How do I break this down to factor it?
2x^2+13xy+15y^2
2x^2+13xy+15y^2
=(2x + 3y)(x + 5y)
you have to combine like terms for starters like x's and x's
To factor the expression 2x^2 + 13xy + 15y^2, we look for two binomials that, when multiplied, give us the original expression. Here's how you can break it down step by step:
Step 1: Look at the coefficients of the x^2, xy, and y^2 terms and see if they have any common factors. In this case, the coefficients 2, 13, and 15 do not have any common factors other than 1.
Step 2: We want to find two binomials in the form (ax + by)(cx + dy), which, when multiplied, give us the original expression: 2x^2 + 13xy + 15y^2. The coefficients a, b, c, and d are the unknowns we need to find.
Step 3: Expand the product (ax + by)(cx + dy) to obtain the terms:
(ac)x^2 + (ad)xy + (bc)xy + (bd)y^2
This simplifies to:
acx^2 + (ad + bc)xy + bdy^2.
Step 4: Since we want this expanded form to match our original expression, we have the following conditions:
ac = 2 (from the x^2 term) (Equation 1)
ad + bc = 13 (from the xy term) (Equation 2)
bd = 15 (from the y^2 term) (Equation 3)
Step 5: Factor 2 by considering its possible factors, which are 1 and 2. You need to consider all possible combinations of a and c, such as ac = 1 * 2 or ac = 2 * 1.
Here are the possible pairs of a and c:
(a = 1 and c = 2) or (a = 2 and c = 1)
Step 6: Now, look at equation 1, where ac = 2. Consider the options of a and c for this equation:
Option 1: (a = 1 and c = 2)
In this case, equation 1 becomes:
1 * 2 = 2
Option 2: (a = 2 and c = 1)
In this case, equation 1 becomes:
2 * 1 = 2
Both options give us ac = 2, which satisfies equation 1.
Step 7: Let's move to equation 3: bd = 15.
Since we already have a = 1 and c = 2, we substitute these values into equation 3 and solve for b and d:
(1)(d) = 15
d = 15
Step 8: After finding d = 15, we look at equation 2: ad + bc = 13.
Substitute a = 1, c = 2, and d = 15 into equation 2 and solve for b:
(1)(15) + b(2) = 13
15 + 2b = 13
2b = 13 - 15
2b = -2
b = -1
Step 9: We now have a = 1, b = -1, c = 2, and d = 15. Substituting these values back into the expression (ax + by)(cx + dy):
(1x - 1y)(2x + 15y)
Step 10: Simplify:
(x - y)(2x + 15y)
So, the factored form of 2x^2 + 13xy + 15y^2 is (x - y)(2x + 15y).