Which of the following are quadratic equations in x?
`(i)" "x^(2)-x+3=0" "(ii)" "2x^(2)+(5)/(2)x-sqrt(3)=0`
`(iii)" "sqrt(2)x^(2)+7x+5sqrt(2)=0" "(iv)" "(1)/(3)x^(2)+(1)/(5)x-2=0`
`(v)" "x^(2)-3x-sqrt(x)+4=0" "(vi)x-(6)/(x)=3`
`(vii)" "x+(2)/(x)=x^(2)" "(viii)" "x^(2)-(1)/(x^(2))=5`
`(ix)" "(x+2)^(3)=x^(3)-8" "(x)" "(2x+3)(3x+2)=6(x-1)(x-2)`
(xi)`" "(x+(1)/(x))^(2)=2(x+(1)/(x))+3`

Answer

To determine which of the given equations are quadratic equations in \( x \), we need to check if they can be expressed in the standard form of a quadratic equation, which is:

\[ Ax^2 + Bx + C = 0 \]

where \( A \), \( B \), and \( C \) are constants and \( A \neq 0 \).

Let's analyze each equation step by step:

### Step 1: Analyze each equation

1. **Equation (i)**: \( x^2 - x + 3 = 0 \)
- Here, \( A = 1 \), \( B = -1 \), \( C = 3 \).
- This is a quadratic equation.

2. **Equation (ii)**: \( 2x^2 + \frac{5}{2}x - \sqrt{3} = 0 \)
- Here, \( A = 2 \), \( B = \frac{5}{2} \), \( C = -\sqrt{3} \).
- This is a quadratic equation.

3. **Equation (iii)**: \( \sqrt{2}x^2 + 7x + 5\sqrt{2} = 0 \)
- Here, \( A = \sqrt{2} \), \( B = 7 \), \( C = 5\sqrt{2} \).
- This is a quadratic equation.

4. **Equation (iv)**: \( \frac{1}{3}x^2 + \frac{1}{5}x - 2 = 0 \)
- Here, \( A = \frac{1}{3} \), \( B = \frac{1}{5} \), \( C = -2 \).
- This is a quadratic equation.

5. **Equation (v)**: \( x^2 - 3x - \sqrt{x} + 4 = 0 \)
- This contains \( -\sqrt{x} \), which is not a linear term in \( x \).
- This is **not** a quadratic equation.

6. **Equation (vi)**: \( x - \frac{6}{x} = 3 \)
- Multiply through by \( x \) (assuming \( x \neq 0 \)):
\[ x^2 - 6 = 3x \implies x^2 - 3x - 6 = 0 \]
- This is a quadratic equation.

7. **Equation (vii)**: \( x + \frac{2}{x} = x^2 \)
- Multiply through by \( x \) (assuming \( x \neq 0 \)):
\[ x^2 + 2 = x^3 \implies x^3 - x^2 - 2 = 0 \]
- This is **not** a quadratic equation (it is cubic).

8. **Equation (viii)**: \( x^2 - \frac{1}{x^2} = 5 \)
- Multiply through by \( x^2 \) (assuming \( x \neq 0 \)):
\[ x^4 - 5x^2 - 1 = 0 \]
- This is **not** a quadratic equation (it is quartic).

9. **Equation (ix)**: \( (x + 2)^3 = x^3 - 8 \)
- Expand \( (x + 2)^3 \):
\[ x^3 + 6x^2 + 12x + 8 = x^3 - 8 \]
- Rearranging gives:
\[ 6x^2 + 12x + 16 = 0 \]
- This is a quadratic equation.

10. **Equation (x)**: \( (2x + 3)(3x + 2) = 6(x - 1)(x - 2) \)
- Expand both sides:
\[ 6x^2 + 12x + 9 = 6(x^2 - 3x + 2) \]
- Rearranging gives:
\[ 6x^2 + 12x + 9 - 6x^2 + 18x - 12 = 0 \implies 30x - 3 = 0 \]
- This is **not** a quadratic equation (it is linear).

11. **Equation (xi)**: \( \left(x + \frac{1}{x}\right)^2 = 2\left(x + \frac{1}{x}\right) + 3 \)
- Let \( y = x + \frac{1}{x} \):
\[ y^2 = 2y + 3 \implies y^2 - 2y - 3 = 0 \]
- This is a quadratic equation in \( y \), but not in \( x \).

### Step 2: Summary of Quadratic Equations
The following equations are quadratic equations in \( x \):
- (i) \( x^2 - x + 3 = 0 \)
- (ii) \( 2x^2 + \frac{5}{2}x - \sqrt{3} = 0 \)
- (iii) \( \sqrt{2}x^2 + 7x + 5\sqrt{2} = 0 \)
- (iv) \( \frac{1}{3}x^2 + \frac{1}{5}x - 2 = 0 \)
- (vi) \( x^2 - 3x - 6 = 0 \)
- (ix) \( 6x^2 + 12x + 16 = 0 \)