how would i factor
15x^2+7xy-2y^2
15x^2+7xy-2y^2
=15x^2+10xy-3xy-2y^2
=5x(3x+2y)-y(3x+2y)
=(3x+2y)(5x-y)
15x^2 + 7xy - 2y^2
(3x + 2y)(5x - y)
How did you get 15x^2+10xy_3xy_2y^2
To factor the expression 15x^2 + 7xy - 2y^2, we need to look for common factors and use factoring techniques. Here's a step-by-step explanation of how to factor this expression:
Step 1: Check for a common factor.
First, check if there is a common factor among the coefficients of each term. In this case, there is no common factor among 15, 7, and -2.
Step 2: Look for a pattern in the expression.
Next, look for a pattern that might help you factor the trinomial. In this case, we have a quadratic trinomial in the form of ax^2 + bxy + cy^2. Pay attention to the signs of the coefficients: 15x^2 has a positive coefficient, 7xy has a positive coefficient, and -2y^2 has a negative coefficient.
Step 3: Use factoring techniques.
We can use factoring techniques like trial and error or the quadratic formula, but in this case, we will use another approach called "AC method" or "splitting the middle term."
The AC method involves finding two numbers whose product is equal to the product of the coefficient of the x^2 term (a = 15) and the constant term (c = -2). In this case, 15 * -2 = -30. We are looking for two numbers that multiply to -30 but add up to the coefficient of the xy term (b = 7).
The pair of numbers that fit these criteria is 10 and -3. (10 * -3 = -30 and 10 + -3 = 7).
Step 4: Split the middle term.
Rewrite the 7xy term (bxy) using the pair of numbers from Step 3. Replace the 7xy with 10xy - 3xy.
15x^2 + 10xy - 3xy - 2y^2
Step 5: Grouping and factoring by grouping.
Group the terms into two pairs.
(15x^2 + 10xy) + (-3xy - 2y^2)
Factor out the common factors from each group.
5x(3x + 2y) - y(3x + 2y)
Step 6: Final factorization.
Notice that we have a common factor: (3x + 2y). Factor out this common factor.
(5x - y)(3x + 2y)
Therefore, the factored form of the expression 15x^2 + 7xy - 2y^2 is (5x - y)(3x + 2y).