Answer:
21, 23 and -21, 23
Step-by-step explanation:
The difference between two consecutive odd numbers is 2.
Let the smaller number be x. Then, the larger number is x + 2.
Now add their squares and set it equal to 970. Then solve for x.
smaller number: x
larger number: x + 2
sum of the squares of the two numbers: x^2 + (x + 2)^2
equation: x^2 + (x + 2)^2 = 970
Solution of the equation:
x^2 + (x + 2)^2 = 970
Expand the square of the binomial:
x^2 + x^2 + 4x + 4 = 970
Combine like terms on the left side:
2x^2 + 4x + 4 = 970
Subtract 970 from both sides:
2x^2 + 4x - 966 = 0
Divide both sides by 2:
x^2 + 2x - 483 = 0
Now we need to factor the binomial. We need two number whose sum is 2 and whose product is -483. Look at 483. It is not divisible by 2, but it is divisible by 3.
483/3 = 161
161 is divisible by 7.
161/7 = 23
23 is prime, so the prime factorization of 483 is
483 = 3 * 7 * 23
Notice that 3 * 7 = 21, so
483 = 21 * 23
Use -21 and 23:
-21 + 23 = 2
-21 * 23 = -483
x^2 + 2x - 483 factors into (x - 21)(x + 23)
Now we continue solving the equation.
x^2 + 2x - 483 = 0
(x - 21)(x + 23) = 0
x - 21 = 0 or x + 23 = 0
x = 21 or x = -23
We let x equal the smaller of the two integers, so now we add 2 to find the second integer.
x = 21; x + 2 = 23
x = -23; x + 2 = -21
There are two sets of consecutive odd numbers that satisfy this problem.
21, 23
-23, -21
The problem did not state positive or negative consecutive odd numbers, so both solutions are valid.
