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Rewrite the function by completing the square. {:[f(x)=x^(2)+4x-60],[f(x)=(x+◻)^(2)+◻]:}

Rewrite the function by completing the square.\newlinef(x)=x2+4x−60 f(x) = x^2 + 4x - 60 \newlinef(x)=(x+□)2+□ f(x) = (x + \Box)^2 + \Box

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Q. Rewrite the function by completing the square.\newlinef(x)=x2+4x−60 f(x) = x^2 + 4x - 60 \newlinef(x)=(x+□)2+□ f(x) = (x + \Box)^2 + \Box
  1. Function and Form: We start with the function f(x)=x2+4x−60f(x) = x^2 + 4x - 60 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+4xx^2 + 4x, creates a perfect square trinomial.
  2. Finding the Number: First, we take the coefficient of xx, which is 44, divide it by 22, and square it to find the number to complete the square. This number will be added and subtracted inside the parentheses to maintain the equality.\newline(42)2=22=4\left(\frac{4}{2}\right)^2 = 2^2 = 4
  3. Completing the Square: Now we add and subtract this number inside the parentheses to complete the square:\newlinef(x) = x2+4x+4−4−60x^2 + 4x + 4 - 4 - 60
  4. Factoring and Combining: We can now factor the perfect square trinomial and combine the constants:\newlinef(x) = (x+2)2−64(x + 2)^2 - 64
  5. Final Form: The function is now rewritten in the completed square form:\newlinef(x) = (x + 22)^22 - 6464

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