Answer :
Let's analyze each equation step-by-step to determine which ones are quadratic functions.
### Step-by-Step Solution:
1. Equation 1: [tex]\(2b(b-7) + b = 0\)[/tex]
- Start by expanding and simplifying the expression on the left-hand side:
[tex]\[ 2b(b - 7) + b = 2b^2 - 14b + b = 2b^2 - 13b \][/tex]
- This can be written as:
[tex]\[ 2b^2 - 13b = 0 \][/tex]
- Since the highest power of [tex]\( b \)[/tex] is 2, this is a quadratic equation.
2. Equation 2: [tex]\((4a + 2)(2a - 1) + 1 = 0\)[/tex]
- Begin by expanding the expression:
[tex]\[ (4a + 2)(2a - 1) + 1 = 8a^2 - 4a + 4a - 2 + 1 = 8a^2 - 1 \][/tex]
- This simplifies to:
[tex]\[ 8a^2 - 1 = 0 \][/tex]
- Since the highest power of [tex]\( a \)[/tex] is 2, this is a quadratic equation.
3. Equation 3: [tex]\(2y + 2(3y - 5) = 0\)[/tex]
- Expand and simplify the expression:
[tex]\[ 2y + 2(3y - 5) = 2y + 6y - 10 = 8y - 10 \][/tex]
- This results in:
[tex]\[ 8y - 10 = 0 \][/tex]
- The highest power of [tex]\( y \)[/tex] is 1, so this is a linear equation, not a quadratic equation.
4. Equation 4: [tex]\(8 - 5x = 4(3x - 1)\)[/tex]
- Expand and simplify the expression:
[tex]\[ 8 - 5x = 12x - 4 \][/tex]
- Combine like terms:
[tex]\[ 8 + 4 = 12x + 5x \implies 12 = 17x \][/tex]
- The highest power of [tex]\( x \)[/tex] is 1, so this is also a linear equation, not a quadratic equation.
### Conclusion:
The quadratic equations among the given options are:
- [tex]\(2b(b-7) + b = 0\)[/tex]
- [tex]\((4a + 2)(2a - 1) + 1 = 0\)[/tex]
So, the quadratic functions are:
1. Equation 1: [tex]\(2b^2 - 13b = 0\)[/tex]
2. Equation 2: [tex]\(8a^2 - 1 = 0\)[/tex]
Thus, the quadratic functions are the ones represented by [tex]\( \text{eq1} \)[/tex] and [tex]\( \text{eq2} \)[/tex].
### Step-by-Step Solution:
1. Equation 1: [tex]\(2b(b-7) + b = 0\)[/tex]
- Start by expanding and simplifying the expression on the left-hand side:
[tex]\[ 2b(b - 7) + b = 2b^2 - 14b + b = 2b^2 - 13b \][/tex]
- This can be written as:
[tex]\[ 2b^2 - 13b = 0 \][/tex]
- Since the highest power of [tex]\( b \)[/tex] is 2, this is a quadratic equation.
2. Equation 2: [tex]\((4a + 2)(2a - 1) + 1 = 0\)[/tex]
- Begin by expanding the expression:
[tex]\[ (4a + 2)(2a - 1) + 1 = 8a^2 - 4a + 4a - 2 + 1 = 8a^2 - 1 \][/tex]
- This simplifies to:
[tex]\[ 8a^2 - 1 = 0 \][/tex]
- Since the highest power of [tex]\( a \)[/tex] is 2, this is a quadratic equation.
3. Equation 3: [tex]\(2y + 2(3y - 5) = 0\)[/tex]
- Expand and simplify the expression:
[tex]\[ 2y + 2(3y - 5) = 2y + 6y - 10 = 8y - 10 \][/tex]
- This results in:
[tex]\[ 8y - 10 = 0 \][/tex]
- The highest power of [tex]\( y \)[/tex] is 1, so this is a linear equation, not a quadratic equation.
4. Equation 4: [tex]\(8 - 5x = 4(3x - 1)\)[/tex]
- Expand and simplify the expression:
[tex]\[ 8 - 5x = 12x - 4 \][/tex]
- Combine like terms:
[tex]\[ 8 + 4 = 12x + 5x \implies 12 = 17x \][/tex]
- The highest power of [tex]\( x \)[/tex] is 1, so this is also a linear equation, not a quadratic equation.
### Conclusion:
The quadratic equations among the given options are:
- [tex]\(2b(b-7) + b = 0\)[/tex]
- [tex]\((4a + 2)(2a - 1) + 1 = 0\)[/tex]
So, the quadratic functions are:
1. Equation 1: [tex]\(2b^2 - 13b = 0\)[/tex]
2. Equation 2: [tex]\(8a^2 - 1 = 0\)[/tex]
Thus, the quadratic functions are the ones represented by [tex]\( \text{eq1} \)[/tex] and [tex]\( \text{eq2} \)[/tex].
