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Rewrite the function by completing the square. {:[f(x)=x^(2)+4x+41],[f(x)=(x+◻)^(2)+◻]:}

Rewrite the function by completing the square.\newlinef(x)=x2+4x+41 f(x) = x^2 + 4x + 41 \newlinef(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+41 f(x) = x^2 + 4x + 41 \newlinef(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square
  1. Step 11: Starting with the function: We start with the function f(x)=x2+4x+41f(x) = x^2 + 4x + 41 and want to rewrite it in the form f(x)=(x+□)2+□f(x) = (x + \square)^2 + \square. To complete the square, we need to find a number that, when added and subtracted to the x2+4xx^2 + 4x part, completes the square.
  2. Step 22: Finding the number to add and subtract: First, we take the coefficient of xx, which is 44, divide it by 22, and square it to find the number to add and subtract. This gives us (42)2=22=4(\frac{4}{2})^2 = 2^2 = 4.
  3. Step 33: Adding and subtracting to complete the square: We add and subtract 44 to the expression x2+4xx^2 + 4x, which gives us x2+4x+4−4+41x^2 + 4x + 4 - 4 + 41. We add 44 to complete the square and subtract 44 to keep the expression equivalent to the original.
  4. Step 44: Rewriting the expression: Now we can rewrite the expression as (x2+4x+4)+(41−4)(x^2 + 4x + 4) + (41 - 4). The first part is a perfect square trinomial.
  5. Step 55: Factoring the perfect square trinomial: The perfect square trinomial (x2+4x+4)(x^2 + 4x + 4) can be factored into (x+2)2(x + 2)^2. The second part simplifies to 3737.
  6. Step 66: Final rewritten function: Putting it all together, we get f(x)=(x+2)2+37f(x) = (x + 2)^2 + 37. This is the function rewritten by completing the square.

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