Question - Solving a Quadratic Equation with Radicals
Solution:
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:\[ x = -2 - 2\sqrt{3} \]
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