Question

Which of the following is a quadratic equation?

• A. \( (x+2)(x-2)=x^2-4x^3 \)
• B. \( (x+2)^2=3(x+4) \)
• C. \( (2x^2+3)=(5+x)(2x^2-3) \)
• D. \( 2x+\dfrac{1}{2x}=4x^2 \)

Hint:

To identify a quadratic, first clear denominators if any, expand, and collect like terms. If the highest power of \(x\) is exactly \(2\), it’s quadratic.

Answer 1

The Correct Option is B

Step 1: Test each option by bringing all terms to one side and checking the highest power of \(x\).
(1) \((x+2)(x-2)=x^2-4x^3 \Rightarrow x^2-4=x^2-4x^3 \Rightarrow 4x^3-4=0\).
Highest degree \(=3\) (cubic) \(\Rightarrow\) Not quadratic.
(2) \((x+2)^2=3(x+4) \Rightarrow x^2+4x+4=3x+12 \Rightarrow x^2+x-8=0\).
Highest degree \(=2\) (quadratic) \(\Rightarrow\) Quadratic.
(3) \((2x^2+3)=(5+x)(2x^2-3)\). RHS contains \(x\cdot 2x^2=2x^3\) \(\Rightarrow\) degree \(3\). After simplifying, equation is cubic \(\Rightarrow\) Not quadratic.
(4) \(2x+\dfrac{1}{2x}=4x^2\). Multiply by \(2x\): \(4x^2+1=8x^3 \Rightarrow 8x^3-4x^2-1=0\).
Highest degree \(=3\) (cubic) \(\Rightarrow\) Not quadratic.
Step 2: Conclude.
Only option \((2)\) reduces to a degree-\(2\) equation, hence it is quadratic.