To find the value of [tex]\( a^2 + b^2 + c^2 \)[/tex] given the equations [tex]\( a + b + c = 15 \)[/tex] and [tex]\( ab + bc + ca = 75 \)[/tex], we can utilize an important algebraic identity.
The identity we will use is:
[tex]\[
(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)
\][/tex]
Let's break it down step-by-step:
1. Start with the given equations:
[tex]\[
a + b + c = 15
\][/tex]
[tex]\[
ab + bc + ca = 75
\][/tex]
2. Substitute the known values into the identity:
[tex]\[
(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)
\][/tex]
3. Plug in [tex]\( a + b + c = 15 \)[/tex]:
[tex]\[
15^2 = a^2 + b^2 + c^2 + 2 \cdot 75
\][/tex]
4. Calculate [tex]\( 15^2 \)[/tex]:
[tex]\[
225 = a^2 + b^2 + c^2 + 150
\][/tex]
5. Now, solve for [tex]\( a^2 + b^2 + c^2 \)[/tex]:
[tex]\[
225 = a^2 + b^2 + c^2 + 150
\][/tex]
6. Subtract 150 from both sides to isolate [tex]\( a^2 + b^2 + c^2 \)[/tex]:
[tex]\[
225 - 150 = a^2 + b^2 + c^2
\][/tex]
7. Simplify the result:
[tex]\[
75 = a^2 + b^2 + c^2
\][/tex]
Therefore, the value of [tex]\( a^2 + b^2 + c^2 \)[/tex] is [tex]\( \boxed{75} \)[/tex].
