document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210612>Question 1210612</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Fill in the blanks to make an equation that is linear in disguise, and has a solution.<BR>\n" );
document.write( "sqrt(2x - 3) + __x + ___ = ___x - ___ + ____<BR>\n" );
document.write( "8 9 10 11 12 13 14 15 16 17 18 19 20  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854388 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</B></A><img src=\"/images/stars/stars-13.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=854388\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To make the equation \"linear in disguise,\" we need to eliminate the square root $\sqrt{2x - 3}$. The most common way to do this is to ensure the other terms allow you to isolate the root and then square both sides, or more simply, to choose coefficients that force the $x$ terms to cancel out in a way that leaves a solvable linear-like state.\r<BR>\n" );
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document.write( "However, the classic \"in disguise\" trick for this specific structure is to set it up so that when you isolate the radical and square it, the $x^2$ terms on both sides are identical, leaving you with a linear equation.\r<BR>\n" );
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document.write( "Here is a set of numbers from your list that works:\r<BR>\n" );
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document.write( "### The Equation<BR>\n" );
document.write( "$$\sqrt{2x - 3} + \mathbf{8}x + \mathbf{11} = \mathbf{8}x + \mathbf{9} + \sqrt{\dots}$$<BR>\n" );
document.write( "*(Note: Since you only provided one square root in the prompt structure, we will isolate it.)*\r<BR>\n" );
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document.write( "Let's fill the blanks to create a solvable linear path:<BR>\n" );
document.write( "**$\sqrt{2x - 3} + 10x + 15 = 10x + 18$**\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### Step-by-Step Solution:<BR>\n" );
document.write( "1.  **Subtract $10x$ from both sides:**<BR>\n" );
document.write( "    The $x$ terms cancel out immediately, which is the ultimate \"disguise.\"<BR>\n" );
document.write( "    $$\sqrt{2x - 3} + 15 = 18$$\r<BR>\n" );
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document.write( "2.  **Isolate the radical:**<BR>\n" );
document.write( "    Subtract $15$ from both sides:<BR>\n" );
document.write( "    $$\sqrt{2x - 3} = 3$$\r<BR>\n" );
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document.write( "3.  **Square both sides:**<BR>\n" );
document.write( "    $$2x - 3 = 9$$\r<BR>\n" );
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document.write( "4.  **Solve for $x$:**<BR>\n" );
document.write( "    $$2x = 12$$<BR>\n" );
document.write( "    $$x = 6$$\r<BR>\n" );
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document.write( "### Why these numbers?<BR>\n" );
document.write( "* **10 (Blanks 1 & 3):** By putting the same coefficient for $x$ on both sides, the equation becomes a simple radical equation that reduces to linear form.<BR>\n" );
document.write( "* **15 and 18 (Blanks 2 & 4):** We chose these so that $18 - 15 = 3$. When squared, this gives $9$, and since $9 + 3 = 12$ (an even number), it ensures $x$ is a clean integer ($6$).\r<BR>\n" );
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document.write( "**Filled Blanks:**<BR>\n" );
document.write( "$\sqrt{2x - 3} + \underline{\mathbf{10}}x + \underline{\mathbf{15}} = \underline{\mathbf{10}}x + \underline{\mathbf{18}}$<BR>\n" );
document.write( "*(The 5th blank in your prompt `+ ____` was not used in this specific solution, but could be filled with **0** or combined with the other constants).*\r<BR>\n" );
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document.write( "Would you like to try a version where the $x$ terms don't cancel immediately, requiring you to square a binomial? </SPAN>\n" );
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