Question

The difference of two numbers is 3 and the difference of their square is 69. Find the numbers.

Answer 1

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Answer 2

in this problem larger numberbe x and smaller number be y
difference between the numbers x-y=3
x-y=3 add y on both sides
x-y+y=3+y
x=3+y
square of the smaller number=4 times the larger number
y^2=4*x
y^2=4(3+y)
y^2=12+4y
subtract 12 and 4y on both sides
y^2-4y-12=12+4y-12-4y
y^2-4y-12=0
on adding -6 and +2 we get -4 and multiplying -6and 2 we get -12
y^2 -4y-12=0 can be written as
y^2-6y+2y-12=0
y(y-6)+2(y-6)=0 here 1st two terms y is common term and last two terms 2 is common term
now y0--6 is common term
(y-6)(y+2)=0 here equating these two terms with zero we get y value
y-6=0
adding 6 on both sides
y-6+6=0+6
y=6
y+2=0
subtract 2 on both sides
y+2-2=0-2
y=-2
here we take only positive values
y=6 only
now x=3+y=3+6=9
x=9 and y=6
check this problem
difference of these two numbers=9-6=3 

two positive numbers differ by 5 means
here positive numbers are x and y
x-y=5
their sum is x+y
square of their sum=(x+y)^2
(x+y)^2=169
here 169 can be written as 13^2
(x+y)^2=(13)^2
here powers equal then bases equal
x+y=13 and x-y=5
solving these two equations we get x and y values
x+y=13
x-y=5
-------
2x=18
dividing both sides by 2
2x/2=18/2
x=9
x+y=13
9+y=13
subtract 9 on both sides
9+y-9=13-9
y=4
x=9 and y=4 
check:
x+y=13
9+4=13