how do i work out the answer to:
"how many integer values are there for k for which 4x^2 + kxy - 9y^2 is factorable?"
what is everyday uses of fractions
To determine the number of integer values for k for which the expression 4x^2 + kxy - 9y^2 is factorable, we need to understand the properties of a factorable quadratic expression.
A quadratic expression is factorable if it can be written in the form (ax + by)(cx + dy), where a, b, c, and d are constants. In this case, we have the quadratic expression 4x^2 + kxy - 9y^2.
To express it in the form of a factorable quadratic expression, we need to find factors for 4 and -9 that satisfy the relation ad - bc = k.
For positive values of k, we have:
- When k = 1, we can factor 4 as 4 x 1 or 2 x 2, and -9 as -9 x 1 or -3 x 3. However, none of these combinations satisfy the relation ad - bc = 1.
- When k = 2, we can factor 4 as 4 x 1 or 2 x 2, and -9 as -9 x 1 or -3 x 3. Again, none of these combinations satisfy the relation ad - bc = 2.
- We can continue this pattern for larger values of k, systematically checking combinations of factors for 4 and -9 until we find a set of factors that satisfy the relation ad - bc = k.
For negative values of k, we can follow the same approach, checking combinations of negative factors for 4 and -9.
Keep in mind that zero is also considered an integer value for k. So, we need to check if there are any factor combinations for 4 and -9 that satisfy the relation ad - bc = 0.
By systematically going through the possible factor combinations for 4 and -9, you can determine the integer values for k that make the expression 4x^2 + kxy - 9y^2 factorable.