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Rewrite the function by completing the square. {:[f(x)=x^(2)+6x-78],[f(x)=(x+◻)^(2)+◻]:}

Rewrite the function by completing the square.\newlinef(x)=x2+6x−78 f(x) = x^2 + 6x - 78 \newlinef(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square

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Full solution

Q. Rewrite the function by completing the square.\newlinef(x)=x2+6x−78 f(x) = x^2 + 6x - 78 \newlinef(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square
  1. Identifying quadratic and linear terms: We start by identifying the quadratic and linear terms in the function f(x)=x2+6x−78f(x) = x^2 + 6x - 78 that we will use to complete the square.
  2. Finding the perfect square trinomial: To complete the square, we need to find a number that, when added and subtracted to the quadratic and linear terms, forms a perfect square trinomial. This number is (b2)2(\frac{b}{2})^2, where bb is the coefficient of the xx term. In this case, b=6b = 6, so (62)2=32=9(\frac{6}{2})^2 = 3^2 = 9.
  3. Adding and subtracting to create a perfect square trinomial: We add and subtract 99 to the function inside the parentheses to create a perfect square trinomial. We must also subtract 99 outside the parentheses to keep the equation balanced.\newlinef(x) = (x^22 + 66x + 99) - 99 - 7878
  4. Factoring the perfect square trinomial: Now we factor the perfect square trinomial inside the parentheses.\newlinef(x) = (x+3)2−9−78(x + 3)^2 - 9 - 78
  5. Combining constants to simplify the function: Combine the constants outside the parentheses to simplify the function.\newlinef(x) = (x+3)2−87(x + 3)^2 - 87

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