Let's analyze each equation to determine whether it is quadratic. Quadratic equations are in the 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].
1. Equation: [tex]\(5x^2 + 15x = 0\)[/tex]
This equation is already in the standard form [tex]\(5x^2 + 15x = 0\)[/tex], which matches the quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 5 \)[/tex], [tex]\( b = 15 \)[/tex], and [tex]\( c = 0 \)[/tex].
Therefore, [tex]\(5x^2 + 15x = 0\)[/tex] is a quadratic equation.
2. Equation: [tex]\(6x - 1 = 4x + 7\)[/tex]
Simplifying this equation:
[tex]\[
6x - 4x - 1 = 7 \implies 2x - 1 = 7 \implies 2x = 8 \implies x = 4
\][/tex]
This is a linear equation because it does not contain an [tex]\( x^2 \)[/tex] term.
Therefore, [tex]\(6x - 1 = 4x + 7\)[/tex] is not a quadratic equation.
3. Equation: [tex]\(x^2 - 4x = 4x + 7\)[/tex]
Simplifying this equation:
[tex]\[
x^2 - 4x - 4x = 7 \implies x^2 - 8x = 7 \implies x^2 - 8x - 7 = 0
\][/tex]
This is in the standard form of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = -7 \)[/tex].
Therefore, [tex]\(x^2 - 4x = 4x + 7\)[/tex] is a quadratic equation.
4. Equation: [tex]\(2x - 1 = 0\)[/tex]
This equation is in the form [tex]\(2x - 1 = 0\)[/tex], which is a linear equation with no [tex]\( x^2 \)[/tex] term.
Therefore, [tex]\(2x - 1 = 0\)[/tex] is not a quadratic equation.
5. Equation: [tex]\(3x^2 + 5x - 7 = 0\)[/tex]
This equation is already in the standard form [tex]\(3x^2 + 5x - 7 = 0\)[/tex], which matches the quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex] with [tex]\( a = 3 \)[/tex], [tex]\( b = 5 \)[/tex], and [tex]\( c = -7 \)[/tex].
Therefore, [tex]\(3x^2 + 5x - 7 = 0\)[/tex] is a quadratic equation.
6. Equation: [tex]\(x^3 - 2x^2 + 1 = 0\)[/tex]
This equation is in the form [tex]\(x^3 - 2x^2 + 1 = 0\)[/tex], which contains a cubic term [tex]\( x^3 \)[/tex]. Since it has a term with [tex]\( x \)[/tex] raised to the third power, it is not a quadratic equation.
Therefore, [tex]\(x^3 - 2x^2 + 1 = 0\)[/tex] is not a quadratic equation.
In summary, the following equations are quadratic:
- [tex]\(5x^2 + 15x = 0\)[/tex]
- [tex]\(x^2 - 4x = 4x + 7\)[/tex]
- [tex]\(3x^2 + 5x - 7 = 0\)[/tex]
So, the quadratic equations are:
1. [tex]\(5 x^2 + 15 x = 0\)[/tex]
2. [tex]\(x^2 - 4 x = 4 x + 7\)[/tex]
3. [tex]\(3 x^2 + 5 x - 7 = 0\)[/tex]
