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
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