Complete the Square
f(x)= -x^2 + 2x
f(x)= -x^2 + 2x
= -1(x^2 - 2x +1 - 1 )
= -( (x-1)^2 - 1)
= - (x-1)^2 + 1
Thank You
To complete the square for the given quadratic function f(x) = -x^2 + 2x, follow these steps:
Step 1: Ensure that the leading coefficient of the quadratic term (x^2) is not negative. In this case, it is already negative, so we move on to the next step.
Step 2: Take half of the coefficient of the linear term (2x) and square it. Half of 2x is x, and x^2 is x^2.
Step 3: Add the result from Step 2 to the quadratic expression inside the function.
f(x) = -x^2 + 2x + x^2
Now, the quadratic term has been completed.
Step 4: Rearrange the terms within the expression by grouping the quadratic term with the constant term separately.
f(x) = (x^2 - 2x) + x^2
Step 5: Factor out the common factor from each group separately.
f(x) = x(x - 2) + x^2
Step 6: Simplify the expression.
f(x) = x^2 - 2x + x^2
f(x) = 2x^2 - 2x
The expression has been completed. The square has been completed by adding (x^2 - 2x) to both sides of the equation. Thus, the completed square form of the given function f(x) = -x^2 + 2x is f(x) = 2x^2 - 2x.