Answer :
Let's identify which of the given equations are quadratic (can be expressed in the form [tex]\( ax^2 + bx + c = 0 \)[/tex]) and which are not.
### Step-by-step Solution:
1. Equation: [tex]\( 3n + 8 = 15 \)[/tex]
[tex]\[ 3n + 8 = 15 \][/tex]
Simplify:
[tex]\[ 3n = 15 - 8 \][/tex]
[tex]\[ 3n = 7 \][/tex]
This is a linear equation, not a quadratic equation.
2. Equation: [tex]\( x^2 - 5x + 10 = 0 \)[/tex]
[tex]\[ x^2 - 5x + 10 = 0 \][/tex]
This is already in the quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 1 \)[/tex], [tex]\( b = -5 \)[/tex], and [tex]\( c = 10 \)[/tex]. Therefore, it is a quadratic equation.
3. Equation: [tex]\( 12 - 4x = 0 \)[/tex]
[tex]\[ 12 - 4x = 0 \][/tex]
Simplify:
[tex]\[ -4x = -12 \][/tex]
[tex]\[ x = 3 \][/tex]
This is a linear equation, not a quadratic equation.
4. Equation: [tex]\( 2t^2 - 7t = 12 \)[/tex]
[tex]\[ 2t^2 - 7t = 12 \][/tex]
Move all terms to one side:
[tex]\[ 2t^2 - 7t - 12 = 0 \][/tex]
This is in the quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 2 \)[/tex], [tex]\( b = -7 \)[/tex], and [tex]\( c = -12 \)[/tex]. Therefore, it is a quadratic equation.
5. Equation: [tex]\( 6 - 2x + 3x^2 = 0 \)[/tex]
[tex]\[ 6 - 2x + 3x^2 = 0 \][/tex]
Rewriting it in standard form:
[tex]\[ 3x^2 - 2x + 6 = 0 \][/tex]
This is in the quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 3 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = 6 \)[/tex]. Therefore, it is a quadratic equation.
### Next Set of Equations:
6. Equation: [tex]\( 25 - r^2 = 4r \)[/tex]
[tex]\[ 25 - r^2 = 4r \][/tex]
Move all terms to one side:
[tex]\[ -r^2 - 4r + 25 = 0 \][/tex]
Rewriting:
[tex]\[ r^2 + 4r - 25 = 0 \][/tex]
This is in the quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -25 \)[/tex]. Therefore, it is a quadratic equation.
7. Equation: [tex]\( 3x(x - 2) = -7 \)[/tex]
[tex]\[ 3x(x - 2) = -7 \][/tex]
Expand and move all terms to one side:
[tex]\[ 3x^2 - 6x + 7 = 0 \][/tex]
This is in the quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 3 \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = 7 \)[/tex]. Therefore, it is a quadratic equation.
8. Equation: [tex]\( \frac{1}{2}(h - 6) = 0 \)[/tex]
[tex]\[ \frac{1}{2}(h - 6) = 0 \][/tex]
Simplify:
[tex]\[ h - 6 = 0 \][/tex]
[tex]\[ h = 6 \][/tex]
This is a linear equation, not a quadratic equation.
9. Equation: [tex]\( (x + 2)^2 = 0 \)[/tex]
[tex]\[ (x + 2)^2 = 0 \][/tex]
This is simplified to:
[tex]\[ x^2 + 4x + 4 = 0 \][/tex]
This is already in the quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = 4 \)[/tex]. Therefore, it is a quadratic equation.
10. Equation: [tex]\( (\omega - 8)(\omega + 5) = 14 \)[/tex]
[tex]\[ (\omega - 8)(\omega + 5) = 14 \][/tex]
Expand and simplify:
[tex]\[ \omega^2 + 5\omega - 8\omega - 40 = 14 \][/tex]
[tex]\[ \omega^2 - 3\omega - 40 = 14 \][/tex]
Move all terms to one side:
[tex]\[ \omega^2 - 3\omega - 54 = 0 \][/tex]
This is in the quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 1 \)[/tex], [tex]\( b = -3 \)[/tex], and [tex]\( c = -54 \)[/tex]. Therefore, it is a quadratic equation.
### Summary:
Quadratic Equations:
- [tex]\( x^2 - 5x + 10 = 0 \)[/tex]
- [tex]\( 2t^2 - 7t = 12 \)[/tex]
- [tex]\( 6 - 2x + 3x^2 = 0 \)[/tex]
- [tex]\( 25 - r^2 = 4r \)[/tex]
- [tex]\( 3x(x - 2) = -7 \)[/tex]
- [tex]\( (x + 2)^2 = 0 \)[/tex]
- [tex]\( (\omega - 8)(\omega + 5) = 14 \)[/tex]
Non-Quadratic Equations:
- [tex]\( 3n + 8 = 15 \)[/tex]
- [tex]\( 12 - 4x = 0 \)[/tex]
- [tex]\( \frac{1}{2}(h - 6) = 0 \)[/tex]
### Step-by-step Solution:
1. Equation: [tex]\( 3n + 8 = 15 \)[/tex]
[tex]\[ 3n + 8 = 15 \][/tex]
Simplify:
[tex]\[ 3n = 15 - 8 \][/tex]
[tex]\[ 3n = 7 \][/tex]
This is a linear equation, not a quadratic equation.
2. Equation: [tex]\( x^2 - 5x + 10 = 0 \)[/tex]
[tex]\[ x^2 - 5x + 10 = 0 \][/tex]
This is already in the quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 1 \)[/tex], [tex]\( b = -5 \)[/tex], and [tex]\( c = 10 \)[/tex]. Therefore, it is a quadratic equation.
3. Equation: [tex]\( 12 - 4x = 0 \)[/tex]
[tex]\[ 12 - 4x = 0 \][/tex]
Simplify:
[tex]\[ -4x = -12 \][/tex]
[tex]\[ x = 3 \][/tex]
This is a linear equation, not a quadratic equation.
4. Equation: [tex]\( 2t^2 - 7t = 12 \)[/tex]
[tex]\[ 2t^2 - 7t = 12 \][/tex]
Move all terms to one side:
[tex]\[ 2t^2 - 7t - 12 = 0 \][/tex]
This is in the quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 2 \)[/tex], [tex]\( b = -7 \)[/tex], and [tex]\( c = -12 \)[/tex]. Therefore, it is a quadratic equation.
5. Equation: [tex]\( 6 - 2x + 3x^2 = 0 \)[/tex]
[tex]\[ 6 - 2x + 3x^2 = 0 \][/tex]
Rewriting it in standard form:
[tex]\[ 3x^2 - 2x + 6 = 0 \][/tex]
This is in the quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 3 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = 6 \)[/tex]. Therefore, it is a quadratic equation.
### Next Set of Equations:
6. Equation: [tex]\( 25 - r^2 = 4r \)[/tex]
[tex]\[ 25 - r^2 = 4r \][/tex]
Move all terms to one side:
[tex]\[ -r^2 - 4r + 25 = 0 \][/tex]
Rewriting:
[tex]\[ r^2 + 4r - 25 = 0 \][/tex]
This is in the quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -25 \)[/tex]. Therefore, it is a quadratic equation.
7. Equation: [tex]\( 3x(x - 2) = -7 \)[/tex]
[tex]\[ 3x(x - 2) = -7 \][/tex]
Expand and move all terms to one side:
[tex]\[ 3x^2 - 6x + 7 = 0 \][/tex]
This is in the quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 3 \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = 7 \)[/tex]. Therefore, it is a quadratic equation.
8. Equation: [tex]\( \frac{1}{2}(h - 6) = 0 \)[/tex]
[tex]\[ \frac{1}{2}(h - 6) = 0 \][/tex]
Simplify:
[tex]\[ h - 6 = 0 \][/tex]
[tex]\[ h = 6 \][/tex]
This is a linear equation, not a quadratic equation.
9. Equation: [tex]\( (x + 2)^2 = 0 \)[/tex]
[tex]\[ (x + 2)^2 = 0 \][/tex]
This is simplified to:
[tex]\[ x^2 + 4x + 4 = 0 \][/tex]
This is already in the quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = 4 \)[/tex]. Therefore, it is a quadratic equation.
10. Equation: [tex]\( (\omega - 8)(\omega + 5) = 14 \)[/tex]
[tex]\[ (\omega - 8)(\omega + 5) = 14 \][/tex]
Expand and simplify:
[tex]\[ \omega^2 + 5\omega - 8\omega - 40 = 14 \][/tex]
[tex]\[ \omega^2 - 3\omega - 40 = 14 \][/tex]
Move all terms to one side:
[tex]\[ \omega^2 - 3\omega - 54 = 0 \][/tex]
This is in the quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 1 \)[/tex], [tex]\( b = -3 \)[/tex], and [tex]\( c = -54 \)[/tex]. Therefore, it is a quadratic equation.
### Summary:
Quadratic Equations:
- [tex]\( x^2 - 5x + 10 = 0 \)[/tex]
- [tex]\( 2t^2 - 7t = 12 \)[/tex]
- [tex]\( 6 - 2x + 3x^2 = 0 \)[/tex]
- [tex]\( 25 - r^2 = 4r \)[/tex]
- [tex]\( 3x(x - 2) = -7 \)[/tex]
- [tex]\( (x + 2)^2 = 0 \)[/tex]
- [tex]\( (\omega - 8)(\omega + 5) = 14 \)[/tex]
Non-Quadratic Equations:
- [tex]\( 3n + 8 = 15 \)[/tex]
- [tex]\( 12 - 4x = 0 \)[/tex]
- [tex]\( \frac{1}{2}(h - 6) = 0 \)[/tex]
