To solve this problem, let's proceed step by step as follows:
1. Given Information:
- [tex]\( a + b = 2 \)[/tex]
- [tex]\( a^2 + b^2 = 6 \)[/tex]
2. Objective:
- Show that [tex]\( ab = -1 \)[/tex]
- Find the value of [tex]\( (a - b)^2 \)[/tex]
3. Using the identity for the square of a sum:
We know that:
[tex]\[
(a + b)^2 = a^2 + b^2 + 2ab
\][/tex]
Substitute the given values:
[tex]\[
(2)^2 = 6 + 2ab
\][/tex]
Simplify the left-hand side:
[tex]\[
4 = 6 + 2ab
\][/tex]
4. Solve for [tex]\( ab \)[/tex]:
[tex]\[
4 = 6 + 2ab
\][/tex]
Subtract 6 from both sides:
[tex]\[
4 - 6 = 2ab
\][/tex]
Simplify:
[tex]\[
-2 = 2ab
\][/tex]
Divide both sides by 2:
[tex]\[
ab = -1
\][/tex]
Therefore, we have shown that [tex]\( ab = -1 \)[/tex].
5. Find the value of [tex]\( (a - b)^2 \)[/tex]:
Using the identity for the square of a difference:
[tex]\[
(a - b)^2 = a^2 + b^2 - 2ab
\][/tex]
Substitute the known values:
[tex]\[
(a - b)^2 = 6 - 2(-1)
\][/tex]
Simplify:
[tex]\[
(a - b)^2 = 6 + 2
\][/tex]
[tex]\[
(a - b)^2 = 8
\][/tex]
6. Final Answer:
- We have shown that [tex]\( ab = -1 \)[/tex].
- The value of [tex]\( (a - b)^2 \)[/tex] is [tex]\( 8 \)[/tex].
