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Rewrite the function by completing the square. {:[f(x)=x^(2)-8x-2],[f(x)=(x+◻)^(2)+◻]:}

Rewrite the function by completing the square.\newlinef(x)=x2−8x−2 f(x) = x^2 - 8x - 2 , f(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square

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Q. Rewrite the function by completing the square.\newlinef(x)=x2−8x−2 f(x) = x^2 - 8x - 2 , f(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square
  1. Identifying the Coefficient: We start with the function f(x)=x2−8x−2f(x) = x^2 - 8x - 2 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 x2−8xx^2 - 8x part, completes the square.
  2. Completing the Square: First, we identify the coefficient of xx, which is −8-8. To complete the square, we take half of this coefficient and square it. This gives us (−82)2=(−4)2=16\left(\frac{-8}{2}\right)^2 = (-4)^2 = 16. We will add and subtract this value inside the parentheses to complete the square.
  3. Rewriting the Function: We rewrite the function by adding and subtracting 1616 inside the xx terms: f(x)=(x2−8x+16)−16−2f(x) = (x^2 - 8x + 16) - 16 - 2. This allows us to factor the first three terms into a perfect square.
  4. Factoring the Perfect Square: Now we factor the perfect square: f(x)=(x−4)2−16−2f(x) = (x - 4)^2 - 16 - 2. The term (x−4)2(x - 4)^2 is the completed square, and we combine the constants −16-16 and −2-2 to simplify the function.
  5. Simplifying the Function: Combining the constants gives us f(x)=(x−4)2−18f(x) = (x - 4)^2 - 18. This is the function in completed square form.

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