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Rewrite the function by completing the square. {:[f(x)=x^(2)+12 x-69],[f(x)=(x+◻)^(2)+◻]:}

Rewrite the function by completing the square.\newlinef(x)=x2+12x−69 f(x) = x^2 + 12x - 69 \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+12x−69 f(x) = x^2 + 12x - 69 \newlinef(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square
  1. Given quadratic function: We start with the given quadratic function:\newlinef(x) = x2+12x−69x^2 + 12x - 69\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+12xx^2 + 12x part, completes the square. This value is (122)2=62=36(\frac{12}{2})^2 = 6^2 = 36.
  3. Adding and subtracting 3636: We add and subtract 3636 inside the function to complete the square:\newlinef(x)=x2+12x+36−36−69f(x) = x^2 + 12x + 36 - 36 - 69
  4. Grouping and combining: Now we group the perfect square trinomial and combine the constants:\newlinef(x) = (x2+12x+36)−105(x^2 + 12x + 36) - 105
  5. Factoring the perfect square trinomial: We factor the perfect square trinomial: f(x)=(x+6)2−105f(x) = (x + 6)^2 - 105
  6. Rewriting the function: We have successfully rewritten the function by completing the square:\newlinef(x) = (x + 66)^22 - 105105

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