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Rewrite the function by completing the square. {:[f(x)=x^(2)-16 x-100],[f(x)=(x+◻)^(2)+◻]:}

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

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Q. Rewrite the function by completing the square.\newlinef(x)=x2−16x−100 f(x) = x^2 - 16x - 100 \newlinef(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square
  1. Given quadratic function: We start with the given quadratic function:\newlinef(x) = x2−16x−100x^2 - 16x - 100\newlineOur goal is to rewrite this function in the form of (x+a)2+b(x + a)^2 + b.
  2. Completing the square: To complete the square, we need to find a value that, when added and subtracted to the x2−16xx^2 - 16x part, completes the square. This value is found by taking half of the coefficient of xx, squaring it, and then adding it to both sides.\newlineThe coefficient of xx is −16-16, so half of −16-16 is −8-8, and (−8)2=64(-8)^2 = 64.
  3. Adding and subtracting: We add and subtract 6464 to the function to complete the square:\newlinef(x)=x2−16x+64−64−100f(x) = x^2 - 16x + 64 - 64 - 100\newlineNow we have a perfect square trinomial x2−16x+64x^2 - 16x + 64 and a constant −164-164.
  4. Factoring the perfect square trinomial: We factor the perfect square trinomial:\newlinef(x) = (x - 88)^22 - 164164\newlineNow the function is in the completed square form.
  5. Checking the work: We check our work by expanding (x−8)2(x - 8)^2 to ensure it gives us back the original x2−16xx^2 - 16x term:\newline(x−8)2=x2−16x+64(x - 8)^2 = x^2 - 16x + 64\newlineThis matches the x2−16x+64x^2 - 16x + 64 part of our function, so there is no math error.

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