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Rewrite the function by completing the square. {:[f(x)=x^(2)-2x+17],[f(x)=(x+◻)^(2)+◻]:}

Rewrite the function by completing the square.\newlinef(x)=x2−2x+17 f(x) = x^2 - 2x + 17 , f(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square

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Q. Rewrite the function by completing the square.\newlinef(x)=x2−2x+17 f(x) = x^2 - 2x + 17 , f(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square
  1. Identify coefficient of x: We start with the function f(x) = x^22 - 22x + 1717 and want to rewrite it in the form f(x) = (x + a)^22 + b. To complete the square, we need to find a value that, when added and subtracted to the x^22 - 22x part, creates a perfect square trinomial.
  2. Find value to complete the square: First, we identify the coefficient of xx, which is −2-2. To find the value to complete the square, we take half of this coefficient and square it. This value is (−22)2=(−1)2=1\left(\frac{-2}{2}\right)^2 = (-1)^2 = 1.
  3. Add and subtract to create perfect square trinomial: We add and subtract this value inside the function to create a perfect square trinomial. We have to be careful to maintain the equality of the function, so we add 11 and subtract 11 inside the function.\newlinef(x)=(x2−2x+1)+17−1f(x) = (x^2 - 2x + 1) + 17 - 1
  4. Factor and simplify: Now we factor the perfect square trinomial and simplify the constants.\newlinef(x)=(x−1)2+16f(x) = (x - 1)^2 + 16
  5. Rewritten function using completed square: We have successfully rewritten the function by completing the square. The completed square form of the function is f(x)=(x−1)2+16f(x) = (x - 1)^2 + 16.

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