Which of the following are quadratic equations?
(i)`x^(2)8x+12=0` (ii) `x+(1)/(x)=5` (iii) `x+(5)/(x)=x^(2)`
(iv) `x^(2)-5sqrtx+7=0` (v) `x^(2)-5x-sqrtx+4=0` (vi) `x^(2)-(1)/(x^(2))=4`
(vii) `5x^(2)-7x=3x^(2)-7x+3` (viii) `(1)/(4)x^(2)+(7)/(6)x-2=0`

Answer

To determine which of the given equations are quadratic equations, 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 \( a \neq 0 \).

Let's analyze each equation step by step:

### Step 1: Analyze each equation

**(i)** \( x^2 + 8x + 12 = 0 \)

This is already in the standard form \( ax^2 + bx + c = 0 \) where \( a = 1, b = 8, c = 12 \).

**Conclusion:** This is a quadratic equation.

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**(ii)** \( x + \frac{1}{x} = 5 \)

To convert this into standard form, multiply through by \( x \) (assuming \( x \neq 0 \)):

\[ x^2 + 1 = 5x \]

Rearranging gives:

\[ x^2 - 5x + 1 = 0 \]

**Conclusion:** This is a quadratic equation.

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**(iii)** \( x + \frac{5}{x} = x^2 \)

Multiply through by \( x \) (assuming \( x \neq 0 \)):

\[ x^2 + 5 = x^3 \]

Rearranging gives:

\[ x^3 - x^2 - 5 = 0 \]

The highest power of \( x \) is 3.

**Conclusion:** This is NOT a quadratic equation.

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**(iv)** \( x^2 - 5\sqrt{x} + 7 = 0 \)

The term \( \sqrt{x} \) indicates that this is not a standard quadratic form since it involves a square root.

**Conclusion:** This is NOT a quadratic equation.

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**(v)** \( x^2 - 5x - \sqrt{x} + 4 = 0 \)

Similar to the previous case, the term \( \sqrt{x} \) makes this not a standard quadratic equation.

**Conclusion:** This is NOT a quadratic equation.

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**(vi)** \( x^2 - \frac{1}{x^2} = 4 \)

Multiply through by \( x^2 \) (assuming \( x \neq 0 \)):

\[ x^4 - 4x^2 - 1 = 0 \]

The highest power of \( x \) is 4.

**Conclusion:** This is NOT a quadratic equation.

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**(vii)** \( 5x^2 - 7x = 3x^2 - 7x + 3 \)

Rearranging gives:

\[ 5x^2 - 3x^2 - 7x + 7x - 3 = 0 \]

This simplifies to:

\[ 2x^2 - 3 = 0 \]

**Conclusion:** This is a quadratic equation.

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**(viii)** \( \frac{1}{4}x^2 + \frac{7}{6}x - 2 = 0 \)

This is already in the standard form \( ax^2 + bx + c = 0 \) where \( a = \frac{1}{4}, b = \frac{7}{6}, c = -2 \).

**Conclusion:** This is a quadratic equation.

### Summary of Results:
- (i) Quadratic Equation
- (ii) Quadratic Equation
- (iii) Not a Quadratic Equation
- (iv) Not a Quadratic Equation
- (v) Not a Quadratic Equation
- (vi) Not a Quadratic Equation
- (vii) Quadratic Equation
- (viii) Quadratic Equation