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Let's go through each equation to determine whether it is a quadratic equation and identify the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] for those that are.
A quadratic equation is of 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. [tex]\(x^2 - 6x + 2 = 0\)[/tex]
- This is already in the standard form of a quadratic equation.
- [tex]\(a = 1\)[/tex], [tex]\(b = -6\)[/tex], [tex]\(c = 2\)[/tex]
2. [tex]\(3x + 5 = 0\)[/tex]
- This is a linear equation, not quadratic. No [tex]\(x^2\)[/tex] term is present.
3. [tex]\(2x^2 + 7x = 15\)[/tex]
- Rearranging it to the standard form: [tex]\(2x^2 + 7x - 15 = 0\)[/tex]
- [tex]\(a = 2\)[/tex], [tex]\(b = 7\)[/tex], [tex]\(c = -15\)[/tex]
4. [tex]\((x+1)^2 = 2(x-3)\)[/tex]
- Expanding and rearranging: [tex]\(x^2 + 2x + 1 = 2x - 6\)[/tex]
- Bringing all terms to one side: [tex]\(x^2 + 2x + 1 - 2x + 6 = 0\)[/tex]
- Simplifying: [tex]\(x^2 + 7 = 0\)[/tex]
- [tex]\(a = 1\)[/tex], [tex]\(b = 0\)[/tex], [tex]\(c = 7\)[/tex]
5. [tex]\(x^2 + 2x + 1 = 5x + 6\)[/tex]
- Rearranging it to the standard form: [tex]\(x^2 + 2x + 1 - 5x - 6 = 0\)[/tex]
- Simplifying: [tex]\(x^2 - 3x - 5 = 0\)[/tex]
- [tex]\(a = 1\)[/tex], [tex]\(b = -3\)[/tex], [tex]\(c = -5\)[/tex]
6. [tex]\((x-1)(x+2) = x(x+5)\)[/tex]
- Expanding both sides: [tex]\(x^2 + x - 2 = x^2 + 5x\)[/tex]
- Bringing all terms to one side: [tex]\(x^2 + x - 2 - x^2 - 5x = 0\)[/tex]
- Simplifying: [tex]\(-4x - 2 = 0\)[/tex]
- This simplifies to a linear equation.
7. [tex]\((x+2)(x-3) = 5\)[/tex]
- Expanding and rearranging: [tex]\(x^2 - x - 6 = 5\)[/tex]
- Bringing all terms to one side: [tex]\(x^2 - x - 6 - 5 = 0\)[/tex]
- Simplifying: [tex]\(x^2 - x - 11 = 0\)[/tex]
- [tex]\(a = 1\)[/tex], [tex]\(b = -1\)[/tex], [tex]\(c = -11\)[/tex]
8. [tex]\(x(x^2 + 3x - 10) = 0\)[/tex]
- Expanding: [tex]\(x^3 + 3x^2 - 10x = 0\)[/tex]
- This is a cubic equation, not quadratic.
9. [tex]\((x-1)^2 + 3 = 2x + 1\)[/tex]
- Expanding and rearranging: [tex]\(x^2 - 2x + 1 + 3 = 2x + 1\)[/tex]
- Bringing all terms to one side: [tex]\(x^2 - 2x + 1 + 3 - 2x - 1 = 0\)[/tex]
- Simplifying: [tex]\(x^2 - 4x + 3 = 0\)[/tex]
- [tex]\(a = 1\)[/tex], [tex]\(b = -4\)[/tex], [tex]\(c = 3\)[/tex]
10. [tex]\((x+2)^3 = x(x^2 - 10x + 25)\)[/tex]
- Expanding both sides: [tex]\(x^3 + 6x^2 + 12x + 8 = x^3 - 10x^2 + 25x\)[/tex]
- Bringing all terms to one side: [tex]\(x^3 + 6x^2 + 12x + 8 - x^3 + 10x^2 - 25x = 0\)[/tex]
- Simplifying: [tex]\(16x^2 - 13x + 8 = 0\)[/tex]
- [tex]\(a = 16\)[/tex], [tex]\(b = -13\)[/tex], [tex]\(c = 8\)[/tex]
Here is the final categorization of which equations are quadratic and their corresponding coefficients:
1. [tex]\(x^2 - 6x + 2 = 0\)[/tex] : [tex]\(a = 1\)[/tex], [tex]\(b = -6\)[/tex], [tex]\(c = 2\)[/tex]
3. [tex]\(2x^2 + 7x = 15\)[/tex] : [tex]\(a = 2\)[/tex], [tex]\(b = 7\)[/tex], [tex]\(c = -15\)[/tex]
4. [tex]\((x+1)^2 = 2(x-3)\)[/tex] : [tex]\(a = 1\)[/tex], [tex]\(b = 0\)[/tex], [tex]\(c = 7\)[/tex]
5. [tex]\(x^2 + 2x + 1 = 5x + 6\)[/tex] : [tex]\(a = 1\)[/tex], [tex]\(b = -3\)[/tex], [tex]\(c = -5\)[/tex]
7. [tex]\((x+2)(x-3) = 5\)[/tex] : [tex]\(a = 1\)[/tex], [tex]\(b = -1\)[/tex], [tex]\(c = -11\)[/tex]
9. [tex]\((x-1)^2 + 3 = 2x + 1\)[/tex] : [tex]\(a = 1\)[/tex], [tex]\(b = -4\)[/tex], [tex]\(c = 3\)[/tex]
10. [tex]\((x+2)^3 = x(x^2 - 10x + 25)\)[/tex] : [tex]\(a = 16\)[/tex], [tex]\(b = -13\)[/tex], [tex]\(c = 8\)[/tex]
A quadratic equation is of 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. [tex]\(x^2 - 6x + 2 = 0\)[/tex]
- This is already in the standard form of a quadratic equation.
- [tex]\(a = 1\)[/tex], [tex]\(b = -6\)[/tex], [tex]\(c = 2\)[/tex]
2. [tex]\(3x + 5 = 0\)[/tex]
- This is a linear equation, not quadratic. No [tex]\(x^2\)[/tex] term is present.
3. [tex]\(2x^2 + 7x = 15\)[/tex]
- Rearranging it to the standard form: [tex]\(2x^2 + 7x - 15 = 0\)[/tex]
- [tex]\(a = 2\)[/tex], [tex]\(b = 7\)[/tex], [tex]\(c = -15\)[/tex]
4. [tex]\((x+1)^2 = 2(x-3)\)[/tex]
- Expanding and rearranging: [tex]\(x^2 + 2x + 1 = 2x - 6\)[/tex]
- Bringing all terms to one side: [tex]\(x^2 + 2x + 1 - 2x + 6 = 0\)[/tex]
- Simplifying: [tex]\(x^2 + 7 = 0\)[/tex]
- [tex]\(a = 1\)[/tex], [tex]\(b = 0\)[/tex], [tex]\(c = 7\)[/tex]
5. [tex]\(x^2 + 2x + 1 = 5x + 6\)[/tex]
- Rearranging it to the standard form: [tex]\(x^2 + 2x + 1 - 5x - 6 = 0\)[/tex]
- Simplifying: [tex]\(x^2 - 3x - 5 = 0\)[/tex]
- [tex]\(a = 1\)[/tex], [tex]\(b = -3\)[/tex], [tex]\(c = -5\)[/tex]
6. [tex]\((x-1)(x+2) = x(x+5)\)[/tex]
- Expanding both sides: [tex]\(x^2 + x - 2 = x^2 + 5x\)[/tex]
- Bringing all terms to one side: [tex]\(x^2 + x - 2 - x^2 - 5x = 0\)[/tex]
- Simplifying: [tex]\(-4x - 2 = 0\)[/tex]
- This simplifies to a linear equation.
7. [tex]\((x+2)(x-3) = 5\)[/tex]
- Expanding and rearranging: [tex]\(x^2 - x - 6 = 5\)[/tex]
- Bringing all terms to one side: [tex]\(x^2 - x - 6 - 5 = 0\)[/tex]
- Simplifying: [tex]\(x^2 - x - 11 = 0\)[/tex]
- [tex]\(a = 1\)[/tex], [tex]\(b = -1\)[/tex], [tex]\(c = -11\)[/tex]
8. [tex]\(x(x^2 + 3x - 10) = 0\)[/tex]
- Expanding: [tex]\(x^3 + 3x^2 - 10x = 0\)[/tex]
- This is a cubic equation, not quadratic.
9. [tex]\((x-1)^2 + 3 = 2x + 1\)[/tex]
- Expanding and rearranging: [tex]\(x^2 - 2x + 1 + 3 = 2x + 1\)[/tex]
- Bringing all terms to one side: [tex]\(x^2 - 2x + 1 + 3 - 2x - 1 = 0\)[/tex]
- Simplifying: [tex]\(x^2 - 4x + 3 = 0\)[/tex]
- [tex]\(a = 1\)[/tex], [tex]\(b = -4\)[/tex], [tex]\(c = 3\)[/tex]
10. [tex]\((x+2)^3 = x(x^2 - 10x + 25)\)[/tex]
- Expanding both sides: [tex]\(x^3 + 6x^2 + 12x + 8 = x^3 - 10x^2 + 25x\)[/tex]
- Bringing all terms to one side: [tex]\(x^3 + 6x^2 + 12x + 8 - x^3 + 10x^2 - 25x = 0\)[/tex]
- Simplifying: [tex]\(16x^2 - 13x + 8 = 0\)[/tex]
- [tex]\(a = 16\)[/tex], [tex]\(b = -13\)[/tex], [tex]\(c = 8\)[/tex]
Here is the final categorization of which equations are quadratic and their corresponding coefficients:
1. [tex]\(x^2 - 6x + 2 = 0\)[/tex] : [tex]\(a = 1\)[/tex], [tex]\(b = -6\)[/tex], [tex]\(c = 2\)[/tex]
3. [tex]\(2x^2 + 7x = 15\)[/tex] : [tex]\(a = 2\)[/tex], [tex]\(b = 7\)[/tex], [tex]\(c = -15\)[/tex]
4. [tex]\((x+1)^2 = 2(x-3)\)[/tex] : [tex]\(a = 1\)[/tex], [tex]\(b = 0\)[/tex], [tex]\(c = 7\)[/tex]
5. [tex]\(x^2 + 2x + 1 = 5x + 6\)[/tex] : [tex]\(a = 1\)[/tex], [tex]\(b = -3\)[/tex], [tex]\(c = -5\)[/tex]
7. [tex]\((x+2)(x-3) = 5\)[/tex] : [tex]\(a = 1\)[/tex], [tex]\(b = -1\)[/tex], [tex]\(c = -11\)[/tex]
9. [tex]\((x-1)^2 + 3 = 2x + 1\)[/tex] : [tex]\(a = 1\)[/tex], [tex]\(b = -4\)[/tex], [tex]\(c = 3\)[/tex]
10. [tex]\((x+2)^3 = x(x^2 - 10x + 25)\)[/tex] : [tex]\(a = 16\)[/tex], [tex]\(b = -13\)[/tex], [tex]\(c = 8\)[/tex]
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