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Let a and b be real numbers. For this problem, assume that a - b = 4 and a^2 - b^2 = 8.

(a) Find all possible values of ab
(b) Find all possible values of a+b
(c) Find all possible values of a and b

Sagot :

chibabykalu16

Answer:

Given the equations:

[tex] \sf a - b = 4 [/tex]

[tex] \sf a^2 - b^2 = 8 [/tex]

Part (a): Find all possible values of [tex] \sf ab [/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]

Part (b): Find all possible values of [tex] \sf a + b [/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]

Part (c): Find all possible values of [tex] \sf a [/tex] and [tex] \sf b [/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]

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