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To determine which of the given equations is in standard form, we need to recall the standard form of a quadratic equation, which is [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].
Let's examine each of the given equations to check if they are in the standard form:
1. Equation: [tex]\(x^2 - 4 = 0\)[/tex]
- This can be written as [tex]\(x^2 + 0x - 4 = 0\)[/tex], which is of the form [tex]\(ax^2 + bx + c = 0\)[/tex] with [tex]\(a = 1\)[/tex], [tex]\(b = 0\)[/tex], and [tex]\(c = -4\)[/tex].
- Therefore, this equation is in standard form.
2. Equation: [tex]\(x^2 = -4x - 4\)[/tex]
- Rearrange it to the standard form: [tex]\(x^2 + 4x + 4 = 0\)[/tex].
- This can now be written as [tex]\(ax^2 + bx + c = 0\)[/tex] with [tex]\(a = 1\)[/tex], [tex]\(b = 4\)[/tex], and [tex]\(c = 4\)[/tex].
- After rearranging, it is in standard form, but it was not originally presented in standard form.
3. Equation: [tex]\(x^2 - 6x = 10\)[/tex]
- Rearrange it to the standard form: [tex]\(x^2 - 6x - 10 = 0\)[/tex].
- This can now be written as [tex]\(ax^2 + bx + c = 0\)[/tex] with [tex]\(a = 1\)[/tex], [tex]\(b = -6\)[/tex], and [tex]\(c = -10\)[/tex].
- After rearranging, it is in standard form, but it was not originally presented in standard form.
4. Equation: [tex]\(9 = 6x - x^2\)[/tex]
- Rearrange it to the standard form: [tex]\(-x^2 + 6x - 9 = 0\)[/tex].
- This can be written as [tex]\(ax^2 + bx + c = 0\)[/tex] with [tex]\(a = -1\)[/tex], [tex]\(b = 6\)[/tex], and [tex]\(c = -9\)[/tex].
- Although it fits the general form, in conventional standard form, the leading coefficient [tex]\(a\)[/tex] should be positive. To match the positive leading coefficient criterion, it should be presented as [tex]\(x^2 - 6x + 9 = 0\)[/tex], if multiplied by [tex]\(-1\)[/tex]. Thus, it is not initially in the standard form.
Given these examinations, the equation [tex]\(x^2 - 4 = 0\)[/tex] is already presented in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex] without needing any rearrangements.
Therefore, the correct answer is:
[tex]\[ x^2 - 4 = 0 \][/tex]
Let's examine each of the given equations to check if they are in the standard form:
1. Equation: [tex]\(x^2 - 4 = 0\)[/tex]
- This can be written as [tex]\(x^2 + 0x - 4 = 0\)[/tex], which is of the form [tex]\(ax^2 + bx + c = 0\)[/tex] with [tex]\(a = 1\)[/tex], [tex]\(b = 0\)[/tex], and [tex]\(c = -4\)[/tex].
- Therefore, this equation is in standard form.
2. Equation: [tex]\(x^2 = -4x - 4\)[/tex]
- Rearrange it to the standard form: [tex]\(x^2 + 4x + 4 = 0\)[/tex].
- This can now be written as [tex]\(ax^2 + bx + c = 0\)[/tex] with [tex]\(a = 1\)[/tex], [tex]\(b = 4\)[/tex], and [tex]\(c = 4\)[/tex].
- After rearranging, it is in standard form, but it was not originally presented in standard form.
3. Equation: [tex]\(x^2 - 6x = 10\)[/tex]
- Rearrange it to the standard form: [tex]\(x^2 - 6x - 10 = 0\)[/tex].
- This can now be written as [tex]\(ax^2 + bx + c = 0\)[/tex] with [tex]\(a = 1\)[/tex], [tex]\(b = -6\)[/tex], and [tex]\(c = -10\)[/tex].
- After rearranging, it is in standard form, but it was not originally presented in standard form.
4. Equation: [tex]\(9 = 6x - x^2\)[/tex]
- Rearrange it to the standard form: [tex]\(-x^2 + 6x - 9 = 0\)[/tex].
- This can be written as [tex]\(ax^2 + bx + c = 0\)[/tex] with [tex]\(a = -1\)[/tex], [tex]\(b = 6\)[/tex], and [tex]\(c = -9\)[/tex].
- Although it fits the general form, in conventional standard form, the leading coefficient [tex]\(a\)[/tex] should be positive. To match the positive leading coefficient criterion, it should be presented as [tex]\(x^2 - 6x + 9 = 0\)[/tex], if multiplied by [tex]\(-1\)[/tex]. Thus, it is not initially in the standard form.
Given these examinations, the equation [tex]\(x^2 - 4 = 0\)[/tex] is already presented in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex] without needing any rearrangements.
Therefore, the correct answer is:
[tex]\[ x^2 - 4 = 0 \][/tex]
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