document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209763>Question 1209763</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the constant k so that<BR>\n" );
document.write( "2x^2 + 5xy - 8y^2 + 7x + 25y + k<BR>\n" );
document.write( "can be expressed as the product of two linear factors of the form ax + by + c.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850204 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850204\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the given expression be<BR>\n" );
document.write( "$$2x^2 + 5xy - 8y^2 + 7x + 25y + k$$<BR>\n" );
document.write( "We want to express this as a product of two linear factors of the form $(ax + by + c)(dx + ey + f)$.<BR>\n" );
document.write( "First, factor the quadratic terms:<BR>\n" );
document.write( "$$2x^2 + 5xy - 8y^2 = (2x - y)(x + 8y)$$<BR>\n" );
document.write( "So, we expect the linear factors to be of the form:<BR>\n" );
document.write( "$$(2x - y + c_1)(x + 8y + c_2)$$<BR>\n" );
document.write( "Expanding this, we get:<BR>\n" );
document.write( "$$2x^2 + 16xy + 2c_2x - xy - 8y^2 - c_2y + c_1x + 8c_1y + c_1c_2$$<BR>\n" );
document.write( "$$2x^2 + 15xy - 8y^2 + (2c_2 + c_1)x + (8c_1 - c_2)y + c_1c_2$$<BR>\n" );
document.write( "We are given that the expression is<BR>\n" );
document.write( "$$2x^2 + 5xy - 8y^2 + 7x + 25y + k$$<BR>\n" );
document.write( "Comparing the coefficients of $xy$, we see that $15xy \neq 5xy$, so the factorization $(2x - y)(x + 8y)$ is incorrect.<BR>\n" );
document.write( "Let's try a different factorization of $2x^2 + 5xy - 8y^2$.<BR>\n" );
document.write( "We need to find two numbers that multiply to $2(-8) = -16$ and add to $5$. These numbers are $8$ and $-2$, so we can rewrite the middle term as $8xy - 3xy$.<BR>\n" );
document.write( "$$2x^2 + 8xy - 3xy - 12y^2 = 2x(x + 4y) - 3y(x + 4y) = (2x - 3y)(x + 4y)$$<BR>\n" );
document.write( "So we expect the linear factors to be of the form:<BR>\n" );
document.write( "$$(2x - 3y + c_1)(x + 4y + c_2)$$<BR>\n" );
document.write( "Expanding this, we get:<BR>\n" );
document.write( "$$2x^2 + 8xy + 2c_2x - 3xy - 12y^2 - 3c_2y + c_1x + 4c_1y + c_1c_2$$<BR>\n" );
document.write( "$$2x^2 + 5xy - 12y^2 + (2c_2 + c_1)x + (4c_1 - 3c_2)y + c_1c_2$$<BR>\n" );
document.write( "Comparing coefficients:<BR>\n" );
document.write( "$$2c_2 + c_1 = 7$$<BR>\n" );
document.write( "$$4c_1 - 3c_2 = 25$$<BR>\n" );
document.write( "$$c_1c_2 = k$$<BR>\n" );
document.write( "Multiply the first equation by 4:<BR>\n" );
document.write( "$$8c_2 + 4c_1 = 28$$<BR>\n" );
document.write( "Subtract the second equation from this:<BR>\n" );
document.write( "$$8c_2 + 4c_1 - (4c_1 - 3c_2) = 28 - 25$$<BR>\n" );
document.write( "$$11c_2 = 3$$<BR>\n" );
document.write( "$$c_2 = \frac{3}{11}$$<BR>\n" );
document.write( "Substitute $c_2 = \frac{3}{11}$ into $2c_2 + c_1 = 7$:<BR>\n" );
document.write( "$$2(\frac{3}{11}) + c_1 = 7$$<BR>\n" );
document.write( "$$\frac{6}{11} + c_1 = 7$$<BR>\n" );
document.write( "$$c_1 = 7 - \frac{6}{11} = \frac{77 - 6}{11} = \frac{71}{11}$$<BR>\n" );
document.write( "Now, find $k = c_1c_2$:<BR>\n" );
document.write( "$$k = (\frac{71}{11})(\frac{3}{11}) = \frac{213}{121}$$\r<BR>\n" );
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document.write( "Thus, the constant $k$ is $\frac{213}{121}$.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{213/121}$<BR>\n" );
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