SOLUTION: the product of two numbers is 84. One number is 13 more than 2 times the other. Find both numbers

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Question 610698: the product of two numbers is 84. One number is 13 more than 2 times the other. Find both numbers
Answer by Maths68(1474) About Me  (Show Source):
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Let x and y be the two numbers

The product of two numbers is 84.
xy=84.........(1)
One number is 13 more than 2 times the other.
x=2y+13..............(2)
Substitute the value of x in (1)
(2y+13)y=84
2y^2+13y=84
2y^2+13y-84=0
Solve for y
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ay%5E2%2Bby%2Bc=0 (in our case 2y%5E2%2B13y%2B-84+=+0) has the following solutons:

y%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=%2813%29%5E2-4%2A2%2A-84=841.

Discriminant d=841 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-13%2B-sqrt%28+841+%29%29%2F2%5Ca.

y%5B1%5D+=+%28-%2813%29%2Bsqrt%28+841+%29%29%2F2%5C2+=+4
y%5B2%5D+=+%28-%2813%29-sqrt%28+841+%29%29%2F2%5C2+=+-10.5

Quadratic expression 2y%5E2%2B13y%2B-84 can be factored:
2y%5E2%2B13y%2B-84+=+2%28y-4%29%2A%28y--10.5%29
Again, the answer is: 4, -10.5. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B13%2Ax%2B-84+%29

So
y=4 or y=-10.5(inadmissible)
Put the value of y=4 in (1)
xy=84
4x=84
x=84/4
x=21
x,y = 21,4

Check
======
xy=84
21*4=84
84=84

x=2y+13
21=2(4)+13
21=8+13
21=21