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Rewrite the function by completing the square. {:[f(x)=x^(2)+20 x+40],[f(x)=(x+◻)^(2)+◻]:}

Rewrite the function by completing the square.\newlinef(x)=x2+20x+40 f(x) = x^2 + 20x + 40 , 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+20x+40 f(x) = x^2 + 20x + 40 , f(x)=(x+□)2+□ f(x) = (x+\square)^2 + \square
  1. Given quadratic function: We start with the given quadratic function:\newlinef(x)=x2+20x+40f(x) = x^2 + 20x + 40\newlineWe want to rewrite this function in the form of (x+a)2+b(x + a)^2 + b.\newlineTo complete the square, we need to find a value that, when added and subtracted to the x2+20xx^2 + 20x part, forms a perfect square trinomial.
  2. Completing the square: The coefficient of x is 2020. To form a perfect square trinomial, we take half of the coefficient of x, which is 202=10\frac{20}{2} = 10, and then square it to get 102=10010^2 = 100.\newlineWe add and subtract this value inside the function to complete the square.\newlinef(x)=x2+20x+100−100+40f(x) = x^2 + 20x + 100 - 100 + 40
  3. Factoring the perfect square trinomial: Now we have the perfect square trinomial x2+20x+100x^2 + 20x + 100 and the constants −100+40-100 + 40 combined.\newlineWe can factor the perfect square trinomial to get (x+10)2(x + 10)^2.\newlinef(x) = (x+10)2−100+40(x + 10)^2 - 100 + 40
  4. Simplifying the function: Combine the constants −100-100 and +40+40 to simplify the function.\newlinef(x)=(x+10)2−60f(x) = (x + 10)^2 - 60\newlineNow the function is written in the completed square form.

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