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Rewrite the function by completing the square. {:[f(x)=x^(2)-2x-95],[f(x)=(x+◻)^(2)+◻]:}

Rewrite the function by completing the square.\newlinef(x)=x2−2x−95 f(x) = x^2 - 2x - 95 , 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−95 f(x) = x^2 - 2x - 95 , f(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square
  1. Starting with the function: We start with the function f(x)=x2−2x−95f(x) = x^2 - 2x - 95 and want to rewrite it in the form f(x)=(x+□)2+□f(x) = (x + \square)^2 + \square. To complete the square, we need to find a number that, when added to x2−2xx^2 - 2x, creates a perfect square trinomial.
  2. Finding the number to add: First, we take the coefficient of xx, which is −2-2, divide it by 22, and square the result to find the number to add to both sides. This will be our □\square value inside the parentheses.\newline(−22)2=(−1)2=1\left(\frac{-2}{2}\right)^2 = (-1)^2 = 1\newlineSo, we add 11 to x2−2xx^2 - 2x to complete the square.
  3. Completing the square: Now we add and subtract 11 inside the function to balance the equation. This gives us:\newlinef(x) = (x2−2x+1)−1−95(x^2 - 2x + 1) - 1 - 95
  4. Factoring the perfect square trinomial: Next, we factor the perfect square trinomial (x2−2x+1)(x^2 - 2x + 1) to get (x−1)2(x - 1)^2.\newlinef(x) = (x−1)2−1−95(x - 1)^2 - 1 - 95
  5. Simplifying the function: Finally, we combine the constants to simplify the function. f(x)=(x−1)2−96f(x) = (x - 1)^2 - 96

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