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Question 313927: The product of two consecutive positive integers is 109 more than their sum. Find the integers
Answer by rapaljer(4671)   (Show Source):
You can put this solution on YOUR website! This word problem is a "Consecutive Integer" problem, but since it calls for the PRODUCT of the numbers, it will require you to solve a QUADRATIC EQUATION. So TWO skills will be required.
First, when you have two consecutive integers,
let x= first integer
x+1 = second integer
x(x+1) = the product of the integers
x+x+1 = 2x +1 = the sum of the integers
The equation is
"the product is 109 more than the sum of the integers"
x(x+1)= 2x+1 + 109
x^2 + x = 2x + 110
Since this is a quadratic equation, set it equal to zero, by adding -2x-110 to each side.
x^2+x-2x-110=0
x^2-x-110=0
Factor!
(x-11)(x+10)=0
x=11 or x=-10
Since this problem calls for a POSITIVE number, you can reject the x=-10.
The answer is x=11
x+1 = 12.
Check: The product of the numbers is 11*12=132
The sum of the numbers is 11+12 = 23.
It is true that the product 132 is 109 more than the sum of 23.
For additional help with Word Problems, please see my own website. Click on my tutor name "rapaljer" anywhere in algebra.com. There you will see the link to my Homepage. Near the top of my Homepage, look for the link "Basic, Intermediate, and College Algebra: One Step at a Time." Choose "Basic Algebra", and look for "Chapter 1", "Section 1.10 Part I". For help with Solving Quadratic Equations and Factoring", look in Chapter 2 for these topics. In these chapters, you will find my own non-traditional explanation, complete with examples, exercises, and ALL the answers. Many of these exercises are also solved IN COLOR on my "Math in Living Color" pages that go along with these sections.
Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus
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