document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209974>Question 1209974</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For how many integer values of a does the equation<BR>\n" );
document.write( "x^2 + ax + 12a = 5x + 8<BR>\n" );
document.write( "have integer solutions for x? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851300 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=851300\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Alright, let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**1. Rearrange the Equation**\r<BR>\n" );
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document.write( "First, let's rearrange the given equation into a standard quadratic form:\r<BR>\n" );
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document.write( "$$x^2 + ax + 12a = 5x + 8$$<BR>\n" );
document.write( "$$x^2 + (a - 5)x + (12a - 8) = 0$$\r<BR>\n" );
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document.write( "**2. Apply the Quadratic Formula**\r<BR>\n" );
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document.write( "For the equation to have integer solutions for *x*, the discriminant must be a perfect square. The quadratic formula is:\r<BR>\n" );
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document.write( "$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$\r<BR>\n" );
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document.write( "In our case, *a* = 1, *b* = (a - 5), and *c* = (12a - 8). So:\r<BR>\n" );
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document.write( "$$x = \frac{-(a - 5) \pm \sqrt{(a - 5)^2 - 4(1)(12a - 8)}}{2(1)}$$\r<BR>\n" );
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document.write( "$$x = \frac{5 - a \pm \sqrt{a^2 - 10a + 25 - 48a + 32}}{2}$$\r<BR>\n" );
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document.write( "$$x = \frac{5 - a \pm \sqrt{a^2 - 58a + 57}}{2}$$\r<BR>\n" );
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document.write( "**3. Analyze the Discriminant**\r<BR>\n" );
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document.write( "For *x* to be an integer, the discriminant (the expression inside the square root) must be a perfect square. Let's call the discriminant *D*:\r<BR>\n" );
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document.write( "$$D = a^2 - 58a + 57$$\r<BR>\n" );
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document.write( "We need to find integer values of *a* such that *D* is a perfect square. Let *D* = *k*², where *k* is a non-negative integer.\r<BR>\n" );
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document.write( "$$a^2 - 58a + 57 = k^2$$\r<BR>\n" );
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document.write( "Complete the square for the *a* terms:\r<BR>\n" );
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document.write( "$$(a - 29)^2 - 29^2 + 57 = k^2$$\r<BR>\n" );
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document.write( "$$(a - 29)^2 - 841 + 57 = k^2$$\r<BR>\n" );
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document.write( "$$(a - 29)^2 - 784 = k^2$$\r<BR>\n" );
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document.write( "$$(a - 29)^2 - k^2 = 784$$\r<BR>\n" );
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document.write( "$$(a - 29 - k)(a - 29 + k) = 784$$\r<BR>\n" );
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document.write( "**4. Factor 784**\r<BR>\n" );
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document.write( "We need to find pairs of factors of 784. The prime factorization of 784 is 2⁴ * 7².\r<BR>\n" );
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document.write( "The pairs of factors of 784 are:\r<BR>\n" );
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document.write( "(1, 784), (2, 392), (4, 196), (7, 112), (8, 98), (14, 56), (16, 49), (28, 28)\r<BR>\n" );
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document.write( "We also need to consider the negative pairs:\r<BR>\n" );
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document.write( "(-1, -784), (-2, -392), (-4, -196), (-7, -112), (-8, -98), (-14, -56), (-16, -49), (-28, -28)\r<BR>\n" );
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document.write( "**5. Solve for 'a'**\r<BR>\n" );
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document.write( "Let's set up a system of equations for each pair of factors.\r<BR>\n" );
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document.write( "Let:\r<BR>\n" );
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document.write( "* a - 29 - k = m<BR>\n" );
document.write( "* a - 29 + k = n\r<BR>\n" );
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document.write( "Then, m * n = 784. Adding the two equations, we get:\r<BR>\n" );
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document.write( "2a - 58 = m + n<BR>\n" );
document.write( "a = (m + n + 58) / 2\r<BR>\n" );
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document.write( "Since *a* must be an integer, *m + n + 58* must be even. This means *m* and *n* must have the same parity (both even or both odd). Since 784 is even, all pairs of factors will have the same parity.\r<BR>\n" );
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document.write( "Now, we calculate the values of *a* for each pair of factors:\r<BR>\n" );
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document.write( "* (1, 784): a = (1 + 784 + 58) / 2 = 421.5 (not an integer)<BR>\n" );
document.write( "* (2, 392): a = (2 + 392 + 58) / 2 = 226<BR>\n" );
document.write( "* (4, 196): a = (4 + 196 + 58) / 2 = 129<BR>\n" );
document.write( "* (7, 112): a = (7 + 112 + 58) / 2 = 88.5 (not an integer)<BR>\n" );
document.write( "* (8, 98): a = (8 + 98 + 58) / 2 = 82<BR>\n" );
document.write( "* (14, 56): a = (14 + 56 + 58) / 2 = 64<BR>\n" );
document.write( "* (16, 49): a = (16 + 49 + 58) / 2 = 61.5 (not an integer)<BR>\n" );
document.write( "* (28, 28): a = (28 + 28 + 58) / 2 = 57<BR>\n" );
document.write( "* (-1, -784): a = (-1 - 784 + 58) / 2 = -363.5 (not an integer)<BR>\n" );
document.write( "* (-2, -392): a = (-2 - 392 + 58) / 2 = -168<BR>\n" );
document.write( "* (-4, -196): a = (-4 - 196 + 58) / 2 = -71<BR>\n" );
document.write( "* (-7, -112): a = (-7 - 112 + 58) / 2 = -30.5 (not an integer)<BR>\n" );
document.write( "* (-8, -98): a = (-8 - 98 + 58) / 2 = -24<BR>\n" );
document.write( "* (-14, -56): a = (-14 - 56 + 58) / 2 = -6<BR>\n" );
document.write( "* (-16, -49): a = (-16 - 49 + 58) / 2 = -3.5 (not an integer)<BR>\n" );
document.write( "* (-28, -28): a = (-28 - 28 + 58) / 2 = 1\r<BR>\n" );
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document.write( "The integer values of *a* are: 226, 129, 82, 64, 57, -168, -71, -24, -6, 1.\r<BR>\n" );
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document.write( "There are 10 integer values of *a*.\r<BR>\n" );
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document.write( "**Final Answer**\r<BR>\n" );
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document.write( "There are 10 integer values of *a*.<BR>\n" );
document.write( " </SPAN>\n" );
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