document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=73989>Question 73989</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve each of the following quadratic equations by completing the square.<BR>\n" );
document.write( "sect:10.2 # 22<BR>\n" );
document.write( "x^2-6x-3=0\r<BR>\n" );
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document.write( "Section 10.2 # 28\r<BR>\n" );
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document.write( "2x^2+10x+11=0\r<BR>\n" );
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document.write( "Sect. 10.2 # 38\r<BR>\n" );
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document.write( "Number problems. Find two consecutive positive integers such that the sum of<BR>\n" );
document.write( "their squares is 85. <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 #53012 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=53012\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given:  x^2-6x-3=0<BR>\n" );
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document.write( "Solve by completing the square<BR>\n" );
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document.write( "The first thing to do is to make sure the coefficient (the multiplier) of the x^2 term is 1.<BR>\n" );
document.write( "In this case it is. So we can move on to the next step. Use parentheses to separate the<BR>\n" );
document.write( "two terms that contain x from the rest of the equation.  This becomes:<BR>\n" );
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document.write( "(x^2 - 6x     )-3 = 0<BR>\n" );
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document.write( "Divide the coefficient of the x term by 2.  In this case it is -6/2 = -3.  Square this result<BR>\n" );
document.write( "to get (-3)^2 = 9. Add 9 inside the parentheses, but in order to do this without changing the <BR>\n" );
document.write( "equation you need to subtract 9 outside the parentheses.  This makes the equation become:<BR>\n" );
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document.write( "(x^2 - 6x + 9) - 3 - 9 = 0<BR>\n" );
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document.write( "Combine the two numbers outside the parentheses to get:<BR>\n" );
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document.write( "(x^2 - 6x + 9) - 12 = 0<BR>\n" );
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document.write( "Add +12 to both sides to eliminate the -12 on the left side and get:<BR>\n" );
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document.write( "(x^2 - 6x + 9) = 12<BR>\n" );
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document.write( "Note that the trinomial in the parentheses is a perfect square. It can be written as<BR>\n" );
document.write( "(x - 3)^2.  Substitute this into the equation and you get:<BR>\n" );
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document.write( "(x-3)^2 = 12<BR>\n" );
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document.write( "Now take the square root of both sides.  I'm going to use +- to mean plus and minus.  The <BR>\n" );
document.write( "square root of both sides results in:<BR>\n" );
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document.write( "x-3 = +- sqrt(12) <BR>\n" );
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document.write( "Eliminate the -3 on the left side by adding 3 to both sides to get:<BR>\n" );
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document.write( "x = 3 +- sqrt(12)<BR>\n" );
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document.write( "As a final simplification note that sqrt(12) = sqrt(4*3) = sqrt(4)* sqrt(3) = 2*sqrt(3).<BR>\n" );
document.write( "Substitute this for sqrt(12) and the answer becomes:<BR>\n" );
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document.write( "x = 3 +- 2*sqrt(3)<BR>\n" );
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document.write( "On to the next problem: solve by completing the squares: 2x^2+10x+11=0 <BR>\n" );
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document.write( "Recall that I mentioned you want the coefficient of the x^2 term to be 1.  In this case<BR>\n" );
document.write( "it is not.  So divide the entire equation (all terms on both sides by 2 to reduce it to:<BR>\n" );
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document.write( "x^2 + 5x + 11/2 = 0<BR>\n" );
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document.write( "Convert the 11/2 term to 5.5 by dividing 2 into 11.<BR>\n" );
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document.write( "Again use parentheses to isolate the terms containing x and get:<BR>\n" );
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document.write( "(x^2 + 5x     ) + 5.5 = 0<BR>\n" );
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document.write( "Take half of the coefficient of the x term and square it.  Half of 5 is 2.5 which squares<BR>\n" );
document.write( "to 6.25.  Add it inside the parentheses and subtract it outside the parentheses to get:<BR>\n" );
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document.write( "(x^2 + 5x + 6.25) + 5.5 - 6.25 = 0<BR>\n" );
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document.write( "Combine the two constants outside the parentheses to get:<BR>\n" );
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document.write( "(x^2 + 5x + 6.25) - 0.75 = 0<BR>\n" );
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document.write( "Eliminate the - 0.75 from the left side by adding 0.75 to both sides and you have:<BR>\n" );
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document.write( "(x^2 + 5x + 6.25) = 0.75<BR>\n" );
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document.write( "The term in the parentheses is a perfect square so it can be written as such and the equation becomes:<BR>\n" );
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document.write( "(x + 2.5)^2 = 0.75<BR>\n" );
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document.write( "Take the square root of both sides and the result is:<BR>\n" );
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document.write( "x+2.5 = +- sqrt(0.75)<BR>\n" );
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document.write( "Eliminate the 2.5 on the left side by subtracting 2.5 from both sides to get:<BR>\n" );
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document.write( "x = -2.5 +- sqrt(0.75) <BR>\n" );
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document.write( "Note that the sqrt(0.75) = sqrt(0.25*3) = sqrt(0.25)*sqrt(3) = 0.5*sqrt(3)<BR>\n" );
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document.write( "Substitute this and the answer becomes:<BR>\n" );
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document.write( "x = -2.5 +- 0.5*sqrt(3)<BR>\n" );
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document.write( "And on to the final problem. Let one of the integers be N. Then the next consecutive<BR>\n" );
document.write( "integer is N + 1.  The sum of their squares can be written as:<BR>\n" );
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document.write( "N^2 + (N+1)^2<BR>\n" );
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document.write( "and the problem says the sum of these two squares equals 85. So the equation becomes:<BR>\n" );
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document.write( "N^2 + (N+1)^2 = 85<BR>\n" );
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document.write( "Square the term in parentheses on the left side to get:<BR>\n" );
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document.write( "N^2 + N^2 + 2N + 1 = 85<BR>\n" );
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document.write( "Combine the N^2 terms and the equation becomes:<BR>\n" );
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document.write( "2N^2 + 2N + 1 = 85<BR>\n" );
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document.write( "To make things a little easier let's eliminate the +1 on the left side by adding -1 to both<BR>\n" );
document.write( "sides to get:<BR>\n" );
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document.write( "2N^2 + 2N = 84<BR>\n" );
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document.write( "We need a coefficient of 1 on the N^2 term so divide the entire equation (all terms on<BR>\n" );
document.write( "both sides by 2 to get:<BR>\n" );
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document.write( "N^2 + N = 42<BR>\n" );
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document.write( "Just noticed something here.  We don't need to solve this one by completing the square.  <BR>\n" );
document.write( "Subtract 42 from both sides to get:<BR>\n" );
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document.write( "N^2 + N - 42 = 0<BR>\n" );
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document.write( "This can be factored into:<BR>\n" );
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document.write( "(N+7)*(N-6) = 0<BR>\n" );
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document.write( "And this equation will be true if either N = -7 or N = +6 because those are the two values<BR>\n" );
document.write( "that make one of the factors equal to zero. The problem limits the answer to positive<BR>\n" );
document.write( "integers so we can forget N = -7.  The two integers are +6 and the next consecutive<BR>\n" );
document.write( "integer of +7.<BR>\n" );
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document.write( "Check.<BR>\n" );
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document.write( "6^2 + 7^2 = 36 + 49 = 85 .  Yup. It works. \r<BR>\n" );
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document.write( "If you must work it by completing the square you can use the same process as in the preceding<BR>\n" );
document.write( "two problems, and you know that you should get an answer of 6 for N.<BR>\n" );
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document.write( "Hope this helps you to understand and solve these types of problems.<BR>\n" );
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