If \(\left( x – \dfrac{1}{2} \right)^2 – \left( x- \dfrac{3}{2} \right) ^2 =…
2021
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
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Correct answer: B
Key step: apply the difference of squares a^2 - b^2 = (a - b)(a + b).
Let a = x - 1/2 and b = x - 3/2. Then (x - 1/2)^2 - (x - 3/2)^2 = (a - b)(a + b).
Compute a - b: (x - 1/2) - (x - 3/2) = 1.
Compute a + b: (x - 1/2) + (x - 3/2) = 2x - 2 = 2(x - 1).
Therefore the left side equals 1 * 2(x - 1) = 2x - 2.
Set this equal to the right side: 2x - 2 = x + 2. Solve: 2x - x = 2 + 2, so x = 4.
Check: For x = 4, left = (3.5)^2 - (2.5)^2 = 12.25 - 6.25 = 6 and right = 4 + 2 = 6, so x = 4 is correct.