funzo
the numbers are x and x-1
a.
"product of them is equal to 35 more than 11 times sum of them" translated to maths is
(x)(x-1)=35+11(x+x-1)
we need to get it into standard form ([tex]0=ax^2+bx+c[/tex])
expand
[tex]x^2-x=35+11(2x-1)[/tex]
[tex]x^2-x=35+22x-11[/tex]
[tex]x^2-x=22x+24[/tex]
[tex]x^2-23x-24=0[/tex]
b.
factor
what 2 numbers multiply to get -24 and add to get -23?
-24 and 1
[tex](x-24)(x+1)=0[/tex]
set each equal to 0
x-24=0, x=24
x+1=0, x=-1
the solutions are 24 and -1
c.
if x=24, then x-1=23, the 2 numbers are 24 and 23
if x=-1, then x-1=-2, the 2 numbers are -1 and -2
since the question did not specify if the numbers had to be of a certain sign (positive or negative) or something like that, there are 2 pairs of consecutive numbers that satisfy the equation
a) The numbers are represented by x and x-1, so their product is x(x-1).
Their sum will be (x +(x-1)), and 11 times that is 11(x +(x-1)). The quantity that is 35 more than this value is 35+11(x +(x-1)).
The problem statement tells us these two functions of the numbers are equal, so we can write ...
... x(x-1) = 35 + 11(x +(x-1))
b) When simplified and put in standard form, the equation is ...
... x² -x = 35 +22x -11
... x² -23x -24 = 0 . . . . . subtract the right side from both sides
... (x -24)(x +1) = 0 . . . . factor
Solutions are the values of x that make these factors zero: x=24, x=-1.
c) Since the numbers are x and x-1, the solution x=24 means the numbers are 24 and 24-1 = 23. The solution x=-1 means the numbers are -1 and -1-1 = -2.
The numbers are {23, 24} or {-2, -1}.
_____
Check
For 23, 24
The product is 23·24 = 552
The sum is 23+24 = 47, and 11 times that is 517. 35 more than this is 552, so this answer checks OK.
For -2, -1
The product is (-2)·(-1) = 2.
The sum is -2-1 = -3, and 11 times that is -33. 35 more than this is 35-33 = 2, so this answer checks OK.
