Get the answer to "Given x+y=2 and x3+y3=5, what is x2+y2?"

Lorelai Woods

Answered question

2022-02-04

Given $x + y = 2$ and $x^{3} + y^{3} = 5$ , what is $x^{2} + y^{2} ?$

Madison Randolph

Beginner2022-02-05Added 20 answers

Step 1
From

x + y = 2

we have that

x + y = 2 \Rightarrow {(x + y)}^{2} = 2^{2} \Rightarrow x^{2} + y^{2} + 2 x y = 4

From

x^{3} + y^{3} = 5

we have that

x^{3} + y^{3} = 5 \Rightarrow (x + y) \times (x^{2} + y^{2} - x y) = 5 \Rightarrow x^{2} + y^{2} - x y = \frac{5}{2}

So we know that

x^{2} + y^{2} + 2 x y = 4

(1)
and

x^{2} + y^{2} - x y = \frac{5}{2} \Rightarrow 2 \times (x^{2} + y^{2}) - 2 x y = 5

(2)
If you add (1) and (2) you get

3 \times (x^{2} + y^{2}) = 9 \Rightarrow x^{2} + y^{2} = 3

Madison Randolph

Beginner2022-02-06Added 20 answers

Step 1
By applying the sum of cubes formula together with the given equations:

5 = x^{3} + y^{3} = (x + y) (x^{2} - x y + y^{2}) = 2 (x^{2} + y^{2} - x y)

\Rightarrow x^{2} + y^{2} - x y = \frac{5}{2}

\Rightarrow x^{2} + y^{2} = \frac{5}{2} + x y

(1)
Now, by cubing the first given equation, we get

{(x + y)}^{3} = 2^{3}

\Rightarrow x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3} = 8

\Rightarrow (x^{3} + y^{3}) + 3 x y (x + y) = 8

Substituting in our known values for

x + y

and

x^{3} + y^{3}

, we obtain

\Rightarrow 5 + 3 x y \times 2 = 8

\Rightarrow 6 x y = 3

\Rightarrow x y = \frac{1}{2}

Substituting this back into (1)

x^{2} + y^{2} = \frac{5}{2} + \frac{1}{2}

∴ x^{2} + y^{2} = 3

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