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Rewrite the function by completing the square. {:[f(x)=x^(2)-12 x-29],[f(x)=(x+◻)^(2)+◻]:}

Rewrite the function by completing the square.\newlinef(x)=x2−12x−29 f(x) = x^2 - 12x - 29 , 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−29 f(x) = x^2 - 12x - 29 , 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−29x^2 - 12x - 29\newlineOur goal is to rewrite this function in the form of (x+a)2+b(x + a)^2 + b.
  2. Completing the square: First, we need to complete the square for the x2−12xx^2 - 12x part of the function. To do this, we find the value that needs to be added and subtracted to complete the square. This value is (b2)2(\frac{b}{2})^2, where bb is the coefficient of xx, which in this case is −12-12.\newline(−122)2=36\left(\frac{-12}{2}\right)^2 = 36
  3. Adding and subtracting to complete the square: We add and subtract 3636 inside the function to complete the square:\newlinef(x) = (x2−12x+36)−36−29(x^2 - 12x + 36) - 36 - 29
  4. Factoring the quadratic expression: Now we factor the quadratic expression in the parentheses:\newlinef(x) = (x−6)2−36−29(x - 6)^2 - 36 - 29
  5. Simplifying the function: Combine the constants to simplify the function: f(x)=(x−6)2−65f(x) = (x - 6)^2 - 65

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