can you solve this?

Question

problem 1

the product of two consecutive positive integers is 55 more than their sum. find the integers

problem 2

if 6 is added to the square of a number, the result is 22. find all such numbers.

problem 3

the product of the page numbers on two facing pages of a book is 600. find the page numbers

Answer 1

Problem 1
x(x + 1) = x + (x + 1) + 55
x² + x = x + x + 1 + 55
x² - x - 56 = 0
(x - 8)(x + 7) = 0
x = 8 , x = - 7
Answer is 8 and 9

Problem 2
x ² + 6 = 22
x ² = 16
x = ± 4

Problem 3
x(x + 1) = 600
x² + x - 600 = 0
(x + 25)(x - 24) = 0
x = 24 is accepted
Pages are 24 and 25

Answer 2

Hai David,

1. the product of two consecutive positive integers is 55 more than their sum. find the integers

Let the positive integers be n & n+1.
Then, n(n+1) = n + (n+1) + 55
=> n^2 + n = 2n + 1 + 55
=> n^2 - n - 56 = 0
=> n^2 - 8n + 7n - 56 = 0
=> n(n-8) + 7(n-8) =0
=> (n-8)(n+7)=0
=> n=8 or (-)7
Taking only the positive integer, we get n = 8.
The integers are 7 & 8.

2. if 6 is added to the square of a number, the result is 22. find all such numbers.

Let N be the number.
Then, N^2 + 6 = 22
=> N^2 = 22-6 = 16
=> N = sq. rt. 16 = +4 or (-)4
The numbers are (+)4 & (-)4

3. the product of the page numbers on two facing pages of a book is 600. find the page

Facing pages indicates that the page numbers are consecutive.
Let the numbers be m & m+1
Then, m(m+1) = 600
=> m^2 + m - 600 = 0
=> m^2 + 25m - 24m - 600 = 0
=> m(m+25) - 24(m+25) = 0
=> (m+25)(m-24) = 0
=> m = (+)24 or (-)25
Taking only the positive value, m = 24
=> m+1 = 25
So, the page numbers are 24 & 25.

Answer 3

problem 1:

Suppose 1st integer =x

So x * (x+1) = x +(x+1) +55
x^2 + x = 2x + 56
x^2 - x - 56 = 0
Use factorisation.
x^2 -7x +8x - 56=0
x(x-7) +8(x-7)=0
(x+8)(x-7)=0
The possible values of x are 7, -8, but since it's given that x is a positive integer, it can only be 7.

So the two integers are 7 & 8.

problem 2

Suppose the number is x
So x^2 + 6 = 22
x^2 = 16
x = 4, -4

problem 3

Let the left page number = x
So the right page number = x+1

x(x+1) = 600
x^2 + x - 600 = 0
x^2 +25x-24x - 600 = 0
x(x+25) -24(x+25)=0
(x-24)(x+25)=0

The possible values of x are 24 and -25, but since a page number can't be negative, the answer is 24.

So the page numbers are 24 and 25.

Answer 4

problem 1

Let x = 1st integer, x + 1 = 2nd integer.

x(x + 1) = x + x + 1 + 55
x^2 + x = 2x + 56
x^2 - x - 56 = 0
(x - 8)(x + 7) = 0
x = 8, x = - 7

Answer: 1st set: 1st integer is 8, the 2nd is 9; or 2nd set: 1st integer is - 7, the second is - 6

Proof (1st set: 8 and 9):
8 * 9 = 8 + 9 + 55
72 = 72

Proof (2nd set: - 7 and - 6):
- 7 * - 6 = - 7 + (- 6) + 55
42 = - 7 - 6 + 55
42 = 42

problem 2
Let x = the number

x^2 + 6 = 22
x^2 = 16
x = 4

Answer: the number is 4

Proof:
= 4^2 + 6
= 16 + 6
= 22

problem 3

Let x = 1st page; x + 1 = opposite page

x(x + 1) = 600
x^2 + x - 600 = 0
(x - 24)(x + 25) = 0

Answer: 1st page is 24, opposite page is 25.

Proof:
= 24 * 25
= 600

Answer 5

1) set up an algebraic equation: x(x+1)=55+x+(x+1)
next simplify both sides: x^2+x=56+2x
subtract 56+2x from both sides: x^2-x-56=0
factor: (x-8)(x+7)=0
solve: x=8 or -7
since the problems says a positive integer, the answer must be 8 and 9

2)again set up an algebraic equation: 6+x^2=22
now subtract 6 from both sides: x^2=16
now take the square root of both sides:x=4 or -4
the answer is 4 or -4

3)Set up an equation: x(x+1)=600
multiply through: x^2+x=600
subtract 600 from both sides: x^2+x-600=0
factor: (x-24)(x+25)=0
solve: x=24 or -25
book pages can't be negative, so the page numbers must be 24 and 25.

Answer 6

Problem-1

let the two consecutive positive integeres be n and n+1

n(n+1) = n+n+1+55
n^2+n = 2n + 56
n^2 - n - 56 = 0
n^2 - 8n + 7n - 56 = 0
n(n-8)+7(n-8) = 0
n=8 or -7 (cancelled as it can't be -ve)
hence the numbers are 8 & 9

Problem-2

let the number be n

n^2 + 6 = 22
n^ = 16
n = 4 or -4

problem-3

let the number of first of the two pages be n and then the next would be n+1

n(n+1) = 600
n^2+n-600 = 0
n^2+25n-24n-600 = 0
n(n+25)-24(n+25) = 0
n = 25 & -25 (cancelled as page number can't be negative)
hence the numbers are 24 & 25

Answer 7

ans1 is 8 & 9

ans2 is +4 & -4
&
ans3 is 24 & 25
i m correct na?

Answer 8

ans1 is 8& 9.
ans2 is +4and -4
ans3 is 24and 25.