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If [tex]$a + b = 2$[/tex] and [tex]$a^2 + b^2 = 6$[/tex], show that [tex][tex]$ab = -1$[/tex][/tex], and find the value of [tex]$(a - b)^2[/tex].



Answer :

GinnyAnswer
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].

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