Sure, let's carefully determine which of the given equations are quadratic equations and which are not. A quadratic equation is any equation that can be written in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are constants and [tex]\( a \neq 0 \)[/tex]. Essentially, it must have a variable raised to the second power. Let's analyze each one:
1. Equation 1: [tex]\( 3m + 8 = 15 \)[/tex]
- This equation does not have any term with [tex]\( m \)[/tex] raised to the second power. It is a linear equation, not a quadratic equation.
- Conclusion: Not quadratic.
2. Equation 2: [tex]\( x^2 - 5x + 10 = 0 \)[/tex]
- This equation has a term [tex]\( x^2 \)[/tex], which means that there is an [tex]\( x \)[/tex] raised to the second power.
- Conclusion: It is a quadratic equation.
3. Equation 3: [tex]\( 12 - 4x = 0 \)[/tex]
- This equation does not have any term with [tex]\( x \)[/tex] raised to the second power. It is a linear equation.
- Conclusion: Not quadratic.
4. Equation 4: [tex]\( 2t^2 - 7t = 12 \)[/tex]
- This equation can be rewritten as [tex]\( 2t^2 - 7t - 12 = 0 \)[/tex] and has a term [tex]\( t^2 \)[/tex], which means it includes a variable raised to the second power.
- Conclusion: It is a quadratic equation.
5. Equation 5: [tex]\( 6 - 2x + 3x^2 = 0 \)[/tex]
- This equation can be rearranged as [tex]\( 3x^2 - 2x + 6 = 0 \)[/tex] and has a term [tex]\( x^2 \)[/tex], which means it includes a variable raised to the second power.
- Conclusion: It is a quadratic equation.
Summarizing the results:
- Equation 1 ([tex]\( 3m + 8 = 15 \)[/tex]): Not quadratic.
- Equation 2 ([tex]\( x^2 - 5x + 10 = 0 \)[/tex]): Quadratic.
- Equation 3 ([tex]\( 12 - 4x = 0 \)[/tex]): Not quadratic.
- Equation 4 ([tex]\( 2t^2 - 7t = 12 \)[/tex]): Quadratic.
- Equation 5 ([tex]\( 6 - 2x + 3x^2 = 0 \)[/tex]): Quadratic.
The visual representation as a list of whether each equation is quadratic (1) or not (0) is:
[tex]\[ [0, 1, 0, 1, 1] \][/tex]
