Question

Two consecutive numbers can be represented by x and x - 1. The product of two consecutive numbers is equal to 35 more than eleven times the sum of the numbers. a. Write a quadratic equation to represent the situation. b. What are the solutions to the equation? c. What are the two consecutive numbers?

Answer 1

• a) x(x-1) = 35 + 11(x +(x-1))
• b) x=24 or x=-1
• c) 23, 24 . . . or . . . -2, -1

Step-by-step explanation:

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.

Answer 2

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 (
`0=ax^2+bx+c`)

expand

`x^2-x=35+11(2x-1)`

`x^2-x=35+22x-11`

`x^2-x=22x+24`

`x^2-23x-24=0`

b.

factor

what 2 numbers multiply to get -24 and add to get -23?

-24 and 1

`(x-24)(x+1)=0`

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