Question 222965
Consecutive integers are 1 apart from each other. So if one integer is "x" then the next one is "x+1". Now let's use these to translate
"the product of two consecutive integers is 1 more than their sum"<br>
Product means multiplication so "the product of two consecutive integers" is (x)(x+1). Sum means addition so "their sum" is (x) + (x+1)<br>
So "the product of two consecutive integers is 1 more than their sum" translates into:
{{{(x)(x+1) = 1 + (x) + (x+1)}}}
Now we just need to solve this. Start by simplifying each side:
{{{x^2 + x = 2x + 2}}}
Next, since this is a quadratic equation (because of the {{{x^2}}}), we will get one side equal to zero by subtracting the entire right side from both sides:
{{{x^2 - x - 2 = 0}}}
Now we can solve this by factoring (or by using the quadratic formula):
{{{x - 2)(x + 1) = 0}}}
The Zero Product Property tells us that if this or any product is equal to zero then one or more of the factors must be zero. So:
{{{x-2 = 0}}} or {{{x+1 = 0}}}
Solving each of these simple equations we get:
{{{x = 2}}} or {{{x = -1}}}
Now remember what "x" represents. (Look back at the beginning.) It represents the first (or smaller) of the two consecutive integers. And (x+1) represents the second (or larger) of the two. We have found two values for x: 2 and -1. So we have found two pairs of consecutive integers:<ul><li>2 and 3</li><li>-1 amd 0</li></ul>
You can check them out and they both work:<ul><li>2*3 is one more than 2+3</li><li>-1*0 is one more than -1+0</li><ul>