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If [tex]\( x = \frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}+\sqrt{2}} \)[/tex] and [tex]\( y = \frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}} \)[/tex], find [tex]\( x^2 + x + y + y^2 \)[/tex].

Sagot :

GinnyAnswer
Let's consider the expressions given in the problem:

[tex]\[ x = \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}} \][/tex]
[tex]\[ y = \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}} \][/tex]

We need to find the value of [tex]\( x^2 + x + y + y^2 \)[/tex].

First, let's find [tex]\( x^2 \)[/tex] and [tex]\( y^2 \)[/tex].

### Step 1: Calculating [tex]\( x^2 \)[/tex]

[tex]\[ x = \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}} \][/tex]

[tex]\[ x^2 = \left(\frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}}\right)^2 \][/tex]

[tex]\[ x^2 = \frac{(\sqrt{5} - \sqrt{2})^2}{(\sqrt{5} + \sqrt{2})^2} \][/tex]

### Step 2: Calculating [tex]\( y^2 \)[/tex]

[tex]\[ y = \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}} \][/tex]

[tex]\[ y^2 = \left(\frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}}\right)^2 \][/tex]

[tex]\[ y^2 = \frac{(\sqrt{5} + \sqrt{2})^2}{(\sqrt{5} - \sqrt{2})^2} \][/tex]

### Step 3: Calculating [tex]\( x + y \)[/tex]

[tex]\[ x = \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}} \][/tex]
[tex]\[ y = \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}} \][/tex]

[tex]\[ x + y = \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}} + \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}} \][/tex]

### Step 4: Summing Up [tex]\( x^2 + x + y + y^2 \)[/tex]

Now let's add them together to find the required expression.

[tex]\[ x^2 + x + y + y^2 = \frac{(\sqrt{5} - \sqrt{2})^2}{(\sqrt{5} + \sqrt{2})^2} + \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}} + \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}} + \frac{(\sqrt{5} + \sqrt{2})^2}{(\sqrt{5} - \sqrt{2})^2} \][/tex]

So, we have:

[tex]\[ x^2 = \frac{(\sqrt{2} - \sqrt{5})^2}{(\sqrt{2} + \sqrt{5})^2} \][/tex]
[tex]\[ y^2 = \frac{(\sqrt{2} + \sqrt{5})^2}{(\sqrt{2} - \sqrt{5})^2} \][/tex]

And:

[tex]\[ x + y = \frac{(\sqrt{2} - \sqrt{5})}{(\sqrt{2} + \sqrt{5})} + \frac{(\sqrt{2} + \sqrt{5})}{(\sqrt{2} - \sqrt{5})} \][/tex]

Thus, the desired sum is:

[tex]\[ x^2 + x + y + y^2 = \frac{(\sqrt{2} - \sqrt{5})^2}{(\sqrt{2} + \sqrt{5})^2} + \frac{\sqrt{2} - \sqrt{5}}{\(\sqrt{2} + \sqrt{5}\)} + \frac{\sqrt{2} + \sqrt{5}}{\(\sqrt{2} - \sqrt{5}\)} + \frac{(\sqrt{2} + \sqrt{5})^2}{(\sqrt{2} - \sqrt{5})^2} \][/tex]

Which evaluates to:

[tex]\[ (-\sqrt{2} + \sqrt{5})^2/(\sqrt{2} + \sqrt{5})^2 + (-\sqrt{2} + \sqrt{5})/(\sqrt{2} + \sqrt{5}) + (\sqrt{2} + \sqrt{5})/(-\sqrt{2} + \sqrt{5}) + (\sqrt{2} + \sqrt{5})^2/(-\sqrt{2} + \sqrt{5})^2 \][/tex]

This is the required result.
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