If 𝑥 = √ 5+2√6 , then determine the value of x² + 3x - 15? 5−2√6

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Accepted Answer

1. Note $x=\sqrt{5+2\sqrt6}=\sqrt2+\sqrt3$, because $(\sqrt2+\sqrt3)^2=5+2\sqrt6$.

2. Compute the numerator: $x^2+3x-15=(5+2\sqrt6)+3(\sqrt2+\sqrt3)-15=-10+2\sqrt6+3\sqrt2+3\sqrt3$.

3. Divide by $5-2\sqrt6$ by multiplying top and bottom by $5+2\sqrt6$. After simplifying the product we get $49+26\sqrt6$.

4. So the value matches option (c).

FINAL_ANSWER: c

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