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Rewrite the function by completing the square. {:[f(x)=x^(2)-4x-26],[f(x)=(x+◻)^(2)+◻]:}

Rewrite the function by completing the square.\newlinef(x)=x2−4x−26 f(x) = x^2 - 4x - 26 , f(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square

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Q. Rewrite the function by completing the square.\newlinef(x)=x2−4x−26 f(x) = x^2 - 4x - 26 , f(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square
  1. Find constant term: First, we need to find the constant term that will complete the square. We take the coefficient of xx, which is −4-4, divide it by 22, and then square it.\newline(−42)2=(−2)2=4\left(\frac{-4}{2}\right)^2 = (-2)^2 = 4
  2. Add and subtract constant term: Next, we add and subtract this constant term (44) inside the function to complete the square, making sure the overall value of the function does not change.\newlinef(x)=x2−4x+4−4−26f(x) = x^2 - 4x + 4 - 4 - 26
  3. Rewrite the function: Now we can rewrite the function by grouping the perfect square trinomial and combining the constants.\newlinef(x)=(x2−4x+4)−4−26f(x) = (x^2 - 4x + 4) - 4 - 26
  4. Factor perfect square trinomial: The perfect square trinomial x2−4x+4x^2 - 4x + 4 can be factored into (x−2)2(x - 2)^2.\newlinef(x) = (x−2)2−4−26(x - 2)^2 - 4 - 26
  5. Combine constants: Finally, we combine the constants −4-4 and −26-26 to get −30-30.\newlinef(x) = (x−2)2−30(x - 2)^2 - 30

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