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: Recognise the equation as a difference of two squares.
The equation $4x^{2} - (a-1)^{2} = 0$ can be written as $(2x)^{2} - (a-1)^{2} = 0$, which is in the form $A^{2}-B^{2}=0$ with $A = 2x$ and $B = a-1$.

Step 2: Factor using the identity $A^{2}-B^{2}=(A-B)(A+B)$.
\[ (2x - (a-1))(2x + (a-1)) = 0 \]

Step 3: Set each factor to zero and solve.
From the first factor:
\[ 2x - (a-1) = 0 \implies x = \frac{a-1}{2} \]
From the second factor:
\[ 2x + (a-1) = 0 \implies x = \frac{-(a-1)}{2} = \frac{-a+1}{2} \]

Final Answer:
The two roots are $\dfrac{a-1}{2}$ and $\dfrac{-a+1}{2}$, matching option (B).
\[ \boxed{x = \frac{a-1}{2}, \frac{-a+1}{2}} \]