AMC10 2011 B

AMC10 2011 B · Q19

AMC10 2011 B · Q19. It mainly tests Absolute value, Exponents & radicals.

What is the product of all the roots of the equation $\sqrt{5|x| + 8} = \sqrt{x^2 −16}$.

方程 $\sqrt{5|x| + 8} = \sqrt{x^2 −16}$ 的所有根的乘积是多少。

(A) −64 −64

(B) −24 −24

(D) 24 24

(E) 576 576

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): The right side of the equation is defined only when |x| ≥ 4. If x ≥ 4, the equation is equivalent to 5x + 8 = x² − 16, and the only solution with x ≥ 4 is x = 8. If x ≤ −4, the equation is equivalent to 8 − 5x = x² − 16, and the only solution with x ≤ −4 is x = −8. The product of the solutions is −8 · 8 = −64.

答案 (A)：方程右边定义域为 |x| ≥ 4。若 x ≥ 4，方程等价于 5x + 8 = x² − 16，且唯一满足 x ≥ 4 的解为 x = 8。若 x ≤ −4，方程等价于 8 − 5x = x² − 16，且唯一满足 x ≤ −4 的解为 x = −8。解的乘积为 −8 · 8 = −64。

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