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Rewrite the function by completing the square. {:[f(x)=x^(2)-12 x+50],[f(x)=(x+◻)^(2)+◻]:}

Rewrite the function by completing the square.\newlinef(x)=x2−12x+50 f(x) = x^2 - 12x + 50 , f(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square

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Q. Rewrite the function by completing the square.\newlinef(x)=x2−12x+50 f(x) = x^2 - 12x + 50 , f(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square
  1. Given quadratic function: We start with the given quadratic function:\newlinef(x) = x2−12x+50x^2 - 12x + 50\newlineWe want to rewrite this function in the form of (x+a)2+b(x + a)^2 + b.\newlineTo complete the square, we need to find a value that, when added and subtracted to the x2−12xx^2 - 12x part, completes the square.
  2. Rewriting in desired form: The coefficient of xx is −12-12. To complete the square, we take half of this coefficient and square it. This gives us (−122)2=(−6)2=36\left(\frac{-12}{2}\right)^2 = (-6)^2 = 36.\newlineWe will add and subtract this value inside the function to complete the square.
  3. Completing the square: We add 3636 and subtract 3636 from the function to maintain the equality:\newlinef(x) = (x2−12x+36)−36+50(x^2 - 12x + 36) - 36 + 50\newlineNow, the first three terms form a perfect square trinomial.
  4. Factoring the perfect square trinomial: We factor the perfect square trinomial:\newlinef(x) = (x−6)2−36+50(x - 6)^2 - 36 + 50\newlineNow, we combine the constant terms (−36+50)(-36 + 50) to simplify the function.
  5. Combining constant terms: Combining the constant terms gives us:\newlinef(x) = (x - 66)^22 + 1414\newlineThis is the function in completed square form.

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