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?
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
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 )
To factor the expression 50(2x-5)^2 - 162(3y-2)^2, follow these steps:
Step 1: Identify the common factors in both terms.
Common factors:
- 50(2x-5)^2 -> Common factor: 50
- 162(3y-2)^2 -> Common factor: 162
Step 2: Simplify the common factors.
- 50(2x-5)^2 can be simplified to 2(2x-5)^2
- 162(3y-2)^2 can be simplified to 3(3y-2)^2
Step 3: Write the simplified terms together.
2(2x-5)^2 - 3(3y-2)^2
Step 4: Notice that (2x-5)^2 and (3y-2)^2 are both perfect squares, so this expression can be factored further using the difference of squares formula: a^2 - b^2 = (a+b)(a-b).
Applying this formula, factors can be written as:
(2x-5)^2 = (2x-5)(2x-5)
(3y-2)^2 = (3y-2)(3y-2)
Step 5: Rewrite the expression using factored terms:
2(2x-5)(2x-5) - 3(3y-2)(3y-2)
Step 6: Simplify the expression further if possible.
That's it! The expression 50(2x-5)^2 - 162(3y-2)^2 factors to 2(2x-5)(2x-5) - 3(3y-2)(3y-2).
To factor the given expression, we can start by identifying any common factors in both terms. In this case, the common factors are 50, 2x-5, and 3y-2.
Step 1: Identify common factors
First, let's factor out the common factor 50 from both terms:
50 (2x-5)^2 - 162 (3y-2)^2
= 2 * 25 (2x-5)^2 - 2 * 81 (3y-2)^2
Step 2: Write the expression in a factored form
Now, we can express the expression in a factored form using the common factors:
= 2 [25 (2x-5)^2 - 81 (3y-2)^2]
Step 3: Apply the difference of squares formula
The expression inside the bracket [ ] is a difference of squares. We can use the formula: a^2 - b^2 = (a + b)(a - b) to further factor it.
Let a = 5(2x-5) and b = 9(3y-2), so we have:
= 2 [(5(2x-5))^2 - (9(3y-2))^2]
= 2 [ (5(2x-5) + 9(3y-2))(5(2x-5) - 9(3y-2)) ]
Step 4: Simplify the expression inside the bracket
Now, let's simplify the expression inside the square brackets:
= 2 [ (10x - 25 + 27y - 18)(10x - 25 - 27y + 18) ]
= 2 [ (10x + 27y - 43)(10x - 27y - 7) ]
Therefore, the fully factored expression is:
50 (2x-5)^2 - 162 (3y-2)^2 = 2 (10x + 27y - 43)(10x - 27y - 7)
I hope this step-by-step explanation helps!