SOLUTION: find two consecutive integers whose product is 210.

Question 896758: find two consecutive integers whose product is 210.
Found 3 solutions by vja_1994, richwmiller, MathTherapy:
Answer by vja_1994(6)   (Show Source): You can put this solution on YOUR website!

Simple guess and check.
-Alejandro
Answer by richwmiller(17219)   (Show Source): You can put this solution on YOUR website!
There are several ways to figure this out.
sqrt(210)=abt 14.5
So a first attempt would be 14*15 which works.
and of course the negatives -15*-14=210
x*(x+1)=210
x^2+x-210=0

Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .

Now multiply the first coefficient by the last term to get .

Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?

To find these two numbers, we need to list all of the factors of (the previous product).

Factors of :

1,2,3,5,6,7,10,14,15,21,30,35,42,70,105,210

-1,-2,-3,-5,-6,-7,-10,-14,-15,-21,-30,-35,-42,-70,-105,-210

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to .

1*(-210) = -210
2*(-105) = -210
3*(-70) = -210
5*(-42) = -210
6*(-35) = -210
7*(-30) = -210
10*(-21) = -210
14*(-15) = -210
(-1)*(210) = -210
(-2)*(105) = -210
(-3)*(70) = -210
(-5)*(42) = -210
(-6)*(35) = -210
(-7)*(30) = -210
(-10)*(21) = -210
(-14)*(15) = -210

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :

First Number Second Number Sum
1 -210 1+(-210)=-209
2 -105 2+(-105)=-103
3 -70 3+(-70)=-67
5 -42 5+(-42)=-37
6 -35 6+(-35)=-29
7 -30 7+(-30)=-23
10 -21 10+(-21)=-11
14 -15 14+(-15)=-1
-1 210 -1+210=209
-2 105 -2+105=103
-3 70 -3+70=67
-5 42 -5+42=37
-6 35 -6+35=29
-7 30 -7+30=23
-10 21 -10+21=11
-14 15 -14+15=1

From the table, we can see that the two numbers and add to (the middle coefficient).

So the two numbers and both multiply to and add to

Now replace the middle term with . Remember, and add to . So this shows us that .

Replace the second term with .

Group the terms into two pairs.

Factor out the GCF from the first group.

Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.

Combine like terms. Or factor out the common term

===============================================================

Answer:

So factors to .

In other words, .

Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).

which leads to the same solutions.
x=14 x+1=15
x=-15 x+1=-14

Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!
find two consecutive integers whose product is 210.

Since they're consecutive integers, then their difference is 1.
With the 1st integer being S, the 2nd would be S + 1, and equation would then be: S(S + 1) = 210, or .
You'll need to find two factors of 210 with a difference of 1.
You should get: - 15 and - 14, or 14 and 15
Then do a check!!
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