Solving a Quadratic Equation with Radicals
The given equation is:
\[ x - \sqrt{3}x - 4 = 0 \]
To solve this equation for \( x \), first combine like terms and isolate the \( x \)-terms on one side:
\[ x - \sqrt{3}x = 4 \]
Combine the \( x \) terms together by factoring out \( x \):
\[ x(1 - \sqrt{3}) = 4 \]
Next, divide both sides of the equation by \( (1 - \sqrt{3}) \) to solve for \( x \):
\[ x = \frac{4}{1 - \sqrt{3}} \]
To rationalize the denominator, multiply the numerator and the denominator by the conjugate of the denominator:
\[ x = \frac{4}{1 - \sqrt{3}} \cdot \frac{1 + \sqrt{3}}{1 + \sqrt{3}} \]
\[ x = \frac{4 \cdot (1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} \]
\[ x = \frac{4 + 4\sqrt{3}}{1 - 3} \]
\[ x = \frac{4 + 4\sqrt{3}}{-2} \]
Divide both terms in the numerator by the denominator:
\[ x = \frac{4}{-2} + \frac{4\sqrt{3}}{-2} \]
\[ x = -2 - 2\sqrt{3} \]
So, the solution to the equation is:
\[ x = -2 - 2\sqrt{3} \]
