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We will determine whether each of the given equations is quadratic or not. A quadratic equation is a polynomial equation of degree 2, which generally takes 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].
Let's analyze each equation step-by-step:
1. [tex]\((x+3)+8=0\)[/tex]
Simplifying, we get:
[tex]\(x + 11 = 0\)[/tex], which is a linear equation (degree 1), not a quadratic.
2. [tex]\(x^2 + 10 = 3x\)[/tex]
Rearrange to form a standard polynomial:
[tex]\(x^2 - 3x + 10 = 0\)[/tex], which is a quadratic equation (degree 2).
3. [tex]\(2(x+3)^2=0\)[/tex]
Expanding the equation:
[tex]\(2(x^2 + 6x + 9) = 0\)[/tex]
Simplifies to:
[tex]\(2x^2 + 12x + 18 = 0\)[/tex], which is a quadratic equation (degree 2).
4. [tex]\((x-2)^2 + 5=0\)[/tex]
Expanding the equation:
[tex]\((x^2 - 4x + 4) + 5 = 0\)[/tex]
Simplifies to:
[tex]\(x^2 - 4x + 9 = 0\)[/tex], which is a quadratic equation (degree 2).
5. [tex]\(x^2 - 2x + 5=0\)[/tex]
This is already in the standard form:
[tex]\(x^2 - 2x + 5 = 0\)[/tex], which is a quadratic equation (degree 2).
6. [tex]\(0 = x(x+3)(x-5)\)[/tex]
Expanding the equation:
[tex]\(0 = x(x^2 - 2x - 15)\)[/tex]
Simplifies to:
[tex]\(x^3 - 2x^2 - 15x = 0\)[/tex], which is a cubic equation (degree 3), not a quadratic.
7. [tex]\((x-1)^2=0\)[/tex]
Expanding the equation:
[tex]\(x^2 - 2x + 1 = 0\)[/tex], which is a quadratic equation (degree 2).
8. [tex]\((x+1)(x-2) = 3\)[/tex]
Expanding the left-hand side:
[tex]\(x^2 - x - 2 = 3\)[/tex]
Rearrange to form a standard polynomial:
[tex]\(x^2 - x - 5 = 0\)[/tex], which is a quadratic equation (degree 2).
9. [tex]\(2x^4 + x - 1 = 0\)[/tex]
This is already in polynomial form, but the highest power is:
Degree 4, which is not quadratic (degree 4).
10. [tex]\(\frac{9x}{7} + 7x - 1 = 0\)[/tex]
Combining the terms:
[tex]\(\left(\frac{9}{7} + 7\right)x - 1 = 0\)[/tex]
Just simplifies the coefficients of x, but the degree remains:
Degree 1, which is not quadratic (degree 1).
11. [tex]\(\left[x(x-2)^2 - 3\right] = 7\)[/tex]
Equivalently:
[tex]\(x(x^2 - 4x + 4) - 3 = 7\)[/tex]
Simplifying:
[tex]\(x^3 - 4x^2 + 4x - 3 = 7\)[/tex]
Rearranging:
[tex]\(x^3 - 4x^2 + 4x - 10 = 0\)[/tex], which is a cubic equation (degree 3), not a quadratic.
12. [tex]\(\frac{x^3 + 2}{5} - 6 = 0\)[/tex]
Multiplying by 5 to clear the fraction:
[tex]\(x^3 + 2 - 30 = 0\)[/tex]
Simplifies to:
[tex]\(x^3 - 28 = 0\)[/tex], which is a cubic equation (degree 3), not a quadratic.
Summary:
Quadratic equations from the list are:
- [tex]\(2. x^2 - 3x + 10 = 0\)[/tex]
- [tex]\(3. 2x^2 + 12x + 18 = 0\)[/tex]
- [tex]\(4. x^2 - 4x + 9 = 0\)[/tex]
- [tex]\(5. x^2 - 2x + 5 = 0\)[/tex]
- [tex]\(7. x^2 - 2x + 1 = 0\)[/tex]
- [tex]\(8. x^2 - x - 5 = 0\)[/tex]
Equations that are not quadratic:
- [tex]\(1. x + 11 = 0\)[/tex]
- [tex]\(6. x^3 - 2x^2 - 15x = 0\)[/tex]
- [tex]\(9. 2x^4 + x - 1 = 0\)[/tex]
- [tex]\(10. \left(\frac{9}{7} + 7\right)x - 1 = 0\)[/tex]
- [tex]\(11. x^3 - 4x^2 + 4x - 10 = 0\)[/tex]
- [tex]\(12. x^3 - 28 = 0\)[/tex]
Let's analyze each equation step-by-step:
1. [tex]\((x+3)+8=0\)[/tex]
Simplifying, we get:
[tex]\(x + 11 = 0\)[/tex], which is a linear equation (degree 1), not a quadratic.
2. [tex]\(x^2 + 10 = 3x\)[/tex]
Rearrange to form a standard polynomial:
[tex]\(x^2 - 3x + 10 = 0\)[/tex], which is a quadratic equation (degree 2).
3. [tex]\(2(x+3)^2=0\)[/tex]
Expanding the equation:
[tex]\(2(x^2 + 6x + 9) = 0\)[/tex]
Simplifies to:
[tex]\(2x^2 + 12x + 18 = 0\)[/tex], which is a quadratic equation (degree 2).
4. [tex]\((x-2)^2 + 5=0\)[/tex]
Expanding the equation:
[tex]\((x^2 - 4x + 4) + 5 = 0\)[/tex]
Simplifies to:
[tex]\(x^2 - 4x + 9 = 0\)[/tex], which is a quadratic equation (degree 2).
5. [tex]\(x^2 - 2x + 5=0\)[/tex]
This is already in the standard form:
[tex]\(x^2 - 2x + 5 = 0\)[/tex], which is a quadratic equation (degree 2).
6. [tex]\(0 = x(x+3)(x-5)\)[/tex]
Expanding the equation:
[tex]\(0 = x(x^2 - 2x - 15)\)[/tex]
Simplifies to:
[tex]\(x^3 - 2x^2 - 15x = 0\)[/tex], which is a cubic equation (degree 3), not a quadratic.
7. [tex]\((x-1)^2=0\)[/tex]
Expanding the equation:
[tex]\(x^2 - 2x + 1 = 0\)[/tex], which is a quadratic equation (degree 2).
8. [tex]\((x+1)(x-2) = 3\)[/tex]
Expanding the left-hand side:
[tex]\(x^2 - x - 2 = 3\)[/tex]
Rearrange to form a standard polynomial:
[tex]\(x^2 - x - 5 = 0\)[/tex], which is a quadratic equation (degree 2).
9. [tex]\(2x^4 + x - 1 = 0\)[/tex]
This is already in polynomial form, but the highest power is:
Degree 4, which is not quadratic (degree 4).
10. [tex]\(\frac{9x}{7} + 7x - 1 = 0\)[/tex]
Combining the terms:
[tex]\(\left(\frac{9}{7} + 7\right)x - 1 = 0\)[/tex]
Just simplifies the coefficients of x, but the degree remains:
Degree 1, which is not quadratic (degree 1).
11. [tex]\(\left[x(x-2)^2 - 3\right] = 7\)[/tex]
Equivalently:
[tex]\(x(x^2 - 4x + 4) - 3 = 7\)[/tex]
Simplifying:
[tex]\(x^3 - 4x^2 + 4x - 3 = 7\)[/tex]
Rearranging:
[tex]\(x^3 - 4x^2 + 4x - 10 = 0\)[/tex], which is a cubic equation (degree 3), not a quadratic.
12. [tex]\(\frac{x^3 + 2}{5} - 6 = 0\)[/tex]
Multiplying by 5 to clear the fraction:
[tex]\(x^3 + 2 - 30 = 0\)[/tex]
Simplifies to:
[tex]\(x^3 - 28 = 0\)[/tex], which is a cubic equation (degree 3), not a quadratic.
Summary:
Quadratic equations from the list are:
- [tex]\(2. x^2 - 3x + 10 = 0\)[/tex]
- [tex]\(3. 2x^2 + 12x + 18 = 0\)[/tex]
- [tex]\(4. x^2 - 4x + 9 = 0\)[/tex]
- [tex]\(5. x^2 - 2x + 5 = 0\)[/tex]
- [tex]\(7. x^2 - 2x + 1 = 0\)[/tex]
- [tex]\(8. x^2 - x - 5 = 0\)[/tex]
Equations that are not quadratic:
- [tex]\(1. x + 11 = 0\)[/tex]
- [tex]\(6. x^3 - 2x^2 - 15x = 0\)[/tex]
- [tex]\(9. 2x^4 + x - 1 = 0\)[/tex]
- [tex]\(10. \left(\frac{9}{7} + 7\right)x - 1 = 0\)[/tex]
- [tex]\(11. x^3 - 4x^2 + 4x - 10 = 0\)[/tex]
- [tex]\(12. x^3 - 28 = 0\)[/tex]
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