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Rewrite the function by completing the square. f(x)=x2−16x−100f(x)=x^2-16x-100

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Full solution

Q. Rewrite the function by completing the square. f(x)=x2−16x−100f(x)=x^2-16x-100
  1. Identify coefficients: Identify the coefficients of the quadratic function f(x)=x2−16x−100f(x) = x^2 - 16x - 100. Here, a=1a = 1, b=−16b = -16, and c=−100c = -100.
  2. Calculate perfect square trinomial: To complete the square, we need to find a number to add and subtract to the function that will create a perfect square trinomial from the x2x^2 and xx terms.\newlineThe number to add and subtract is given by (b/2)2(b/2)^2.\newlineCalculate (b/2)2=(−16/2)2=(−8)2=64(b/2)^2 = (-16/2)^2 = (-8)^2 = 64.
  3. Add and subtract number: Add and subtract the calculated number inside the function.\newlinef(x)=x2−16x+64−64−100f(x) = x^2 - 16x + 64 - 64 - 100.
  4. Group trinomial and constants: Group the perfect square trinomial and the constants.\newlinef(x)=(x2−16x+64)−64−100f(x) = (x^2 - 16x + 64) - 64 - 100.
  5. Factor perfect square trinomial: Factor the perfect square trinomial. f(x)=(x−8)2−64−100f(x) = (x - 8)^2 - 64 - 100.
  6. Combine constants: Combine the constants to simplify the function.\newlinef(x)=(x−8)2−164f(x) = (x - 8)^2 - 164.

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