document.write( "Question 143722: 1. Solve $\"x%5E2%2B8x=11\"$ by completing the square. Find exact solutions. \r
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Algebra.Com's Answer #104585 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"x%5E2%2B8x=11\"$ Start with the given equation\r
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\n" ); document.write( "\n" ); document.write( "Take half of the x-coefficient 8 to get 4 (ie $\"%281%2F2%29%2A8=4\"$ )\r
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\n" ); document.write( "\n" ); document.write( "Now square 4 to get 16. (ie $\"4%5E2=16\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2B8x%2B16=11%2B16\"$ Add this result to both sides\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2B8x%2B16=27\"$ Combine like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B4%29%5E2=27\"$ Factor $\"x%5E2%2B8x%2B16\"$ to get $\"%28x%2B4%29%5E2\"$ . (note: if you need help with factoring, check out this solver) \r
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Take the square root of both sides\r
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Subtract 4 from both sides\r
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Simplify $\"sqrt%2827%29\"$ to get $\"3%2Asqrt%283%29\"$ . (note: If you need help with simplifying square roots, check out this solver)\r
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\n" ); document.write( "\n" ); document.write( "So the solutions are\r
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\n" ); document.write( "\n" ); document.write( "a)\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2B8x=11\"$ Start with the given equation\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2B8x-11=0\"$ Subtract 11 from both sides\r
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\n" ); document.write( "\n" ); document.write( "To find the vertex, we first need to find the axis of symmetry (ie the x-coordinate of the vertex)\r
\n" ); document.write( "\n" ); document.write( "To find the axis of symmetry, use this formula:\r
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\n" ); document.write( "\n" ); document.write( " $\"x=-b%2F%282a%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "From the equation $\"y=x%5E2%2B8x-11\"$ we can see that a=1 and b=8\r
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\n" ); document.write( "\n" ); document.write( " $\"x=%28-8%29%2F%282%2A1%29\"$ Plug in b=8 and a=1\r
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\n" ); document.write( "\n" ); document.write( " $\"x=%28-8%29%2F2\"$ Multiply 2 and 1 to get 2\r
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\n" ); document.write( "\n" ); document.write( " $\"x=-4\"$ Reduce\r
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\n" ); document.write( "\n" ); document.write( "So the axis of symmetry is $\"x=-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "So the x-coordinate of the vertex is $\"x=-4\"$ . Lets plug this into the equation to find the y-coordinate of the vertex.\r
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\n" ); document.write( "\n" ); document.write( "Lets evaluate $\"f%28-4%29\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"f%28x%29=x%5E2%2B8x-11\"$ Start with the given polynomial\r
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\n" ); document.write( "\n" ); document.write( " $\"f%28-4%29=%28-4%29%5E2%2B8%28-4%29-11\"$ Plug in $\"x=-4\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"f%28-4%29=%2816%29%2B8%28-4%29-11\"$ Raise -4 to the second power to get 16\r
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\n" ); document.write( "\n" ); document.write( " $\"f%28-4%29=%2816%29%2B-32-11\"$ Multiply 8 by -4 to get -32\r
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\n" ); document.write( "\n" ); document.write( " $\"f%28-4%29=-27\"$ Now combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So the vertex is (-4,-27)\r
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\n" ); document.write( "\n" ); document.write( "b)\r
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\n" ); document.write( "\n" ); document.write( "From part a) we found the axis of symmetry to be $\"x=-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"y=x%5E2%2B8x-11\"$ , we can see that $\"a=1\"$ , $\"b=8\"$ , and $\"c=-11\"$ . Since $\"a%3E0\"$ , this tells us that the parabola opens upward and that there is a minimum. \r
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\n" ); document.write( "\n" ); document.write( "To find the minimum, we only need to look at the vertex. Since the vertex is the point (-4,-27), this means that the minimum is $\"y=-27\"$ \r
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\n" ); document.write( "\n" ); document.write( "Here's a sketch to visually verify our answers\r
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\n" ); document.write( "\n" ); document.write( " $\"+graph%28+500%2C+500%2C+-11%2C+10%2C+-29%2C+10%2C+x%5E2%2B8x-11%29+\"$ Graph of $\"y=x%5E2%2B8x-11\"$ \n" ); document.write( "

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