what is the equation and whats the solution to the equation?

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. Write a quadratic equation to represent the situation. What is the solution to the equation.

Answer 1

Let first no. is x

second consecutive no. is x+1

 

The product of two consecutive numbers is equal to 35 more than eleven times the sum of the numbers.

 

x(x+1)=11(x+x+1)+35

 

x²+x=11(2x+1)+35

 

x²+x=22x+11+35

 

x²+x-22x=46

 

x²-21x-46=0

(x-23)(x+2)=0

x-23=0 or x+2=0

x=23 or x=-2

 

First no.is = 23 and second no. is =24

 

or

 

First no.is =- 2 and second no. is =-1

Answer 2

"The product of two consecutive numbers" is x*(x+1)

 

"35 more than eleven times the sum of the numbers" is 35 + 11*[x + (x+1)]  

 

These two things are equal, so 

 

x*(x+1) = 35 + 11*[x + (x+1)]

x^2 + x = 35 + 11*(2*x + 1)

x^2 + x = 35 + 22*x + 11

x^2 + x = 22*x + 46

x^2 -21*x - 46 = 0          This is the quadratic equation 

 

(x + 2)*(x - 23) = 0

x + 2 = 0   or   x - 23 = 0

x = -2   or   x = 23

 

So the two consecutive numbers could be -2 and -1, or 23 and 24

 

Check:  (-2) * (-1) = 2

            11*(-2 + (-1)) = 11*(-3) = -33

            2 - (-33) = 35 so this solution checks

 

           23 * 24 = 552

           11*(23 + 24) = 11*47 = 517

           552 - 517 = 35 so this solution checks 

Answer 3

Two pairs of numbers comply with the given condition, as it is to expect, since every quadratic equation has two solutions (except when the vertex of its graph is on the x-axis).

 

Let's see:

 

Let the numbers be x and (x+1)  -  They could be x and (x-1) as well!

 

The product of these numbers: x(x+1)

 

Eleven times their sum, plus 35: 11(x + x+1) + 35

 

Now, set up the equation:  x(x + 1) = 11(2x + 1) + 35

 

Expand on both sides:  x² + x = 22x + 11 + 35

 

Simplify and move all terms to the left:  x² + x - 22x - 46 = 0

 

Simplify by adding like-terms:  x² - 21x - 46 = 0

 

Factor as product of two binomials: (x - 23)(x + 2) = 0

 

Then, x = 23, or x = -2

 

The two pairs of numbers are: 23 and 24; and -2 and -1.  Both pairs comply with the given conditions, as you may see by plugging then in the original equation: x(x + 1) = 11(x + x + 1) + 35.

 

Now, you try the same problem using x and (x-1).  

 

Did you get the same results??