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Rewrite the function by completing the square. f(x)=x2−2x+1=0f(x)= x^2-2x+1=0 f(x)=(x+_)2+_f(x)=(x+\_)^2+\_

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Full solution

Q. Rewrite the function by completing the square. f(x)=x2−2x+1=0f(x)= x^2-2x+1=0 f(x)=(x+_)2+_f(x)=(x+\_)^2+\_
  1. Identify coefficients: Identify the coefficients of the quadratic function f(x)=x2−2x+1f(x) = x^2 - 2x + 1.\newlineHere, a=1a = 1, b=−2b = -2, and c=1c = 1.
  2. Complete the square: To complete the square, we need to find a value that when added and subtracted to the function, will not change the function's value but will allow us to write it in the form of (x+p)2+q(x + p)^2 + q.
  3. Calculate value: The general form for completing the square is (x+b2a)2−(b2a)2+c(x + \frac{b}{2a})^2 - (\frac{b}{2a})^2 + c. We already have a=1a = 1 and b=−2b = -2, so we calculate (−2/2⋅1)2(-2/2\cdot1)^2.
  4. Rewrite function: Calculate (−22×1)2=(−1)2=1(-\frac{2}{2}\times 1)^2 = (-1)^2 = 1.
  5. Group terms: Now, rewrite the function by adding and subtracting this value inside the function: f(x)=x2−2x+1+1−1f(x) = x^2 - 2x + 1 + 1 - 1.
  6. Recognize perfect square: Group the perfect square terms and the constant: f(x)=(x2−2x+1)−1f(x) = (x^2 - 2x + 1) - 1.
  7. Write in completed form: Recognize that the grouped terms form a perfect square: (x−1)2(x - 1)^2.
  8. Write in completed form: Recognize that the grouped terms form a perfect square: (x−1)2(x - 1)^2.Write the function in the completed square form: f(x)=(x−1)2−1f(x) = (x - 1)^2 - 1.

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