Answer :
To determine which of the given equations are quadratic, we must identify the equations that take the standard form of a quadratic equation. A quadratic equation generally has the form [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are coefficients and [tex]\( a \neq 0 \)[/tex].
Let’s analyze each given equation step by step:
1. Equation: [tex]\( 2x^2 + 12x = 0 \)[/tex]
- This equation contains the term [tex]\( x^2 \)[/tex] and does not contain any higher power of [tex]\( x \)[/tex] (such as [tex]\( x^3 \)[/tex]).
- Conclusion: It is a quadratic equation.
2. Equation: [tex]\( x^2 - 2x = 4x + 1 \)[/tex]
- Rearrange the equation into the standard form:
- [tex]\( x^2 - 2x - 4x - 1 = 0 \implies x^2 - 6x - 1 = 0 \)[/tex]
- This rearrangement shows the terms [tex]\( x^2 \)[/tex] and [tex]\( x \)[/tex], with no higher powers of [tex]\( x \)[/tex].
- Conclusion: It is a quadratic equation.
3. Equation: [tex]\( x^3 - 6x^2 + 8 = 0 \)[/tex]
- This equation contains the term [tex]\( x^3 \)[/tex], which represents a cubic term.
- Conclusion: It is not a quadratic equation.
4. Equation: [tex]\( 5x - 3 = 0 \)[/tex]
- This equation is a linear equation as it contains only [tex]\( x \)[/tex] without the [tex]\( x^2 \)[/tex] term.
- Conclusion: It is not a quadratic equation.
5. Equation: [tex]\( 5x - 1 = 3x + 8 \)[/tex]
- Rearrange the equation into the standard form:
- [tex]\( 5x - 1 - 3x - 8 = 0 \implies 2x - 9 = 0 \)[/tex]
- This rearrangement shows only the term [tex]\( x \)[/tex], without the [tex]\( x^2 \)[/tex] term.
- Conclusion: It is not a quadratic equation.
6. Equation: [tex]\( 9x^2 + 6x - 3 = 0 \)[/tex]
- This equation contains the term [tex]\( x^2 \)[/tex] and other terms involving [tex]\( x \)[/tex] and constants.
- Conclusion: It is a quadratic equation.
Based on the analysis of each equation, the quadratic equations are:
1. [tex]\( 2x^2 + 12x = 0 \)[/tex]
2. [tex]\( x^2 - 2x = 4x + 1 \)[/tex]
6. [tex]\( 9x^2 + 6x - 3 = 0 \)[/tex]
Therefore, the selected quadratic equations are:
[tex]\[ 2x^2 + 12x = 0, \quad x^2 - 2x = 4x + 1, \quad 9x^2 + 6x - 3 = 0. \][/tex]
Let’s analyze each given equation step by step:
1. Equation: [tex]\( 2x^2 + 12x = 0 \)[/tex]
- This equation contains the term [tex]\( x^2 \)[/tex] and does not contain any higher power of [tex]\( x \)[/tex] (such as [tex]\( x^3 \)[/tex]).
- Conclusion: It is a quadratic equation.
2. Equation: [tex]\( x^2 - 2x = 4x + 1 \)[/tex]
- Rearrange the equation into the standard form:
- [tex]\( x^2 - 2x - 4x - 1 = 0 \implies x^2 - 6x - 1 = 0 \)[/tex]
- This rearrangement shows the terms [tex]\( x^2 \)[/tex] and [tex]\( x \)[/tex], with no higher powers of [tex]\( x \)[/tex].
- Conclusion: It is a quadratic equation.
3. Equation: [tex]\( x^3 - 6x^2 + 8 = 0 \)[/tex]
- This equation contains the term [tex]\( x^3 \)[/tex], which represents a cubic term.
- Conclusion: It is not a quadratic equation.
4. Equation: [tex]\( 5x - 3 = 0 \)[/tex]
- This equation is a linear equation as it contains only [tex]\( x \)[/tex] without the [tex]\( x^2 \)[/tex] term.
- Conclusion: It is not a quadratic equation.
5. Equation: [tex]\( 5x - 1 = 3x + 8 \)[/tex]
- Rearrange the equation into the standard form:
- [tex]\( 5x - 1 - 3x - 8 = 0 \implies 2x - 9 = 0 \)[/tex]
- This rearrangement shows only the term [tex]\( x \)[/tex], without the [tex]\( x^2 \)[/tex] term.
- Conclusion: It is not a quadratic equation.
6. Equation: [tex]\( 9x^2 + 6x - 3 = 0 \)[/tex]
- This equation contains the term [tex]\( x^2 \)[/tex] and other terms involving [tex]\( x \)[/tex] and constants.
- Conclusion: It is a quadratic equation.
Based on the analysis of each equation, the quadratic equations are:
1. [tex]\( 2x^2 + 12x = 0 \)[/tex]
2. [tex]\( x^2 - 2x = 4x + 1 \)[/tex]
6. [tex]\( 9x^2 + 6x - 3 = 0 \)[/tex]
Therefore, the selected quadratic equations are:
[tex]\[ 2x^2 + 12x = 0, \quad x^2 - 2x = 4x + 1, \quad 9x^2 + 6x - 3 = 0. \][/tex]
