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Rewrite the function by completing the square. {:[f(x)=x^(2)+8x-29],[f(x)=(x+◻)^(2)+◻]:}

Rewrite the function by completing the square.\newlinef(x)=x2+8x−29 f(x) = x^2 + 8x - 29 \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+8x−29 f(x) = x^2 + 8x - 29 \newlinef(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square
  1. Given quadratic function: We start with the given quadratic function f(x)=x2+8x−29f(x) = x^2 + 8x - 29. To complete the square, we need to form a perfect square trinomial from the x2x^2 and 8x8x terms.
  2. Forming a perfect square trinomial: To create a perfect square trinomial, we take the coefficient of the x term, which is 88, divide it by 22, and then square it. This gives us (82)2=42=16(\frac{8}{2})^2 = 4^2 = 16.
  3. Adding and subtracting to maintain equality: We add and subtract this number 1616 inside the function to maintain the equality. This gives us f(x)=x2+8x+16−16−29f(x) = x^2 + 8x + 16 - 16 - 29.
  4. Rewriting the function by grouping: Now we can rewrite the function by grouping the perfect square trinomial and combining the constants: f(x)=(x2+8x+16)−16−29f(x) = (x^2 + 8x + 16) - 16 - 29.
  5. Factoring the perfect square trinomial: The perfect square trinomial x2+8x+16x^2 + 8x + 16 can be factored into (x+4)2(x + 4)^2. So, we have f(x)=(x+4)2−16−29f(x) = (x + 4)^2 - 16 - 29.
  6. Combining the constants: Next, we combine the constants −16-16 and −29-29 to get −45-45. This gives us the completed square form of the function: f(x)=(x+4)2−45f(x) = (x + 4)^2 - 45.

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