Question

Which of the equations among the following is/are quadratic equation(s)? \(q_1 : x^2 + x = (x+1)^2\), \(q_2 : x-1 = x^2 - 1\), \(q_3 : x = x^2\), \(q_4 : \sqrt{x} = x^2 \sqrt{x+1}\)

• A. \(q_1\) only
• B. \(q_1, q_2\) and \(q_3\) only
• C. \(q_2\) and \(q_3\) only
• D. \(q_2\) and \(q_4\) only

Hint:

Don't just look for an \(x^2\); ensure that the \(x^2\) term doesn't cancel out during simplification (like in \(q_1\)).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A quadratic equation must be expressible in the standard form \(ax^2 + bx + c = 0\), where \(a \neq 0\). The highest power (degree) of the variable must be exactly 2 after simplification.
Step 2: Key Formula or Approach:
Simplify each equation and check the degree of the polynomial.
Step 3: Detailed Explanation:
1. \(q_1\): \(x^2 + x = x^2 + 2x + 1 \rightarrow x + 1 = 0\). This is Linear (degree 1).
2. \(q_2\): \(x^2 - x = 0\). This is Quadratic (degree 2).
3. \(q_3\): \(x^2 - x = 0\). This is Quadratic (degree 2).
4. \(q_4\): Contains square roots and higher powers after squaring. It is not a polynomial equation in the standard quadratic sense.
Step 4: Final Answer:
The quadratic equations are \(q_2\) and \(q_3\).