Question

The roots of the quadratic equation \(4x^2 - (a - 1)^2 = 0\) are :

• A. \(a - 1, a + 1\)
• B. \(\frac{a - 1}{2}, \frac{-a + 1}{2}\)
• C. \(\frac{a - 1}{2}, \frac{-a - 1}{2}\)
• D. \(\pm (a - 1)\)

Hint:

Using basic algebraic isolation is much faster here than using the full quadratic formula.
Always distribute the negative sign carefully: \(-\frac{a - 1}{2}\) becomes \(\frac{-a + 1}{2}\).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a quadratic equation \(4x^2 - (a - 1)^2 = 0\).
We need to find the roots of this equation in terms of the parameter \(a\).

Step 2: Key Formula or Approach:
The equation is of the form \(A^2 - B^2 = 0\), which can be solved easily by isolating \(x^2\) or by factoring using the identity:
\[ A^2 - B^2 = (A - B)(A + B) \]

Step 3: Detailed Explanation:
1. Write the given equation:
\[ 4x^2 - (a - 1)^2 = 0 \] 2. Isolate the term containing \(x\):
\[ 4x^2 = (a - 1)^2 \] 3. Divide both sides by \(4\):
\[ x^2 = \frac{(a - 1)^2}{4} \] 4. Take the square root of both sides:
\[ x = \pm \sqrt{\frac{(a - 1)^2}{4}} \] \[ x = \pm \frac{a - 1}{2} \] 5. Write down the two distinct solutions:
- First root:
\[ x_1 = \frac{a - 1}{2} \] - Second root:
\[ x_2 = -\left(\frac{a - 1}{2}\right) = \frac{-(a - 1)}{2} = \frac{-a + 1}{2} \] 6. This matches the roots given in option (B).

Step 4: Final Answer:
The correct option is (B).