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Rewrite the function by completing the square. {:[f(x)=x^(2)-6x-60],[f(x)=(x+◻)^(2)+◻]:}

Rewrite the function by completing the square.\newlinef(x)=x2−6x−60 f(x) = x^2 - 6x - 60 , f(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square

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Q. Rewrite the function by completing the square.\newlinef(x)=x2−6x−60 f(x) = x^2 - 6x - 60 , f(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square
  1. Identifying the Quadratic Function: We start with the given quadratic function f(x)=x2−6x−60f(x) = x^2 - 6x - 60. To complete the square, we need to form a perfect square trinomial from the quadratic and linear terms.
  2. Completing the Square: First, we identify the coefficient of the xx term, which is −6-6. We then take half of this coefficient, square it, and add it to and subtract it from the right side of the equation to maintain equality. Half of −6-6 is −3-3, and (−3)2(-3)^2 is 99.
  3. Adding and Subtracting to Maintain Equality: We add and subtract 99 to the right side of the equation:\newlinef(x)=x2−6x+9−9−60.f(x) = x^2 - 6x + 9 - 9 - 60.
  4. Grouping the Perfect Square Trinomial: Now we rewrite the equation grouping the perfect square trinomial and the constants:\newlinef(x) = (x2−6x+9)−69(x^2 - 6x + 9) - 69.
  5. Factoring the Perfect Square Trinomial: The expression (x2−6x+9)(x^2 - 6x + 9) is a perfect square trinomial and can be factored into (x−3)2(x - 3)^2:\newlinef(x)=(x−3)2−69f(x) = (x - 3)^2 - 69.
  6. Rewriting the Function: We have now rewritten the function by completing the square:\newlinef(x) = (x - 33)^22 - 6969.

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