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

$2 x squared, minus 4 x, equals t$

In the equation above, t is a constant. If the equation has no real solutions, which of the following could be the value of t ?

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Explanation

Choice A is correct. The number of solutions to any quadratic equation in the form $a, x squared, plus b x, plus c, equals 0$ , where a, b, and c are constants, can be found by evaluating the expression $b squared, minus 4 a, c$ , which is called the discriminant. If the value of $b squared, minus 4 a, c$ is a positive number, then there will be exactly two real solutions to the equation. If the value of $b squared, minus 4 a, c$ is zero, then there will be exactly one real solution to the equation. Finally, if the value of $b squared, minus 4 a, c$ is negative, then there will be no real solutions to the equation.

The given equation $2 x squared, minus 4 x, equals t$ is a quadratic equation in one variable, where t is a constant. Subtracting t from both sides of the equation gives $2 x squared, minus 4 x, minus t, equals 0$ . In this form, $a, equals 2$ , $b equals negative 4$ , and $c equals negative t$ . The values of t for which the equation has no real solutions are the same values of t for which the discriminant of this equation is a negative value. The discriminant is equal to $open parenthesis, negative 4, close parenthesis, squared, minus 4 times 2, times negative t$ ; therefore, $open parenthesis, negative 4, close parenthesis, squared, minus, 4 times 2, times negative t, is less than 0$ . Simplifying the left side of the inequality gives $16 plus 8, t, is less than 0$ . Subtracting 16 from both sides of the inequality and then dividing both sides by 8 gives $t is less than negative 2$ . Of the values given in the options, $negative 3$ is the only value that is less than $negative 2$ . Therefore, choice A must be the correct answer.

Choices B, C, and D are incorrect and may result from a misconception about how to use the discriminant to determine the number of solutions of a quadratic equation in one variable.