Find the values of a^2 + b^2 and a^4 + b^4 when a + b = 5 and Ab = 4.
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To find the values of a^2 + b^2 and a^4 + b^4 when a + b = 5 and ab = 4, we can use the given equations to solve for the values of a and b, and then substitute those values into the expressions for a^2 + b^2 and a^4 + b^4.
Let's start by solving the system of equations a + b = 5 and ab = 4 to find the values of a and b.
We can use the equation a + b = 5 to solve for b in terms of a. Rearranging the equation, we have b = 5 - a.
Substituting this value of b into the equation ab = 4, we get a(5 - a) = 4.
Expanding the equation, we have 5a - a^2 = 4.
Rearranging the equation, we get a^2 - 5a + 4 = 0.
Now, we can factor the quadratic equation (a^2 - 5a + 4) to solve for a.
The equation factors as (a - 4)(a - 1) = 0.
So, we have two possible solutions: a - 4 = 0 or a - 1 = 0.
Solving for a in each case, we get a = 4 or a = 1.
Now that we have the values of a, we can substitute them back into the equation a + b = 5 to find the values of b.
When a = 4, substituting into a + b = 5, we get 4 + b = 5.
Solving for b, we get b = 5 - 4 = 1.
Similarly, when a = 1, substituting into a + b = 5, we get 1 + b = 5.
Solving for b, we get b = 5 - 1 = 4.
So, we have two sets of values: a = 4 and b = 1, or a = 1 and b = 4.
Now that we have the values of a and b, we can substitute them into the expressions a^2 + b^2 and a^4 + b^4 to find their respective values.
When a = 4 and b = 1, we have:
a^2 + b^2 = 4^2 + 1^2 = 16 + 1 = 17.
a^4 + b^4 = 4^4 + 1^4 = 256 + 1 = 257.
When a = 1 and b = 4, we have:
a^2 + b^2 = 1^2 + 4^2 = 1 + 16 = 17.
a^4 + b^4 = 1^4 + 4^4 = 1 + 256 = 257.
Therefore, the values of a^2 + b^2 and a^4 + b^4 when a + b = 5 and ab = 4 are both equal to 17 and 257, respectively.