Answer :
Alright, let's solve each equation step-by-step by writing them in standard form and identifying whether they are quadratic equations or not. We will also extract the coefficients [tex]\(a, b,\)[/tex] and [tex]\( c\)[/tex] for each equation.
1. Equation 1:
[tex]\[ 5x^2 - 0 \][/tex]
* Standard Form:
[tex]\[ 5x^2 = 0 \implies 5x^2 + 0x + 0 = 0 \][/tex]
* Coefficients:
[tex]\[ a = 5, \quad b = 0, \quad c = 0 \][/tex]
* This is a quadratic equation.
2. Equation 2:
[tex]\[ sc(2c - 3) = 0 \][/tex]
* Expand and simplify:
[tex]\[ sc \cdot 2c - sc \cdot 3 = 0 \implies 2sc^2 - 3sc = 0 \][/tex]
* Standard Form:
[tex]\[ 2sc^2 - 3sc = 0 \][/tex]
* Coefficients:
[tex]\[ a = 2s, \quad b = -3s, \quad c = 0 \][/tex]
* This is a quadratic equation in terms of [tex]\( c \)[/tex] for a fixed value of [tex]\( s \)[/tex].
3. Equation 3:
[tex]\[ -2f(11f - 3) - 0 \][/tex]
* Expand and simplify:
[tex]\[ -2f \cdot 11f + (-2f) \cdot (-3) = 0 \implies -22f^2 + 6f = 0 \][/tex]
* Standard Form:
[tex]\[ -22f^2 + 6f + 0 = 0 \][/tex]
* Coefficients:
[tex]\[ a = -22, \quad b = 6, \quad c = 0 \][/tex]
* This is a quadratic equation.
4. Equation 4:
[tex]\[ 3 + 6 + 4x + 5 \][/tex]
* Simplify the expression:
[tex]\[ 3 + 6 + 4x + 5 \implies 14 + 4x \][/tex]
This is a linear equation (not quadratic) since it does not involve [tex]\(x^2\)[/tex].
Coefficients:
[tex]\[ a = 0, \quad b = 4, \quad c = 14 \][/tex]
5. Equation 5:
[tex]\[ (x - 3)(x + 6) = 0 \][/tex]
* Expand the product:
[tex]\[ x^2 + 6x - 3x - 18 = 0 \implies x^2 + 3x - 18 = 0 \][/tex]
* Standard Form:
[tex]\[ x^2 + 3x - 18 = 0 \][/tex]
* Coefficients:
[tex]\[ a = 1, \quad b = 3, \quad c = -18 \][/tex]
* This is a quadratic equation.
Summary of Coefficients:
1. [tex]\((a, b, c) = (5, 0, 0)\)[/tex]
2. [tex]\((a, b, c) = (2s, -3s, 0)\)[/tex]
3. [tex]\((a, b, c) = (-22, 6, 0)\)[/tex]
4. [tex]\((a, b, c) = (0, 4, 14)\)[/tex] [tex]\( \boxed{\text{Not Quadratic}}\)[/tex]
5. [tex]\((a, b, c) = (1, 3, -18)\)[/tex]
The values of [tex]\( a_1, b_1, c_1 \)[/tex] for the quadratic equations are:
1. [tex]\( a = 5, b = 0, c = 0 \)[/tex]
2. [tex]\( a = 2s, b = -3s, c = 0 \)[/tex]
3. [tex]\( a = -22, b = 6, c = 0 \)[/tex]
4. [tex]\(\text{Not Quadratic}\)[/tex]
5. [tex]\( a = 1, b = 3, c = -18 \)[/tex]
1. Equation 1:
[tex]\[ 5x^2 - 0 \][/tex]
* Standard Form:
[tex]\[ 5x^2 = 0 \implies 5x^2 + 0x + 0 = 0 \][/tex]
* Coefficients:
[tex]\[ a = 5, \quad b = 0, \quad c = 0 \][/tex]
* This is a quadratic equation.
2. Equation 2:
[tex]\[ sc(2c - 3) = 0 \][/tex]
* Expand and simplify:
[tex]\[ sc \cdot 2c - sc \cdot 3 = 0 \implies 2sc^2 - 3sc = 0 \][/tex]
* Standard Form:
[tex]\[ 2sc^2 - 3sc = 0 \][/tex]
* Coefficients:
[tex]\[ a = 2s, \quad b = -3s, \quad c = 0 \][/tex]
* This is a quadratic equation in terms of [tex]\( c \)[/tex] for a fixed value of [tex]\( s \)[/tex].
3. Equation 3:
[tex]\[ -2f(11f - 3) - 0 \][/tex]
* Expand and simplify:
[tex]\[ -2f \cdot 11f + (-2f) \cdot (-3) = 0 \implies -22f^2 + 6f = 0 \][/tex]
* Standard Form:
[tex]\[ -22f^2 + 6f + 0 = 0 \][/tex]
* Coefficients:
[tex]\[ a = -22, \quad b = 6, \quad c = 0 \][/tex]
* This is a quadratic equation.
4. Equation 4:
[tex]\[ 3 + 6 + 4x + 5 \][/tex]
* Simplify the expression:
[tex]\[ 3 + 6 + 4x + 5 \implies 14 + 4x \][/tex]
This is a linear equation (not quadratic) since it does not involve [tex]\(x^2\)[/tex].
Coefficients:
[tex]\[ a = 0, \quad b = 4, \quad c = 14 \][/tex]
5. Equation 5:
[tex]\[ (x - 3)(x + 6) = 0 \][/tex]
* Expand the product:
[tex]\[ x^2 + 6x - 3x - 18 = 0 \implies x^2 + 3x - 18 = 0 \][/tex]
* Standard Form:
[tex]\[ x^2 + 3x - 18 = 0 \][/tex]
* Coefficients:
[tex]\[ a = 1, \quad b = 3, \quad c = -18 \][/tex]
* This is a quadratic equation.
Summary of Coefficients:
1. [tex]\((a, b, c) = (5, 0, 0)\)[/tex]
2. [tex]\((a, b, c) = (2s, -3s, 0)\)[/tex]
3. [tex]\((a, b, c) = (-22, 6, 0)\)[/tex]
4. [tex]\((a, b, c) = (0, 4, 14)\)[/tex] [tex]\( \boxed{\text{Not Quadratic}}\)[/tex]
5. [tex]\((a, b, c) = (1, 3, -18)\)[/tex]
The values of [tex]\( a_1, b_1, c_1 \)[/tex] for the quadratic equations are:
1. [tex]\( a = 5, b = 0, c = 0 \)[/tex]
2. [tex]\( a = 2s, b = -3s, c = 0 \)[/tex]
3. [tex]\( a = -22, b = 6, c = 0 \)[/tex]
4. [tex]\(\text{Not Quadratic}\)[/tex]
5. [tex]\( a = 1, b = 3, c = -18 \)[/tex]
