Let's solve the problem step-by-step:
1. Define the consecutive integers:
Let's denote the first integer by [tex]\(x\)[/tex]. Since the integers are consecutive, the next integer will be [tex]\(x + 1\)[/tex].
2. Write the equation for their squares:
The square of the first integer [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex].
The square of the second integer [tex]\(x + 1\)[/tex] is [tex]\((x + 1)^2\)[/tex].
3. Sum of their squares:
According to the problem, the sum of the squares of these two integers equals 365:
[tex]\[
x^2 + (x + 1)^2 = 365
\][/tex]
4. Simplify the equation:
Expand and simplify the equation:
[tex]\[
x^2 + (x^2 + 2x + 1) = 365
\][/tex]
[tex]\[
x^2 + x^2 + 2x + 1 = 365
\][/tex]
[tex]\[
2x^2 + 2x + 1 = 365
\][/tex]
5. Move all terms to one side of the equation:
[tex]\[
2x^2 + 2x + 1 - 365 = 0
\][/tex]
[tex]\[
2x^2 + 2x - 364 = 0
\][/tex]
6. Solve the quadratic equation:
To solve this quadratic equation, we can use the quadratic formula [tex]\( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \)[/tex] where [tex]\( a = 2 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = -364 \)[/tex].
7. Determine the roots:
The roots of the equation [tex]\( 2x^2 + 2x - 364 = 0 \)[/tex] are:
[tex]\[
x_1 = -14, \quad x_2 = 13
\][/tex]
8. Verify the consecutive integers:
Substituting back, we find the two consecutive integers:
[tex]\[
\text{First integer: } x = -14
\][/tex]
[tex]\[
\text{Second integer: } x + 1 = -13
\][/tex]
9. Compute the difference:
The difference between the two consecutive integers is:
[tex]\[
|-13 - (-14)| = 1
\][/tex]
Therefore, the difference between the two consecutive integers whose squares sum up to 365 is [tex]\(1\)[/tex].
