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Rewrite the function by completing the square. {:[f(x)=x^(2)+16 x-46],[f(x)=(x+◻)^(2)+◻]:}

Rewrite the function by completing the square.\newlinef(x)=x2+16x−46 f(x) = x^2 + 16x - 46 , 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+16x−46 f(x) = x^2 + 16x - 46 , f(x)=(x+□)2+□ f(x) = (x+\square)^2 + \square
  1. Step 11: Rewrite the function: We start with the function f(x)=x2+16x−46f(x) = x^2 + 16x - 46 and want to rewrite it in the form f(x)=(x+a)2+bf(x) = (x + a)^2 + b.\newlineFirst, we need to complete the square for the x2+16xx^2 + 16x part of the function.\newlineTo do this, we take half of the coefficient of xx, which is 1616, and square it.\newline(162)2=82=64(\frac{16}{2})^2 = 8^2 = 64.
  2. Step 22: Complete the square: We then add and subtract this number, 6464, inside the function to complete the square without changing the function's value.\newlinef(x)=x2+16x+64−64−46.f(x) = x^2 + 16x + 64 - 64 - 46.
  3. Step 33: Add and subtract to complete the square: Now we can rewrite the function by factoring the perfect square trinomial and combining the constants.\newlinef(x)=(x+8)2−64−46f(x) = (x + 8)^2 - 64 - 46.
  4. Step 44: Factor the perfect square trinomial: Combine the constants −64-64 and −46-46 to get −110-110.\newlinef(x) = (x + 88)^22 - 110110.

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