Question 80893: the sum of a number and four times its reciprocal is -5. what is the number? Found 2 solutions by sofiyacherni, bucky:Answer by sofiyacherni(99) (Show Source):
You can put this solution on YOUR website! there are two solutions
name your number x, then reciprocal is 1/x
so x+4/x=-5
find common denominator cross multiply you can factor this equation into
(x+4)(x+1)=0 thus, x+4=0 and x+1=0
so your two solutions are
x=-4 and x=-1
good luck
You can put this solution on YOUR website! Given:
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The sum of a number and four times its reciprocal is -5
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The "number" is the unknown, so let's represent it by x.
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The reciprocal of the number by definition is 1 divided by the number, and 4 times the reciprocal
is, therefore:
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So the sum of the number and 4 times its reciprocal is:
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and the problem tells you that this is -5. So set it equal to -5:
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Now let's solve for x. We can do so by multiplying every term (both sides) of this
equation by x to eliminate the denominator. Do that and the equation becomes:
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Add 5x to both sides to eliminate the -5x on the right side and get the equation into the
standard quadratic form of:
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There are several ways that this can be solved (graphing; completing the square or its
equivalent, using the quadratic formula; but in this case factoring is probably the
easiest.) The equation factors to:
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This equation will be true if either of the two factors is equal to zero. So set each factor
equal to zero and solve for the value of x that will make that happen:
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Subtract 1 from both sides of the equation and the result is:
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Then set the second factor equal to zero:
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and subtract 4 from both sides to get:
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So there are two possible values for x that will work ... -1 and -4
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Check them out by evaluating each in the original problem.
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If x = -1, will ? Substitute -1 for x and you get:
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This value of -1 works. Now let's try the second value, x = -4. Substitute for x and you
get:
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This also works. Therefore, your problem has two solutions ... x = -1 and x = -4.
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Hope this helps you to understand the problem and how you can work it to a solution.
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Cheers
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