Question

Hi,I don't know how to use the formulas,can someone use the questions below as example
1.formula:m-(x-n)^2
Quest:10x-x^2

2.Formula:a-(x+b)^2
Quest:5-2x-x^2

Answer 1

Both expressions you must complete the square.

1.

10 x - x²

Write in the standard form a x² + b x + c

10 x - x² = - ( x² - 10 x )

Coefficients are:

a = 1 , b = - 10 , c = 0

Add and subtract ( b / 2 )² = ( - 10 / 2 )² = ( - 5 )²

- [ x² - 10 x + ( - 5 )² - ( - 5 )² ] =

- [ x² - 2 ∙ 5 ∙ x + ( - 5 )² - ( - 5 )² ] =

- [ x² - 2 ∙ 5 ∙ x + ( - 5 )² - 25 ]
_______________________
Remark:

x² - 2 a x + a² = ( x - a )²

so

x² - 2 ∙ 5 x + ( - 5 )² = ( x - 5 )²
_______________________

10 x - x² = - ( x² - 10 x ) = - [ ( x - 5 )² - 25 ] =

- ( x - 5 )² + 25 = - ( x - 5 )² + 25 = 25 - ( x - 5 )²


- x² + 10 x = 25 - ( x - 5 )²


Similarly

2.

5 - 2 x - x² = - x² - 2 x + 5 = - ( x² + 2 x - 5 )

Writen in the standard form a x² + b x + c

Coefficients are:

a = 1 , b = 2 , c = - 5

Add and subtract ( b / 2 )² = ( 2 / 2 )² = ( 1 )²

- [ x² + 2 x - 5 + ( 1 )² - ( 1 )² ] =

- [ x² + 2 ∙ 1 ∙ x + ( 1 )² - 5 - 1 ] =

- [ x² + 2 ∙ 1 ∙ x + ( 1 )² - 6 ] =
____________________
Remark:

x² + 2 a x + a² = ( x + a )²

so

x² - 2 ∙ 1 x + 1² = ( x + 1)²
____________________

5 - 2 x - x² = - ( x² + 2 x - 5 ) = - [ ( x + 1)² - 6 ] = - ( x + 1)² + 6


5 - 2 x - x² = 6 - ( x + 1)²