Question

Find the two successive positive integers such that
the difference between their squares is 117.​

Answer 1

Answer:

6 and 9

Step-by-step explanation:

Let the larger integer be  a  and the smaller integer be  b . Then we are given:

a−ba2+b2=3=117(1)(2)

Squaring  (1) , we have:

a2−2ab+b2=9⇒2ab=(a2+b2)−9(3)

Substituting  (2)  into  (3) , we get

2ab=117−9⇒ab=54(4)

Then, from  (1)  we have  b=a−3,  which we can substitute into  (4)  to get:

a(a−3)=54(5)

From here, we can proceed a couple of ways. We can solve the quadratic

a2−3a−54=0

or we can take an educated guess. We first note from  (5)  that  a  and  a−3  must bracket  54−−√.  Since  54−−√  is between  7  and  8 , the only possible integer values for  a  are  8,   9,  and  10.  By trial and error (or noting that  54=6×9 ) we quickly find that  a=9  solves  (5) . Therefore our two numbers are  6 and 9.

Answer 2

Answer:

this is your answer

Step-by-step explanation:

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