SOLUTION: the sum of two numbers is 30. if the larger number is divided by the smaller number the quotient is equal to the smaller number. find the numbers.

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Question 699126: the sum of two numbers is 30. if the larger number is divided by the smaller number the quotient is equal to the smaller number. find the numbers.
Answer by Stitch(470) About Me  (Show Source):
You can put this solution on YOUR website!
Equation 1: A+%2B+B+=+30
Equation 2:A%2FB+=+B
Solve equation 2 for A by multiplying both sides by B
A+=+B%5E2
Now plug B^2 into equation 1 for A
Equation 1: A+%2B+B+=+30
%28B%5E2%29+%2B+B+=+30
Subtract 30 from both sides to set the equation equal to zero
B%5E2+%2B+B+-+30+=+0
Now you can use the quadratic equation
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aB%5E2%2BbB%2Bc=0 (in our case 1B%5E2%2B1B%2B-30+=+0) has the following solutons:

B%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=%281%29%5E2-4%2A1%2A-30=121.

Discriminant d=121 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-1%2B-sqrt%28+121+%29%29%2F2%5Ca.

B%5B1%5D+=+%28-%281%29%2Bsqrt%28+121+%29%29%2F2%5C1+=+5
B%5B2%5D+=+%28-%281%29-sqrt%28+121+%29%29%2F2%5C1+=+-6

Quadratic expression 1B%5E2%2B1B%2B-30 can be factored:
1B%5E2%2B1B%2B-30+=+1%28B-5%29%2A%28B--6%29
Again, the answer is: 5, -6. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B1%2Ax%2B-30+%29

Note that B can equal 5 or -6.
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Plug 5 into equation 1 for B
Equation 1: A+%2B+B+=+30
A+%2B+%285%29+=+30
Subtract 5 from both sides
highlight%28A+=+25%29
Check your answer:
5 + 25 = 30 That checks
25 / 5 = 5 That checks
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Plug -6 into equation 1 for B
Equation 1: A+%2B+B+=+30
A+%2B+%28-6%29+=+30
Add 6 to both sides
highlight%28A+=+36%29
Check your answer:
-6 + 36 = 30 That checks
36 / -6 = -6 That checks
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So for this problem you have two answers: 5 & 25, or -6 and 36