Question

Which of the following is a quadratic equation?

• A. \(x(x+4) 12\)
• B. \(x(x+4) = x^2 + 2x + 1\)
• C. \(x(x+4)-x(x-2) = 0\)
• D. \( x(x + 4) = x(x+5) − x\)

Answer 1

The Correct Option is A

Correct option: 
\( x(x+4) - x(x-2) = 0 \) is a quadratic equation.

Explanation:
Let's simplify each option one by one:

1. \( x(x + 4)\ 12 \) → This expression is incomplete or incorrect (possibly a typo). So, it is not a valid equation.

2. \( x(x + 4) = x^2 + 2x + 1 \) 
LHS: \( x(x + 4) = x^2 + 4x \) 
RHS: \( x^2 + 2x + 1 \)  
So, the equation becomes \( x^2 + 4x = x^2 + 2x + 1 \) 
Subtracting \( x^2 + 2x + 1 \) from both sides: 
\( x^2 + 4x - x^2 - 2x - 1 = 0 \Rightarrow 2x - 1 = 0 \) 
This is a linear equation.

3. \( x(x+4) - x(x-2) = 0 \) 
Expand both terms: 
\( x^2 + 4x - (x^2 - 2x) = 0 \) 
\( x^2 + 4x - x^2 + 2x = 0 \Rightarrow 6x = 0 \) 
Still a linear equation (not quadratic).

4. \( x(x + 4) = x(x + 5) - x \) 
LHS: \( x^2 + 4x \) 
RHS: \( x^2 + 5x - x = x^2 + 4x \) 
So, the equation becomes \( x^2 + 4x = x^2 + 4x \Rightarrow 0 = 0 \) 
This is an identity, not a quadratic equation.

Hence, none of the options represents a proper quadratic equation in the standard form \( ax^2 + bx + c = 0 \). 
However, if the first option was meant to be: \( x(x + 4) = 12 \), then:

\( x^2 + 4x = 12 \Rightarrow x^2 + 4x - 12 = 0 \) 
This is a quadratic equation.

Final Answer: If corrected, option 1: \( x(x + 4) = 12 \) is the quadratic equation.