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If u2−ut=6u^2-ut=6 and t2−ut=−2t^2-ut=-2, what is the value of 2(u−t)22(u-t)^2?

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Q. If u2−ut=6u^2-ut=6 and t2−ut=−2t^2-ut=-2, what is the value of 2(u−t)22(u-t)^2?
  1. Add Equations: First, we need to add the two given equations to eliminate the term −ut-ut. By adding u2−ut=6u^2 - ut = 6 and t2−ut=−2t^2 - ut = -2, we get u2+t2−2utu^2 + t^2 - 2ut.\newlineCalculation: (u2−ut)+(t2−ut)=6+(−2)(u^2 - ut) + (t^2 - ut) = 6 + (-2)
  2. Simplify Equation: Simplify the left side of the equation by combining like terms and the right side by adding 66 and −2-2.\newlineCalculation: u2+t2−2ut=4u^2 + t^2 - 2ut = 4
  3. Recognize Expansion: Recognize that u2+t2−2utu^2 + t^2 - 2ut is the expansion of (u−t)2(u - t)^2. Therefore, (u−t)2=4(u - t)^2 = 4.\newlineCalculation: (u−t)2=4(u - t)^2 = 4
  4. Multiply by 22: To find the value of 2(u−t)22(u-t)^2, we multiply the equation (u−t)2=4(u - t)^2 = 4 by 22.\newlineCalculation: 2(u−t)2=2×42(u - t)^2 = 2 \times 4
  5. Calculate Final Answer: Calculate the result of the multiplication to get the final answer.\newlineCalculation: 2(u−t)2=82(u - t)^2 = 8

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