Certainly! Let's solve the problem step by step.
We need to find two consecutive positive even numbers such that the sum of their squares is 164.
1. Let's denote the first even number as [tex]\( x \)[/tex].
2. The next consecutive even number will then be [tex]\( x + 2 \)[/tex]. Since even numbers differ by a gap of 2, this is straightforward.
3. Now, let's set up the equation given by the problem:
The sum of the squares of these numbers is 164. Therefore, we can write:
[tex]\[
x^2 + (x + 2)^2 = 164
\][/tex]
4. Expand and simplify the equation:
First, compute [tex]\((x + 2)^2\)[/tex]:
[tex]\[
(x + 2)^2 = x^2 + 4x + 4
\][/tex]
So, substituting back into the equation, we get:
[tex]\[
x^2 + x^2 + 4x + 4 = 164
\][/tex]
Combine like terms:
[tex]\[
2x^2 + 4x + 4 = 164
\][/tex]
5. Rearrange it into standard quadratic form:
[tex]\[
2x^2 + 4x + 4 - 164 = 0
\][/tex]
[tex]\[
2x^2 + 4x - 160 = 0
\][/tex]
Divide the entire equation by 2 to simplify:
[tex]\[
x^2 + 2x - 80 = 0
\][/tex]
6. Solve the quadratic equation:
We need to find the roots of the quadratic equation [tex]\( x^2 + 2x - 80 = 0 \)[/tex]. Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = -80 \)[/tex], we get:
[tex]\[
x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-80)}}{2 \cdot 1}
\][/tex]
Simplify inside the square root:
[tex]\[
x = \frac{-2 \pm \sqrt{4 + 320}}{2}
\][/tex]
[tex]\[
x = \frac{-2 \pm \sqrt{324}}{2}
\][/tex]
[tex]\[
x = \frac{-2 \pm 18}{2}
\][/tex]
This gives us two solutions:
[tex]\[
x = \frac{-2 + 18}{2} = \frac{16}{2} = 8
\][/tex]
[tex]\[
x = \frac{-2 - 18}{2} = \frac{-20}{2} = -10
\][/tex]
7. Select the positive solution:
Since we are looking for positive even numbers, we discard [tex]\( x = -10 \)[/tex]. Therefore, [tex]\( x = 8 \)[/tex].
8. Find the consecutive even number:
The first even number is [tex]\( x = 8 \)[/tex].
The next consecutive even number is [tex]\( x + 2 = 8 + 2 = 10 \)[/tex].
Thus, the two consecutive positive even numbers are 8 and 10.
