SOLUTION: marilyn is three years younger than her friend rosie. in seven years the product of their ages will be 2 less than six times the product of their ages five years ago. how ald are e

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Question 567656: marilyn is three years younger than her friend rosie. in seven years the product of their ages will be 2 less than six times the product of their ages five years ago. how ald are each at present
Answer by Maths68(1474) About Me  (Show Source):
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Let
Present age of Marilyn = m
Present age of Rosie = r


Marilyn is three years younger than her friend Rosie
m=r-3..............(1)
Five years ago
Age of Marilyn was = m-5
Age of Rosie was = r-5
In seven years
Age of Marilyn will be = m+7
Age of Rosie will be = r+7
THEN
In seven years the product of their ages will be 2 less than six times the product of their ages five years ago.
(m+7)(r+7)=6[(m-5)(r-5)]-2
mr+7m+7r+49=6(mr-5m-5r+25)-2
mr+7m+7r+49=6mr-30m-30r+150-2
mr-6mr+7m+30m+30r+7r=150-2-49
-5mr+37m+37r=150-51
-5mr+37m+37r=99..................(2)
Put the value of m from (1) to (2)
-5(r-3)r+37(r-3)+37r=99
-5r^2+15r+37r-111+37r=99
-5r^2+89r-99-111=0
-5r^2+89r-200=0
Now solve for r
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ar%5E2%2Bbr%2Bc=0 (in our case -5r%5E2%2B89r%2B-210+=+0) has the following solutons:

r%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=%2889%29%5E2-4%2A-5%2A-210=3721.

Discriminant d=3721 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-89%2B-sqrt%28+3721+%29%29%2F2%5Ca.

r%5B1%5D+=+%28-%2889%29%2Bsqrt%28+3721+%29%29%2F2%5C-5+=+2.8
r%5B2%5D+=+%28-%2889%29-sqrt%28+3721+%29%29%2F2%5C-5+=+15

Quadratic expression -5r%5E2%2B89r%2B-210 can be factored:
-5r%5E2%2B89r%2B-210+=+-5%28r-2.8%29%2A%28r-15%29
Again, the answer is: 2.8, 15. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-5%2Ax%5E2%2B89%2Ax%2B-210+%29


r=15 or r = 2.8 (reject)
Put r=15 in (1)
m=r-3
m=15-3
m=12

Present age of Marilyn = m = 12 years old
Present age of Rosie = r = 15 years old


Check
======
Five years ago
Age of Marilyn was = 7
Age of Rosie was = 10
In seven years
Age of Marilyn will be = 19
Age of Rosie will be = 22
THEN
In seven years the product of their ages will be 2 less than six times the product of their ages five years ago.
19*22=6(7*10)-2
418=6(70)-2
418=420-2
418=418