Two numbers have a sum of 22 and their product is 103. What are the numbers in simplest radical form?

2 Answers
Jun 19, 2018

11±32

Explanation:

Call the two numbers a and b. Then:
a+b=22 and ab=103.

From the first equation:
b=22a
Substitute into the second:
a(22a)=103
22aa2=103
0=a222a+103

Quadratic formula:
a=12(22±484412)=12(22±72)
=12(22±62)=11±32

So we have two solutions for a. Use these to obtain b from the first formula:
11±32+b=22
b=11¯¯¯¯¯+32

So for the pair of values of a, b always takes the other one of the two values. So the two numbers are simply the pair:
11±32

Double check with the second formula that these are correct:
(11+32)(1132)=103
Note that this is a 'difference of two squares' formula
12192=103
Yep, this checks out.

Jun 19, 2018

a=11+32
b=1132

Explanation:

a+b=22b=22a
ab=103
a(22a)=103
a2+22a=103
a222a+103=0
We have (a11)2121+103=(a11)218
a11=±18=±32
a=11+32
b=1132