Question

Which of the following is a quadratic equation?

• A. \((x + 3)(x - 3) = x^2 - 4x^3\)
• B. \((x+3)^2 = 4(x+4)\)
• C. \((2x-2)^2 = 4x^2 + 7\)
• D. \(4x + \frac{1}{4x} = 4x\)

Hint:

To check if an equation is quadratic, expand all terms and move them to one side. If the highest power of the variable that remains is 2, it is a quadratic equation.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A quadratic equation is a polynomial equation of the second degree, meaning the highest exponent of the variable is 2. The standard form is \(ax^2 + bx + c = 0\), where \(a \neq 0\). We need to simplify each option to see which one fits this form.

Step 2: Detailed Explanation:
(A) \((x + 3)(x - 3) = x^2 - 4x^3 $\Rightarrow$ x^2 - 9 = x^2 - 4x^3 $\Rightarrow$ 4x^3 - 9 = 0\). The highest power is 3, so this is a cubic equation.
(B) \((x+3)^2 = 4(x+4) $\Rightarrow$ x^2 + 6x + 9 = 4x + 16 $\Rightarrow$ x^2 + 2x - 7 = 0\). The highest power is 2, so this is a quadratic equation.
(C) \((2x-2)^2 = 4x^2 + 7 $\Rightarrow$ 4x^2 - 8x + 4 = 4x^2 + 7 $\Rightarrow$ -8x - 3 = 0\). The \(x^2\) terms cancel out, leaving a linear equation.
(D) \(4x + \frac{1}{4x} = 4x $\Rightarrow$ \frac{1}{4x} = 0\). This equation has no solution and is not a polynomial equation. If we multiply by \(4x\), we get \(1 = 0\), which is a contradiction.

Step 3: Final Answer:
The equation that simplifies to a quadratic form is \((x+3)^2 = 4(x+4)\).