Question

If (x + ½)²=3. , then what is x³ +1/x³ ?

• A. 1
• B. √3
• C. 0
• D. 3√3

Answer 1

Answer:

C. 0

If $(x + \frac{1}{x})^2 = 3$, then the value of $x^3 + \frac{1}{x^3}$ is 0.

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If $(x + \frac{1}{x})^2 = 3$, then $x + \frac{1}{x} = \sqrt{3}$.
Whenever $x + \frac{1}{x} = \sqrt{3}$, the value of $x^3 + \frac{1}{x^3}$ is always 0 (and $x^6 = -1$).

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Step 1: Find the value of $x + \frac{1}{x}$
$(x + \frac{1}{x})^2 = 3$
Taking the square root on both sides:
$x + \frac{1}{x} = \sqrt{3}$

Step 2: Use the algebraic identity for cubes
We know the identity:
$x^3 + \frac{1}{x^3} = (x + \frac{1}{x})^3 - 3(x + \frac{1}{x})$

Step 3: Substitute the value
Substitute $(x + \frac{1}{x}) = \sqrt{3}$ into the identity:
$x^3 + \frac{1}{x^3} = (\sqrt{3})^3 - 3(\sqrt{3})$
$x^3 + \frac{1}{x^3} = 3\sqrt{3} - 3\sqrt{3}$
$x^3 + \frac{1}{x^3} = 0$