Let's determine if each of the given equations is a quadratic equation by rewriting them in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex] and identifying the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
### Equation 1: [tex]\(6x^2 - 3x + 9 = 0\)[/tex]
This equation is already in the standard quadratic form [tex]\(ax^2 + bx + c = 0\)[/tex].
- [tex]\(a = 6\)[/tex]
- [tex]\(b = -3\)[/tex]
- [tex]\(c = 9\)[/tex]
Since [tex]\(a \neq 0\)[/tex], it is a quadratic equation.
### Equation 2: [tex]\(12x^2 - 1 = 3x(4x + 2)\)[/tex]
First, let's simplify the right-hand side:
[tex]\[ 3x(4x + 2) = 12x^2 + 6x \][/tex]
Now, rewrite the equation:
[tex]\[ 12x^2 - 1 = 12x^2 + 6x \][/tex]
Subtract [tex]\(12x^2 + 6x\)[/tex] from both sides:
[tex]\[ 12x^2 - 12x^2 - 1 - 6x = 0 \][/tex]
[tex]\[ -6x - 1 = 0 \][/tex]
Now we have it in the standard form:
[tex]\[ 0x^2 - 6x - 1 = 0 \][/tex]
- [tex]\(a = 0\)[/tex]
- [tex]\(b = -6\)[/tex]
- [tex]\(c = -1\)[/tex]
Since [tex]\(a = 0\)[/tex], this is not a quadratic equation.
### Equation 3: [tex]\(x - x^2 + 3 = 4x^2 + 2\)[/tex]
First, let's move everything to one side by subtracting [tex]\(4x^2 + 2\)[/tex] from both sides:
[tex]\[ x - x^2 + 3 - 4x^2 - 2 = 0 \][/tex]
[tex]\[ -5x^2 + x + 1 = 0 \][/tex]
Now we have it in the standard form:
[tex]\[ -5x^2 + x + 1 = 0 \][/tex]
- [tex]\(a = -5\)[/tex]
- [tex]\(b = 1\)[/tex]
- [tex]\(c = 1\)[/tex]
Since [tex]\(a \neq 0\)[/tex], it is a quadratic equation.
### Equation 4: [tex]\(0 = 7x^2 - 2x - 1\)[/tex]
This equation is already in the standard quadratic form [tex]\(ax^2 + bx + c = 0\)[/tex].
- [tex]\(a = 7\)[/tex]
- [tex]\(b = -2\)[/tex]
- [tex]\(c = -1\)[/tex]
Since [tex]\(a \neq 0\)[/tex], it is a quadratic equation.
### Summary of Results
Let's summarize the coefficients and whether each equation is quadratic:
1. [tex]\(6x^2 - 3x + 9 = 0\)[/tex]
- [tex]\(a = 6\)[/tex]
- [tex]\(b = -3\)[/tex]
- [tex]\(c = 9\)[/tex]
- Quadratic: True
2. [tex]\(12x^2 - 1 = 3x(4x + 2)\)[/tex]
- [tex]\(a = 0\)[/tex]
- [tex]\(b = -6\)[/tex]
- [tex]\(c = -1\)[/tex]
- Quadratic: False
3. [tex]\(x - x^2 + 3 = 4x^2 + 2\)[/tex]
- [tex]\(a = -5\)[/tex]
- [tex]\(b = 1\)[/tex]
- [tex]\(c = 1\)[/tex]
- Quadratic: True
4. [tex]\(0 = 7x^2 - 2x - 1\)[/tex]
- [tex]\(a = 7\)[/tex]
- [tex]\(b = -2\)[/tex]
- [tex]\(c = -1\)[/tex]
- Quadratic: True
