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Rewrite the function by completing the square. {:[f(x)=x^(2)+6x+87],[f(x)=(x+◻)^(2)+◻]:}

Rewrite the function by completing the square.\newlinef(x)=x2+6x+87 f(x) = x^2 + 6x + 87 \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+6x+87 f(x) = x^2 + 6x + 87 \newlinef(x)=(x+□)2+□ f(x) = (x + \Box)^2 + \Box
  1. Identifying the Coefficient: To complete the square, we need to form a perfect square trinomial from the quadratic and linear terms of the function f(x)=x2+6x+87f(x) = x^2 + 6x + 87. We will then adjust the constant term to maintain equality.
  2. Adding and Subtracting to Complete the Square: First, we identify the coefficient of the xx term, which is 66. To form a perfect square trinomial, we need to find (62)2\left(\frac{6}{2}\right)^2.\newlineCalculation: (62)2=32=9\left(\frac{6}{2}\right)^2 = 3^2 = 9.\newlineWe will add and subtract this value inside the function to complete the square.
  3. Rewriting the Function: We add 99 and subtract 99 from the function to balance the equation.\newlinef(x)=x2+6x+9−9+87.f(x) = x^2 + 6x + 9 - 9 + 87.
  4. Factoring the Perfect Square Trinomial: Now we can rewrite the function by grouping the perfect square trinomial and combining the constants.\newlinef(x) = (x2+6x+9)−9+87 (x^2 + 6x + 9) - 9 + 87 .
  5. Combining the Constants: The perfect square trinomial (x2+6x+9)(x^2 + 6x + 9) can be factored into (x+3)2(x + 3)^2.\newlinef(x) = (x+3)2−9+87(x + 3)^2 - 9 + 87.
  6. Combining the Constants: The perfect square trinomial (x2+6x+9)(x^2 + 6x + 9) can be factored into (x+3)2(x + 3)^2.\newlinef(x) = (x + 33)^22 - 99 + 8787.Finally, we combine the constants −9-9 and 8787 to get the completed square form of the function.\newlinef(x) = (x + 33)^22 + 7878.

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