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Rewrite the function by completing the square. {:[f(x)=x^(2)-10 x-96],[f(x)=(x+◻)^(2)+◻]:}

Rewrite the function by completing the square.\newlinef(x)=x2−10x−96 f(x) = x^2 - 10x - 96 \newlinef(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square

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Q. Rewrite the function by completing the square.\newlinef(x)=x2−10x−96 f(x) = x^2 - 10x - 96 \newlinef(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square
  1. Identify coefficient of xx: We start with the function f(x)=x2−10x−96f(x) = x^2 - 10x - 96 and want to rewrite it in the form f(x)=(x+a)2+bf(x) = (x + a)^2 + b. To complete the square, we need to find a value that, when added and subtracted to the x2−10xx^2 - 10x part, completes the square.
  2. Complete the square: First, we identify the coefficient of x, which is −10-10. To complete the square, we take half of this coefficient and square it. This gives us (−102)2=(−5)2=25\left(\frac{-10}{2}\right)^2 = (-5)^2 = 25.
  3. Add and subtract to complete the square: We add and subtract this value inside the function to complete the square. We must add 2525 to both sides of the equation to keep it balanced.\newlinef(x) = x2−10x+25−25−96x^2 - 10x + 25 - 25 - 96
  4. Rewrite as a perfect square trinomial: Now we can rewrite the function as a perfect square trinomial plus a constant.\newlinef(x) = (x2−10x+25)−25−96(x^2 - 10x + 25) - 25 - 96
  5. Factor the perfect square trinomial: The perfect square trinomial x2−10x+25x^2 - 10x + 25 can be factored into (x−5)2(x - 5)^2.\newlinef(x) = (x−5)2−25−96(x - 5)^2 - 25 - 96
  6. Combine constants to simplify: Combine the constants to simplify the function. f(x)=(x−5)2−121f(x) = (x - 5)^2 - 121

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