Answer:
Given the equations:
[tex] \sf a - b = 4 [/tex]
[tex] \sf a^2 - b^2 = 8 [/tex]
First, recognize that [tex] \sf a^2 - b^2 [/tex] can be factored as:
[tex] \sf a^2 - b^2 = (a - b)(a + b) [/tex]
Substitute the given value [tex] \sf a - b = 4 [/tex]:
[tex] \sf 8 = 4(a + b) [/tex]
Solve for [tex] \sf a + b [/tex]:
[tex] \sf a + b = \frac{8}{4} = 2 [/tex]
Now, we have two equations:
[tex] \sf a - b = 4 [/tex]
[tex] \sf a + b = 2 [/tex]
Add the two equations to find [tex] \sf a [/tex]:
[tex] \sf (a - b) + (a + b) = 4 + 2 [/tex]
[tex] \sf 2a = 6 [/tex]
[tex] \sf a = 3 [/tex]
Substitute [tex] \sf a = 3 [/tex] back into [tex] \sf a + b = 2 [/tex] to find [tex] \sf b [/tex]:
[tex] \sf 3 + b = 2 [/tex]
[tex] \sf b = 2 - 3 [/tex]
[tex] \sf b = -1 [/tex]
Now, calculate [tex] \sf ab [/tex]:
[tex] \sf ab = 3 \cdot (-1) = -3 [/tex]
So, the correct value of [tex] \sf ab [/tex] is:
[tex] \sf \boxed{-3} [/tex]
From our earlier solution, we found:
[tex] \sf a + b = 2 [/tex]
Therefore, the possible value of [tex] \sf a + b [/tex] is:
[tex] \sf \boxed{2} [/tex]
We use the system of equations:
[tex] \sf a + b = 2 [/tex]
[tex] \sf a - b = 4 [/tex]
Solve these equations simultaneously. Add the two equations:
[tex] \sf (a + b) + (a - b) = 2 + 4 [/tex]
[tex] \sf 2a = 6 [/tex]
[tex] \sf a = 3 [/tex]
Now, substitute [tex] \sf a = 3 [/tex] back into [tex] \sf a + b = 2 [/tex]:
[tex] \sf 3 + b = 2 [/tex]
[tex] \sf b = 2 - 3 [/tex]
[tex] \sf b = -1 [/tex]
So, the possible values of [tex] \sf a [/tex] and [tex] \sf b [/tex] are:
[tex] \sf \boxed{a = 3 \text{ and } b = -1} [/tex]