To solve the problem of finding the value of [tex]\( ab + bc + ca \)[/tex] given the equations [tex]\( a^2 + b^2 + c^2 = 90 \)[/tex] and [tex]\( a + b + c = 20 \)[/tex], we can utilize a well-known algebraic identity. Here's a step-by-step solution:
1. Start with the identity:
[tex]\[
(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)
\][/tex]
This identity expresses the square of the sum of three terms in terms of the squares of the individual terms and their products.
2. Insert the given values into the identity:
- We know that [tex]\( a + b + c = 20 \)[/tex].
- We know that [tex]\( a^2 + b^2 + c^2 = 90 \)[/tex].
Substitute these values into the identity:
[tex]\[
(20)^2 = 90 + 2(ab + bc + ca)
\][/tex]
3. Calculate the square of the sum of the values:
[tex]\[
20^2 = 400
\][/tex]
So the identity now looks like:
[tex]\[
400 = 90 + 2(ab + bc + ca)
\][/tex]
4. Isolate [tex]\( 2(ab + bc + ca) \)[/tex]:
Subtract 90 from both sides of the equation:
[tex]\[
400 - 90 = 2(ab + bc + ca)
\][/tex]
Simplify the left side:
[tex]\[
310 = 2(ab + bc + ca)
\][/tex]
5. Solve for [tex]\( ab + bc + ca \)[/tex]:
Divide both sides of the equation by 2:
[tex]\[
ab + bc + ca = \frac{310}{2}
\][/tex]
Simplify the division:
[tex]\[
ab + bc + ca = 155
\][/tex]
Therefore, the value of [tex]\( ab + bc + ca \)[/tex] is:
[tex]\[
\boxed{155}
\][/tex]
Looking at the available options, the correct answer is:
(b) 155
