Certainly! Let's find the value of [tex]\( x^2 + y^2 \)[/tex] given the equations [tex]\( x + y = 5 \)[/tex] and [tex]\( xy = 10 \)[/tex].
We start with the given equation:
[tex]\[ x + y = 5 \][/tex]
First, let's square both sides of the equation:
[tex]\[ (x + y)^2 = 5^2 \][/tex]
[tex]\[ x^2 + 2xy + y^2 = 25 \][/tex]
We also know:
[tex]\[ xy = 10 \][/tex]
Now, we substitute [tex]\( 2xy \)[/tex] into the squared equation:
[tex]\[ x^2 + 2xy + y^2 = 25 \][/tex]
[tex]\[ 2xy = 2 \times 10 \][/tex]
[tex]\[ 2xy = 20 \][/tex]
So,
[tex]\[ x^2 + y^2 + 20 = 25 \][/tex]
Next, we subtract 20 from both sides to isolate [tex]\( x^2 + y^2 \)[/tex]:
[tex]\[ x^2 + y^2 = 25 - 20 \][/tex]
[tex]\[ x^2 + y^2 = 5 \][/tex]
So, the value of [tex]\( x^2 + y^2 \)[/tex] is:
[tex]\[ \boxed{5} \][/tex]
