SOLUTION: the sum of a number and its reciprocal is 25\12 find the number

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Question 216444: the sum of a number and its reciprocal is 25\12 find the number
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
The sum of a number and its reciprocal is 25/12. Find the number

Step 1. Let x be the number and 1/x be its reciprocal.

Step 2. Then, x%2B1%2Fx=25%2F12 since the sum of a number and its reciprocal is 25/12.

Step 3. Multiply 12x to both sides of the equation to get rid of the denominators.

12x%28x%2B1%2Fx%29=12x%2A25%2F12

12x%5E2%2B12=25x

Step 4. Subtract 25x to put the equation in quadratic form

12x%5E2%2B12-25x=25x-25x

12x%5E2-25x%2B12=0

Step 5. We can use the quadratic formula given as

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

where a=12, b=-25, and c=12.

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 12x%5E2%2B-25x%2B12+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-25%29%5E2-4%2A12%2A12=49.

Discriminant d=49 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--25%2B-sqrt%28+49+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-25%29%2Bsqrt%28+49+%29%29%2F2%5C12+=+1.33333333333333
x%5B2%5D+=+%28-%28-25%29-sqrt%28+49+%29%29%2F2%5C12+=+0.75

Quadratic expression 12x%5E2%2B-25x%2B12 can be factored:
12x%5E2%2B-25x%2B12+=+12%28x-1.33333333333333%29%2A%28x-0.75%29
Again, the answer is: 1.33333333333333, 0.75. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+12%2Ax%5E2%2B-25%2Ax%2B12+%29



So x=3%2F4 or x=4%2F3 (reciprocals of each other)

Check...3%2F4%2B4%2F3=9%2F12%2B16%2F12=25%2F12 which is a true statement.

Step 6. ANSWER: The number is 3%2F4 or 4%2F3 which are reciprocals of each other

I hope the above steps were helpful.

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http://www.FreedomUniversity.TV/courses/IntroAlgebra and for Trigonometry visit
http://www.FreedomUniversity.TV/courses/Trigonometry.

Good luck in your studies!

Respectfully,
Dr J