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Best answer

Given that, $\left(x – \dfrac{1}{2}\right)^{2} – \left(x – \dfrac{3}{2}\right)^{2} = x + 2$

$\implies \left(x – \dfrac{1}{2} – x + \dfrac{3}{2}\right) \left(x – \dfrac{1}{2} + x - \dfrac{3}{2}\right) = x + 2 \quad [\because a^{2} – b^{2} = (a – b)(a + b) ]$

$\implies 2x - 2 = x + 2$

$\implies x = 4$

So, the correct answer is $(B).$

$\implies \left(x – \dfrac{1}{2} – x + \dfrac{3}{2}\right) \left(x – \dfrac{1}{2} + x - \dfrac{3}{2}\right) = x + 2 \quad [\because a^{2} – b^{2} = (a – b)(a + b) ]$

$\implies 2x - 2 = x + 2$

$\implies x = 4$

So, the correct answer is $(B).$

2 votes

**Option B**

$$(x – \frac{1}{2})^2 – (x – \frac{3}{2})^2 = x + 2$$

$$(\frac{2x-1}{2})^2 – (\frac{2x-3}{2})^2 = x+2$$

$$\frac{(2x-1)^2 – (2x – 3)^2}{4} = x + 2$$

$$(2x – 1 + 2x – 3)(2x – 1 – (2x – 3)) = 4(x + 2)$$

$$(4x – 4)(2) = 4(x + 2)$$

$$4(x – 1)(2) = 4(x + 2)$$

$$2x – 2 = x + 2$$

$$x = 4$$

Of course simple substitution would be faster.