To determine which of the given options is a quadratic equation, let's analyze each one step-by-step:
### Option (a) [tex]\( x + \frac{1}{x} = 4 \)[/tex]
Convert this to a standard form of a quadratic equation:
[tex]\[ x + \frac{1}{x} = 4 \][/tex]
Multiply through by [tex]\( x \)[/tex] to clear the fraction:
[tex]\[ x^2 + 1 = 4x \][/tex]
Rearrange to standard form:
[tex]\[ x^2 - 4x + 1 = 0 \][/tex]
This is indeed a quadratic equation because it is of the form [tex]\( ax^2 + bx + c = 0 \)[/tex] where [tex]\( a = 1 \)[/tex], [tex]\( b = -4 \)[/tex], and [tex]\( c = 1 \)[/tex].
### Option (b) [tex]\( x(x-2) \)[/tex]
Expand the expression:
[tex]\[ x(x-2) = x^2 - 2x \][/tex]
This simplifies to [tex]\( x^2 - 2x \)[/tex], which is a quadratic expression. Setting it equal to zero, we get:
[tex]\[ x^2 - 2x = 0 \][/tex]
This is a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex] where [tex]\( a = 1 \)[/tex], [tex]\( b = -2 \)[/tex], and [tex]\( c = 0 \)[/tex].
### Option (c) [tex]\( (x+1)^2 = x^2 - 3x + 5 \)[/tex]
First, expand the left side:
[tex]\[ (x+1)^2 = x^2 + 2x + 1 \][/tex]
Set the equation to standard form:
[tex]\[ x^2 + 2x + 1 = x^2 - 3x + 5 \][/tex]
Subtract [tex]\( x^2 \)[/tex] from both sides:
[tex]\[ 2x + 1 = -3x + 5 \][/tex]
Combine like terms:
[tex]\[ 2x + 3x = 5 - 1 \][/tex]
[tex]\[ 5x = 4 \][/tex]
[tex]\[ x = \frac{4}{5} \][/tex]
This is a linear equation rather than a quadratic equation.
### Option (d) [tex]\( x^2 - \sqrt{x} - 5 = 0 \)[/tex]
This equation has a term [tex]\( \sqrt{x} \)[/tex], which is not in the form of [tex]\( ax^2 + bx + c = 0 \)[/tex].
Thus, it is not a quadratic equation due to the presence of the square root term.
### Conclusion
From the analysis:
- Option (a) is actually a quadratic equation after simplification.
- Option (b) is clearly a quadratic equation.
- Option (c) simplifies to a linear equation.
- Option (d) is not a quadratic equation due to the square root term.
Therefore, the correct answer is:
(b) [tex]\( x(x-2) \)[/tex]
