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

To determine which of the given equations are quadratic equations, we need to evaluate each equation individually and check their forms.

1. Equation \(q_1 : x^2 + x = (x+1)^2\)

The first step is to expand the right-hand side:

\(x^2 + x = (x + 1)^2 = x^2 + 2x + 1\)

Re-arranging the terms gives:

\(x^2 + x = x^2 + 2x + 1 \Rightarrow 0 = x + 1 \Rightarrow x = -1\)

This is not a quadratic equation because, after simplification, we are left with a linear equation \( x = -1 \).

1. Equation \(q_2 : x-1 = x^2 - 1\)

Re-arranging terms gives:

\(x - 1 = x^2 - 1 \Rightarrow x^2 - x = 0 \Rightarrow x(x - 1) = 0\)

This is a quadratic equation because it can be written in the standard form \(ax^2 + bx + c = 0\) where \(a = 1\), \(b = -1\), and \(c = 0\).

1. Equation \(q_3 : x = x^2\)

Re-arranging terms gives:

\(x = x^2 \Rightarrow x^2 - x = 0 \Rightarrow x(x - 1) = 0\)

This is a quadratic equation, similar to \(q_2\).

1. Equation \(q_4 : \sqrt{x} = x^2 \sqrt{x+1}\)

This equation involves a square root and does not lend itself to a quadratic form since isolating \( x^2 \) would still involve irrational terms and the expression does not simplify to a polynomial of degree 2. Therefore, it is not a quadratic equation.

After analyzing each equation, the quadratic equations among the given options are:

• \(q_2\) and \(q_3\)

Therefore, the correct answer is: \(q_2\) and \(q_3\) only.