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Rewrite the function by completing the square. {:[f(x)=x^(2)-4x+70],[f(x)=(x+◻)^(2)+◻]:}

Rewrite the function by completing the square.\newlinef(x)=x2−4x+70 f(x) = x^2 - 4x + 70 \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+70 f(x) = x^2 - 4x + 70 \newlinef(x)=(x+□)2+□ f(x) = (x + \square)^2 + \square
  1. Function and Form: We start with the function f(x)=x2−4x+70f(x) = x^2 - 4x + 70 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. Finding the Completing Square Number: First, we take the coefficient of the x term, which is −4-4, divide it by 22, and square it to find the number to complete the square. This gives us (−42)2=(−2)2=4\left(\frac{-4}{2}\right)^2 = (-2)^2 = 4.
  3. Adding and Subtracting to Complete the Square: We add and subtract this number inside the function to complete the square. We have to add 44 and subtract 44 right after the −4x-4x term to keep the equation balanced.\newlinef(x) = x2−4x+4−4+70x^2 - 4x + 4 - 4 + 70
  4. Grouping the Terms: Now we group the first three terms and the last two terms separately. \newlinef(x)=(x2−4x+4)+(70−4)f(x) = (x^2 - 4x + 4) + (70 - 4)
  5. Factoring the Perfect Square Trinomial: The first three terms now form a perfect square trinomial, which can be factored into (x−2)2(x - 2)^2. The last two terms combine to give 6666.\newlinef(x) = (x−2)2+66(x - 2)^2 + 66

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