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

Question

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 ?

Accepted Answer

Choice A is correct. The number of solutions to any quadratic equation in the form , where a , b , and c are constants, can be found by evaluating the expression , which is called the discriminant. If the value of is a positive number, then there will be exactly two real solutions to the equation. If the value of is zero, then there will be exactly one real solution to the equation. Finally, if the value of is negative, then there will be no real solutions to the equation. The given equation is a quadratic equation in one variable, where t is a constant. Subtracting t from both sides of the equation gives . In this form, , , and . 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 ; therefore, . Simplifying the left side of the inequality gives . Subtracting 16 from both sides of the inequality and then dividing both sides by 8 gives . Of the values given in the options, is the only value that is less than . 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.

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