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Advanced Math / Nonlinear equations in one variable and systems of equations in two variables Difficulty: Hard

$2 x^{2} - 8 x - 7 = 0$

One solution to the given equation can be written as $\frac{8 - \sqrt{k}}{4}$ , where $k$ is a constant. What is the value of $k$ ?

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Explanation

The correct answer is $120$ . The solutions to a quadratic equation of the form $a x^{2} + b x + c = 0$ can be calculated using the quadratic formula and are given by $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ . The given equation is in the form $a x^{2} + b x + c = 0$ , where $a equals 2$ , $b = - 8$ , and $c = - 7$ . It follows that the solutions to the given equation are $x = \frac{8 \pm \sqrt{{(- 8)}^{2} - 4 (2) (- 7)}}{2 (2)}$ , which is equivalent to $x = \frac{8 \pm \sqrt{64 + 56}}{4}$ , or $x = \frac{8 \pm \sqrt{120}}{4}$ . It's given that one solution to the equation $2 x^{2} - 8 x - 7 = 0$ can be written as $\frac{8 - \sqrt{k}}{4}$ . The solution $\frac{8 - \sqrt{120}}{4}$ is in this form. Therefore, the value of $k$ is $120$ .