50 (2x-5)^2 -162 (3y-2)^2
= 2 [25(2x-5)^2 - 81(3y-2)^2 ]
now I see a difference of squares
= 2[ (5(2x-5) + 9(3y-2) )( (5(2x-5) - 9(3y-2) )]
= 2[ (10x-25 + 27y - 18)(10x - 25 - 27y + 18) ]
= 2(10x + 27y - 43)(10x - 27y - 7)
check my arithmetic
Factor:
50 (2x-5)^2 -162 (3y-2)^2
I'm not even sure how to do this. I've tried it for so long and I've been trying multiple ways. I don't know how to factor these. May I get a step to step explanation or work please?
2 answers
50 ( 2x - 5 )² - 162 ( 3y - 2 )²
Make the following substitutions:
k = 2x − 5
m = 3 y − 2
The expression can be rewritten as:
50 ( 2x - 5 )² - 162 ( 3y - 2 )² =
50 k² - 162 m² = 2 ( 25 k² - 81 m² )
Apply difference of squares:
α² - β² = ( α - β ) ( α + β )
with:
α = 5 k and β = 9 m
50 k² - 162 m² = 2 ( 25 k² - 81 m² ) =
2 ( 5 k − 9 m ) ( 5 k + 9 m )
Return to the initial variables:
2 ( 5 k − 9 m ) ( 5 k + 9 m ) =
2 [ 5 ( 2x − 5 ) − 9 ( 3 y − 2 ) ] [ 5 ( 2x − 5 ) + 9 ( 3 y − 2 ) ] =
2 [ 5 ∙ 2x + 5 ∙ ( - 5 ) − 9 ∙ 3 y - 9 ∙ ( − 2 ) ] [ 5 ∙ 2x + 5 ∙ ( - 5 ) + 9 ∙ 3 y + 9 ∙ ( − 2 ) ]
2 ( 10 x - 25 - 27 y + 18 ) ( 10 x - 25 + 27 y - 18 ) =
2 ( 10 x - 27 y - 7 ) ( 10 x + 27 y - 43 )
50 ( 2x - 5 )² - 162 ( 3y - 2 )² = 2 ( 10 x - 27 y - 7 ) ( 10 x + 27 y - 43 )
Make the following substitutions:
k = 2x − 5
m = 3 y − 2
The expression can be rewritten as:
50 ( 2x - 5 )² - 162 ( 3y - 2 )² =
50 k² - 162 m² = 2 ( 25 k² - 81 m² )
Apply difference of squares:
α² - β² = ( α - β ) ( α + β )
with:
α = 5 k and β = 9 m
50 k² - 162 m² = 2 ( 25 k² - 81 m² ) =
2 ( 5 k − 9 m ) ( 5 k + 9 m )
Return to the initial variables:
2 ( 5 k − 9 m ) ( 5 k + 9 m ) =
2 [ 5 ( 2x − 5 ) − 9 ( 3 y − 2 ) ] [ 5 ( 2x − 5 ) + 9 ( 3 y − 2 ) ] =
2 [ 5 ∙ 2x + 5 ∙ ( - 5 ) − 9 ∙ 3 y - 9 ∙ ( − 2 ) ] [ 5 ∙ 2x + 5 ∙ ( - 5 ) + 9 ∙ 3 y + 9 ∙ ( − 2 ) ]
2 ( 10 x - 25 - 27 y + 18 ) ( 10 x - 25 + 27 y - 18 ) =
2 ( 10 x - 27 y - 7 ) ( 10 x + 27 y - 43 )
50 ( 2x - 5 )² - 162 ( 3y - 2 )² = 2 ( 10 x - 27 y - 7 ) ( 10 x + 27 y - 43 )