f(x)= -x^2 +6x -5
complete the square
f(x)= -x^2 +6x -5
= -1(x^2 - 6x - 5)
take 1/2 of the -6, then square it: -----> 9
= -(x^2 - 6x + 9 - 9-5)
= -( (x-3)^2 - 14)
= .....
Reiny's solution is incorrect.
- ( ( x - 3 )² - 14 ) = - ( x² - 2 ∙ x ∙ 3 + 3² - 14 ) =
- ( x² - 6 x + 9 - 14 ) = - ( x² - 6 x - 5 ) = - x² + 6 x + 5
but
- x² + 6 x - 5 = - x² + 6 x - 9 + 4 = ( - x² + 6 x - 9 ) + 4 =
- ( x² - 6 x + 9 ) + 4 = - ( x - 3 )² + 4 = - [ ( x - 3 )² - 4 ]
So:
- x² + 6 x - 5 = - [ ( x - 3 )² - 4 ]
Bosnian is correct, my second line should be
= -1(x^2 - 6x + 5) instead of = -1(x^2 - 6x - 5)
To complete the square for the given quadratic function f(x) = -x^2 + 6x - 5, you can follow these steps:
Step 1: Make sure the coefficient of x^2 is 1. If not, divide the entire equation by the coefficient of x^2 to make it equal to 1. In this case, the coefficient of x^2 is already -1, so no adjustment is needed.
Step 2: Move the constant term (-5) to the other side of the equation. This can be done by adding 5 to both sides of the equation:
f(x) + 5 = -x^2 + 6x
Step 3: To complete the square, you need to take half of the coefficient of the x-term (6) and square it. Half of 6 is 3, and 3 squared is 9. Add 9 to both sides:
f(x) + 5 + 9 = -x^2 + 6x + 9
Step 4: Rearrange the trinomial on the right side of the equation as a perfect square. For the x-terms, group them together:
f(x) + 14 = -(x^2 - 6x + 9)
Step 5: The expression inside the parentheses, x^2 - 6x + 9, is a perfect square trinomial. It can be written as the square of a binomial, which is (x - 3)^2:
f(x) + 14 = -((x - 3)^2)
Step 6: Finally, subtract 14 from both sides of the equation to isolate f(x):
f(x) = -((x - 3)^2) - 14
Thus, the given quadratic function f(x) = -x^2 + 6x - 5 can be rewritten in completed square form as f(x) = -((x - 3)^2) - 14.