Question

If $\left(x-\dfrac{1}{2}\right)^2 - \left(x-\dfrac{3}{2}\right)^2 = x + 2$, then the value of $x$ is:

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Hint:

Using the identity $a^2 - b^2 = (a-b)(a+b)$ can often simplify such expressions faster.

Answer 1

The Correct Option is B

Let's solve the given equation step by step: \((x-\frac{1}{2})^2 - (x-\frac{3}{2})^2 = x + 2\).

1. First, apply the formula for the difference of squares: \((a^2 - b^2 = (a-b)(a+b))\). Here, \(a = x - \frac{1}{2}\) and \(b = x - \frac{3}{2}\).
2. Calculate \((a-b)\): \(((x - \frac{1}{2}) - (x - \frac{3}{2})) = \frac{3}{2} - \frac{1}{2} = 1\).
3. Calculate \((a+b)\): \(((x - \frac{1}{2}) + (x - \frac{3}{2})) = 2x - 2\). 
4. Now substitute these into the difference of squares formula: \((a - b)(a + b) = (1)(2x - 2)\). Thus, \((x - \frac{1}{2})^2 - (x - \frac{3}{2})^2 = 2x - 2\).
5. The original equation becomes: \(2x - 2 = x + 2\).
6. Bring all terms involving \(x\) on one side: \(2x - x = 2 + 2\), which simplifies to \(x = 4\).

Therefore, the value of \(x\) is 4.

Let's verify the options:

• If \(x = 2\), \((1.5)^2 - (0.5)^2 = 2 + 2\), simplifies to \(2 \neq 4\).
• If \(x = 4\), \((3.5)^2 - (2.5)^2 = 4 + 2\), simplifies to \(12 - 6 = 6\) is true and correct.
• If \(x = 6\) or \(x = 8\), the LHS does not equal RHS in similar verification.