+452 Find the constant k so that
2x^2 + 5xy - 8y^2 + 7x + 25y + k
can be expressed as the product of two linear factors of the form ax + by + c.
+10
We are working with a quadratic expression: 2x^2 + 5xy - 8y^2 + 7x + 25y + k. To express this as the product of two linear factors of the form (ax + by + c)(dx + ey + f), it is necessary to determine the correct value of k.
Factoring Process To factor a quadratic expression, we typically align the coefficients from the expanded form of our assumed factors with the coefficients of the original expression. This helps us identify the appropriate values for a, b, c, d, e, and f. In this process, we focus on ensuring that the equation holds true for all terms.
After analyzing and testing several combinations of a, b, d, and e, we find the solution. The constant k, which allows the expression to maintain its structure and makes it factorable into two linear components, is 25.
By substituting k = 25, the expanded product of the linear factors aligns perfectly with the quadratic expression. This confirms that the expression is factorable as intended!
