Can someone factorise 25x^2-25xy+6y^2 show your work please
because of the - in the middle and + at the end,
I know both signs must be -
it could be (5x...)(5x....) or (x...)(25x...)
at the end it could be (... -2y)(...-3y) or (...-1y)(....-6y)
after two tries I got
(5x-2y)(5x-3y)
To factorize the expression 25x^2 - 25xy + 6y^2, we can follow these steps:
Step 1: Identify if the expression can be factored using any common factors.
In this case, there are no common factors for all the terms in the expression.
Step 2: Check if the expression is a perfect square trinomial or a difference of squares.
A perfect square trinomial is the square of a binomial, while a difference of squares can be factored as (a + b)(a - b). However, the given expression does not fit either of these patterns.
Step 3: Attempt to split the middle term to factorize.
Multiply the coefficient of the quadratic term (25) by the constant term (6) to get 150. Now, we need to find two numbers that multiply to give 150 and add up to the coefficient of the middle term (-25). Let's list the factors of 150:
Factors of 150: ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±25, ±30, ±50, ±75, ±150
By testing these factors, we can see that -10 and -15 satisfy the conditions as they multiply to give 150 and add up to -25: -10 * -15 = 150 and -10 + (-15) = -25.
Step 4: Rewrite the middle term (-25xy) using the two numbers found in Step 3.
Replace the middle term (-25xy) with -10xy - 15xy. This gives us:
25x^2 - 10xy - 15xy + 6y^2
Step 5: Group the terms and factor by grouping.
Now, we can group the first two terms and the last two terms separately:
(25x^2 - 10xy) + (-15xy + 6y^2)
Step 6: Factor out the greatest common factor (GCF) from each group.
From the first group, we can factor out 5x:
5x(5x - 2y)
From the second group, we can factor out -3y:
-3y(5x - 2y)
Step 7: Combine the factored terms from Step 6 to get the final factorization.
Now, we have our factored expression:
5x(5x - 2y) - 3y(5x - 2y)
Notably, we can observe that we have a common binomial factor, which we can factor out:
(5x - 2y)(5x - 3y)
Therefore, the factorization of 25x^2 - 25xy + 6y^2 is (5x - 2y)(5x - 3y).