Solution:
The number of solutions to any quadratic equation in the form ax2+bx+c=0, where a, b, and c are constants, can be found by evaluating the expression b2−4ac, which is called the discriminant. If the value of b2−4ac is a positive number, then there will be exactly two real solutions to the equation. If the value of b2−4ac is zero, then there will be exactly onereal solution to the equation. Finally, if the value of b2−4ac is negative, then there will be noreal solutions to the equation.
The given equation 2x2−4x=t is a quadratic equation in one variable, where t is a constant. Subtracting t from both sides of the equation gives 2x2−4x−t=0. In this form, a=2,b=−4, and c=−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 (−4)2−4(2)(−t); therefore, (−4)2−4(2)(−t)<0. Simplifying the left side of the inequality gives 16+8t<0. Subtracting 16 from both sides of the inequality and then dividing both sides by 8 gives t<−2. Of the values given in the options, −3 is the only value that is less than −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.
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