Answer :
Absolutely! Let's write out each given quadratic equation in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex] and identify the corresponding values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex].
### 1. [tex]\( 3x - 2x^2 = 7 \)[/tex]
Rewriting it in standard form:
[tex]\[ -2x^2 + 3x - 7 = 0 \][/tex]
Thus, [tex]\( a = -2 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = -7 \)[/tex].
### 2. [tex]\( 5 - 2x^2 = 6x \)[/tex]
Rewriting it in standard form:
[tex]\[ -2x^2 - 6x + 5 = 0 \][/tex]
Thus, [tex]\( a = -2 \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = 5 \)[/tex].
### 3. [tex]\( (x + 3)(x + 4) = 0 \)[/tex]
Expanding and writing in standard form:
[tex]\[ x^2 + 7x + 12 = 0 \][/tex]
Thus, [tex]\( a = 1 \)[/tex], [tex]\( b = 7 \)[/tex], and [tex]\( c = 12 \)[/tex].
### 4. [tex]\( (2x + 7)(x - 1) = 0 \)[/tex]
Expanding and writing in standard form:
[tex]\[ 2x^2 + 5x - 7 = 0 \][/tex]
Thus, [tex]\( a = 2 \)[/tex], [tex]\( b = 5 \)[/tex], and [tex]\( c = -7 \)[/tex].
### 5. [tex]\( 2x(x - 3) = 15 \)[/tex]
Expanding and writing in standard form:
[tex]\[ 2x^2 - 6x - 15 = 0 \][/tex]
Thus, [tex]\( a = 2 \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = -15 \)[/tex].
### 6. [tex]\( (x + 7)(x - 7) = -3x \)[/tex]
Expanding and writing in standard form:
[tex]\[ x^2 + 3x - 49 = 0 \][/tex]
Thus, [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = -49 \)[/tex].
### 7. [tex]\( (x - 4)^2 + 8 = 0 \)[/tex]
Expanding and writing in standard form:
[tex]\[ x^2 - 8x + 24 = 0 \][/tex]
Thus, [tex]\( a = 1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = 24 \)[/tex].
### 8. [tex]\( (x + 2)^2 = 3(x + 2) \)[/tex]
Expanding and writing in standard form:
[tex]\[ x^2 + x - 2 = 0 \][/tex]
Thus, [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -2 \)[/tex].
### 9. [tex]\( (2x - 1)^2 = (x + 1)^2 \)[/tex]
Expanding and writing in standard form:
[tex]\[ 3x^2 - 6x = 0 \][/tex]
Thus, [tex]\( a = 3 \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = 0 \)[/tex].
### 10. [tex]\( 2x(x + 4) = (x - 3)(x - 3) \)[/tex]
Expanding and writing in standard form:
[tex]\[ x^2 + 14x - 9 = 0 \][/tex]
Thus, [tex]\( a = 1 \)[/tex], [tex]\( b = 14 \)[/tex], and [tex]\( c = -9 \)[/tex].
Here is the summary:
1. [tex]\( a = -2 \)[/tex], [tex]\( b = 3 \)[/tex], [tex]\( c = -7 \)[/tex]
2. [tex]\( a = -2 \)[/tex], [tex]\( b = -6 \)[/tex], [tex]\( c = 5 \)[/tex]
3. [tex]\( a = 1 \)[/tex], [tex]\( b = 7 \)[/tex], [tex]\( c = 12 \)[/tex]
4. [tex]\( a = 2 \)[/tex], [tex]\( b = 5 \)[/tex], [tex]\( c = -7 \)[/tex]
5. [tex]\( a = 2 \)[/tex], [tex]\( b = -6 \)[/tex], [tex]\( c = -15 \)[/tex]
6. [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], [tex]\( c = -49 \)[/tex]
7. [tex]\( a = 1 \)[/tex], [tex]\( b = -8 \)[/tex], [tex]\( c = 24 \)[/tex]
8. [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], [tex]\( c = -2 \)[/tex]
9. [tex]\( a = 3 \)[/tex], [tex]\( b = -6 \)[/tex], [tex]\( c = 0 \)[/tex]
10. [tex]\( a = 1 \)[/tex], [tex]\( b = 14 \)[/tex], [tex]\( c = -9 \)[/tex]
### 1. [tex]\( 3x - 2x^2 = 7 \)[/tex]
Rewriting it in standard form:
[tex]\[ -2x^2 + 3x - 7 = 0 \][/tex]
Thus, [tex]\( a = -2 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = -7 \)[/tex].
### 2. [tex]\( 5 - 2x^2 = 6x \)[/tex]
Rewriting it in standard form:
[tex]\[ -2x^2 - 6x + 5 = 0 \][/tex]
Thus, [tex]\( a = -2 \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = 5 \)[/tex].
### 3. [tex]\( (x + 3)(x + 4) = 0 \)[/tex]
Expanding and writing in standard form:
[tex]\[ x^2 + 7x + 12 = 0 \][/tex]
Thus, [tex]\( a = 1 \)[/tex], [tex]\( b = 7 \)[/tex], and [tex]\( c = 12 \)[/tex].
### 4. [tex]\( (2x + 7)(x - 1) = 0 \)[/tex]
Expanding and writing in standard form:
[tex]\[ 2x^2 + 5x - 7 = 0 \][/tex]
Thus, [tex]\( a = 2 \)[/tex], [tex]\( b = 5 \)[/tex], and [tex]\( c = -7 \)[/tex].
### 5. [tex]\( 2x(x - 3) = 15 \)[/tex]
Expanding and writing in standard form:
[tex]\[ 2x^2 - 6x - 15 = 0 \][/tex]
Thus, [tex]\( a = 2 \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = -15 \)[/tex].
### 6. [tex]\( (x + 7)(x - 7) = -3x \)[/tex]
Expanding and writing in standard form:
[tex]\[ x^2 + 3x - 49 = 0 \][/tex]
Thus, [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = -49 \)[/tex].
### 7. [tex]\( (x - 4)^2 + 8 = 0 \)[/tex]
Expanding and writing in standard form:
[tex]\[ x^2 - 8x + 24 = 0 \][/tex]
Thus, [tex]\( a = 1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = 24 \)[/tex].
### 8. [tex]\( (x + 2)^2 = 3(x + 2) \)[/tex]
Expanding and writing in standard form:
[tex]\[ x^2 + x - 2 = 0 \][/tex]
Thus, [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -2 \)[/tex].
### 9. [tex]\( (2x - 1)^2 = (x + 1)^2 \)[/tex]
Expanding and writing in standard form:
[tex]\[ 3x^2 - 6x = 0 \][/tex]
Thus, [tex]\( a = 3 \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = 0 \)[/tex].
### 10. [tex]\( 2x(x + 4) = (x - 3)(x - 3) \)[/tex]
Expanding and writing in standard form:
[tex]\[ x^2 + 14x - 9 = 0 \][/tex]
Thus, [tex]\( a = 1 \)[/tex], [tex]\( b = 14 \)[/tex], and [tex]\( c = -9 \)[/tex].
Here is the summary:
1. [tex]\( a = -2 \)[/tex], [tex]\( b = 3 \)[/tex], [tex]\( c = -7 \)[/tex]
2. [tex]\( a = -2 \)[/tex], [tex]\( b = -6 \)[/tex], [tex]\( c = 5 \)[/tex]
3. [tex]\( a = 1 \)[/tex], [tex]\( b = 7 \)[/tex], [tex]\( c = 12 \)[/tex]
4. [tex]\( a = 2 \)[/tex], [tex]\( b = 5 \)[/tex], [tex]\( c = -7 \)[/tex]
5. [tex]\( a = 2 \)[/tex], [tex]\( b = -6 \)[/tex], [tex]\( c = -15 \)[/tex]
6. [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], [tex]\( c = -49 \)[/tex]
7. [tex]\( a = 1 \)[/tex], [tex]\( b = -8 \)[/tex], [tex]\( c = 24 \)[/tex]
8. [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], [tex]\( c = -2 \)[/tex]
9. [tex]\( a = 3 \)[/tex], [tex]\( b = -6 \)[/tex], [tex]\( c = 0 \)[/tex]
10. [tex]\( a = 1 \)[/tex], [tex]\( b = 14 \)[/tex], [tex]\( c = -9 \)[/tex]
