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

$2 x squared, minus 2, equals 2 x plus 3$

Which of the following is a solution to the equation above?

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Explanation

Choice D is correct. A quadratic equation in the form $a, x squared, plus b x, plus c, equals 0$ , where a, b, and c are constants, can be solved using the quadratic formula: $x equals, the fraction with numerator negative b plus or minus the square root of b squared, minus 4 a c, end root, and denominator 2 a, end fraction$ . Subtracting $2 x plus 3$ from both sides of the given equation yields $2 x squared, minus 2 x, minus 5, equals 0$ . Applying the quadratic formula, where $a, equals 2$ , $b equals negative 2$ , and $c equals negative 5$ , yields $x equals the fraction with numerator negative, open parenthesis, negative 2, close parenthesis, plus or minus the square root of, open parenthesis, negative 2, close parenthesis, squared, minus 4 times 2, times negative 5, end root, and denominator 2 times 2, end fraction$ . This can be rewritten as $x equals, the fraction with numerator 2 plus or minus the square root of 44, end root, and denominator 4$ . Since $the square root of 44 equals, the square root of 2 squared, times 11, end root$ , or $2 times the square root of 11$ , the equation can be rewritten as $x equals, the fraction with numerator 2 plus or minus 2 times the square root of 11, end root, and denominator 4$ . Dividing 2 from both the numerator and denominator yields $the fraction with numerator 1 plus the square root of 11, end root, and denominator 2$ or $the fraction with numerator 1 minus the square root of 11, end root, and denominator 2$ . Of these two solutions, only $the fraction with numerator 1 plus the square root of 11, end root, and denominator 2$ is present among the choices. Thus, the correct choice is D.

Choice A is incorrect and may result from a computational or conceptual error. Choice B is incorrect and may result from using $x equals, the fraction with numerator negative b plus or minus the square root of b squared, minus 4 a c, end root, and denominator a, end fraction$ instead of $x equals, the fraction with numerator negative b, plus or minus the square root of b squared, minus 4 a c, end root, and denominator 2 a, end fraction$ as the quadratic formula. Choice C is incorrect and may result from rewriting $the square root of 44$ as $4 times the square root of 11$ instead of $2 times the square root of 11$ .