - 4 - 12 + 3 = -8 + 24 - 4 - 12 + 3 = 3 \][/tex]
The result is \( 3 \), which matches the student's result.
4. Expression 4: \( p^2 + 3 q^2 - q^2 + p \)
1. Calculate \( p^2 \):
[tex]\[ p^2 = (-2)^2 = 4 \][/tex]
2. Calculate \( 3 q^2 \):
[tex]\[ 3 q^2 = 3 \times (3)^2 = 3 \times 9 = 27 \][/tex]
3. Calculate \( - q^2 \):
[tex]\[ - q^2 = - (3)^2 = -9 \][/tex]
4. Evaluate \( p \):
[tex]\[ p = -2 \][/tex]
5. Sum up:
[tex]\[ p^2 + 3 q^2 - q^2 + p = 4 + 27 - 9 - 2 = 4 + 18 - 2 = 20 \][/tex]
The result is \( 20 \), not 3.
Therefore, the expression the student evaluated to get a result of 3 is:
[tex]\[ p^3 + 2 p^2 q - p^2 + 2 p q + q \][/tex]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.