Question - Solving a Quadratic Equation with the Quadratic Formula
Solution:
To solve the equation $$ x - (7/x) + 4 = 0 $$, we can multiply through by x to clear the fraction:\[ x^2 - 7 + 4x = 0 \]Rearrange the terms to get a quadratic equation in standard form:\[ x^2 + 4x - 7 = 0 \]This quadratic equation doesn't factor nicely, so we'll need to use the quadratic formula to find the solutions for x. The quadratic formula is given by:\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]For our equation, $$ a = 1 $$, $$ b = 4 $$, and $$ c = -7 $$, so we substitute these values into the formula:\[ x = \frac{{-4 \pm \sqrt{{4^2 - 4(1)(-7)}}}}{2(1)} \]\[ x = \frac{{-4 \pm \sqrt{{16 + 28}}}}{2} \]\[ x = \frac{{-4 \pm \sqrt{{44}}}}{2} \]\[ x = \frac{{-4 \pm 2\sqrt{{11}}}}{2} \]Now simplify by dividing all terms by 2:\[ x = -2 \pm \sqrt{{11}} \]Since the instructions say to list each solution only once and use a comma to separate the answers if needed, the complete and simplified answer is:\[ x = -2 + \sqrt{{11}}, -2 - \sqrt{{11}} \]
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