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Rewrite the function by completing the square. {:[f(x)=x^(2)-6x+43],[f(x)=(x+◻)^(2)+◻]:}

Rewrite the function by completing the square.\newlinef(x)=x2−6x+43 f(x) = x^2 - 6x + 43 , f(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square

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Q. Rewrite the function by completing the square.\newlinef(x)=x2−6x+43 f(x) = x^2 - 6x + 43 , f(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square
  1. Identify coefficient of x term: To complete the square, we need to form a perfect square trinomial from the quadratic and linear terms of the function f(x) = x^22 - 66x + 4343. We will then adjust the constant term to maintain equality.
  2. Find value for perfect square trinomial: First, we identify the coefficient of the xx term, which is −6-6. To form a perfect square trinomial, we need to find the value that, when squared, gives us the constant term to add and subtract inside the function. This value is (−62)2=(−3)2=9\left(-\frac{6}{2}\right)^2 = (-3)^2 = 9.
  3. Add and subtract value inside the function: We add and subtract this value inside the function to complete the square. We add 99 and subtract 99 immediately after to keep the equation balanced.\newlinef(x)=x2−6x+9−9+43f(x) = x^2 - 6x + 9 - 9 + 43
  4. Rewrite function by grouping: Now we can rewrite the function by grouping the perfect square trinomial and combining the constants:\newlinef(x)=(x2−6x+9)+(43−9)f(x) = (x^2 - 6x + 9) + (43 - 9)
  5. Factor perfect square trinomial and simplify constants: The perfect square trinomial (x2−6x+9)(x^2 - 6x + 9) can be factored into (x−3)2(x - 3)^2, and we simplify the constants (43−9)(43 - 9) to get 3434.\newlinef(x)=(x−3)2+34f(x) = (x - 3)^2 + 34

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