To solve for [tex]\( y \)[/tex] in the equation [tex]\( y = (a + b)^2 \)[/tex] given [tex]\( ab = 10 \)[/tex] and [tex]\( a^2 + b^2 = 30 \)[/tex], let's go through the problem step-by-step.
1. Start with the given:
- [tex]\( ab = 10 \)[/tex]
- [tex]\( a^2 + b^2 = 30 \)[/tex]
2. Recall the algebraic identity for the square of a sum:
[tex]\[ (a + b)^2 = a^2 + 2ab + b^2 \][/tex]
3. Substitute the given values into the identity:
[tex]\[ (a + b)^2 = a^2 + b^2 + 2ab \][/tex]
[tex]\[ (a + b)^2 = 30 + 2 \times 10 \][/tex]
4. Perform the multiplication:
[tex]\[ 2 \times 10 = 20 \][/tex]
5. Add the results to find [tex]\((a + b)^2\)[/tex]:
[tex]\[ (a + b)^2 = 30 + 20 = 50 \][/tex]
Therefore, the value of [tex]\( y \)[/tex] in the equation [tex]\( y = (a + b)^2 \)[/tex] is [tex]\( 50 \)[/tex]. The correct answer is:
[tex]\[ \boxed{50} \][/tex]
