document.write( "Question 80893: the sum of a number and four times its reciprocal is -5. what is the number? \n" ); document.write( "

Algebra.Com's Answer #58037 by bucky(2189) $\"\"$ $\"About$

You can put this solution on YOUR website!
Given:
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\n" ); document.write( "The sum of a number and four times its reciprocal is -5
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\n" ); document.write( "The \"number\" is the unknown, so let's represent it by x.
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\n" ); document.write( "The reciprocal of the number by definition is 1 divided by the number, and 4 times the reciprocal
\n" ); document.write( "is, therefore:
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\n" ); document.write( " $\"4%2A%281%2Fx%29+=+4%2Fx\"$
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\n" ); document.write( "So the sum of the number and 4 times its reciprocal is:
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\n" ); document.write( " $\"x+%2B+4%2Fx\"$
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\n" ); document.write( "and the problem tells you that this is -5. So set it equal to -5:
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\n" ); document.write( " $\"x+%2B+4%2Fx+=+-5\"$
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\n" ); document.write( "Now let's solve for x. We can do so by multiplying every term (both sides) of this
\n" ); document.write( "equation by x to eliminate the denominator. Do that and the equation becomes:
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\n" ); document.write( " $\"x%5E2+%2B+4+=+-5x\"$
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\n" ); document.write( "Add 5x to both sides to eliminate the -5x on the right side and get the equation into the
\n" ); document.write( "standard quadratic form of:
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\n" ); document.write( " $\"x%5E2+%2B+5x+%2B+4+=+0\"$
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\n" ); document.write( "There are several ways that this can be solved (graphing; completing the square or its
\n" ); document.write( "equivalent, using the quadratic formula; but in this case factoring is probably the
\n" ); document.write( "easiest.) The equation factors to:
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\n" ); document.write( " $\"%28x+%2B+1%29%2A%28x+%2B+4%29+=+0\"$
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\n" ); document.write( "This equation will be true if either of the two factors is equal to zero. So set each factor
\n" ); document.write( "equal to zero and solve for the value of x that will make that happen:
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\n" ); document.write( " $\"x+%2B+1+=+0\"$
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\n" ); document.write( "Subtract 1 from both sides of the equation and the result is:
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\n" ); document.write( " $\"x+=+-1\"$
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\n" ); document.write( "Then set the second factor equal to zero:
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\n" ); document.write( " $\"x+%2B+4+=+0\"$
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\n" ); document.write( "and subtract 4 from both sides to get:
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\n" ); document.write( " $\"x+=+-4+\"$
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\n" ); document.write( "So there are two possible values for x that will work ... -1 and -4
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\n" ); document.write( "Check them out by evaluating each in the original problem.
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\n" ); document.write( "If x = -1, will $\"x+%2B+4%2A%281%2Fx%29+=+-5\"$ ? Substitute -1 for x and you get:
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\n" ); document.write( " $\"-1+%2B+%284%2F-1%29+=+-1+-4+=+-5\"$
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\n" ); document.write( "This value of -1 works. Now let's try the second value, x = -4. Substitute for x and you
\n" ); document.write( "get:
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\n" ); document.write( " $\"-4+%2B+4%2F-4+=+-4+%2B+%28-1%29+=+-4+-1+=+-5\"$
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\n" ); document.write( "This also works. Therefore, your problem has two solutions ... x = -1 and x = -4.
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\n" ); document.write( "Hope this helps you to understand the problem and how you can work it to a solution.
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\n" ); document.write( "Cheers \n" ); document.write( "

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