Substitution: z = x - 2 / 3
( x - 2 / 3 )² - 16 ( x - 2 / 3 ) + 63 = z² - 16 z + 63
Now you must solve equation:
z² - 16 z + 63 = 0
z1/2 = [ - b ± √ ( b² - 4 a c ) ] / 2 a
In this case:
a = 1 , b = - 16 , c = 63
z1/2 = [ - ( - 16 ) ± √ ( ( - 16 )² - 4 ∙ 1 ∙ 63 ) ] / 2 ∙ 1 =
[ 16 ± √ ( 256 - 252 ) ] / 2 =
( 16 ± √ 4 ) / 2 =
( 16 ± 2 ) / 2 =
8 ± 1
z1 = 8 + 1 = 9
z2 = 8 - 1 = 7
z² - 16 z + 63 = ( z - z1 ) ( z - z2 ) =
z² - 16 z + 63 = ( z - 9 ) ( z - 7 ) =
z² - 16 z + 63 = ( z - 7 ) ( z - 9 )
Return to the initial variables:
z = x - 2 / 3
( x - 2 / 3 )² - 16 ( x - 2 / 3 ) + 63 =
z² - 16 z + 63 =
( z - 7 ) ( z - 9 ) =
( x - 2 / 3 - 7 ) ( x - 2 / 3 - 9 ) =
( x - 2 / 3 - 21 / 3 ) ( x - 2 / 3 - 27 / 3 ) =
( x - 23 / 3 ) ( x - 29 / 3 ) =
( 3 x / 3 - 23 / 3 ) ( 3 x / 3 - 29 / 3 ) =
( 1 / 3 ) ∙ ( 3 x - 23 ) ∙ ( 1 / 3 ) ∙ ( 3 x - 29 ) =
( 1 / 3 ) ∙ ( 1 / 3 ) ∙ ( 3 x - 23 ) ∙ ( 3 x - 29 ) =
( 1 / 9 ) ∙ ( 3 x - 23 ) ∙ ( 3 x - 29 )
( x - 2 / 3 )² - 16 ( x - 2 / 3 ) + 63 = ( 1 / 9 ) ( 3 x - 23 ) ( 3 x - 29 )
Factor completely
(x-2/3)^2-16(x-2/3)+63
1 answer