Factorize:
2x^2-xy-6y^2
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To factorize the expression 2x^2 - xy - 6y^2, let's follow these steps:
Step 1: Check for any common factors.
In this case, there are no common factors among the terms.
Step 2: Look for a common pattern.
Since the coefficient of the x^2 term is not the same as the coefficient of the y^2 term, we cannot use a common pattern to factorize it.
Step 3: Use the product-sum method.
In this method, we need to find two numbers that multiply to give the product of the coefficient of x^2 term and y^2 term (in this case, 2 * -6 = -12) and add up to give the coefficient of the xy term (-1).
So, the two numbers that satisfy this condition are -4 and 3. (-4 * 3 = -12, -4 + 3 = -1)
Step 4: Rewrite the middle term.
Now we rewrite the middle term (-xy) by splitting it into -4xy + 3xy, using the two numbers we found in Step 3.
2x^2 - xy - 6y^2 becomes:
2x^2 - 4xy + 3xy - 6y^2
Step 5: Grouping.
Next, we group the terms in pairs:
(2x^2 - 4xy) + (3xy - 6y^2)
Step 6: Factor out common terms from each pair.
We can factor out 2x from the first pair and -3y from the second pair:
2x(x - 2y) + 3y(x - 2y)
Step 7: Factor out the common binomial.
Now, we have a common binomial (x - 2y) in both terms. We can factor it out:
(x - 2y)(2x + 3y)
Therefore, the factorized form of 2x^2 - xy - 6y^2 is (x - 2y)(2x + 3y).