Math: 7bd10ef3

ID: 7bd10ef3

(SAT Suite Question Bank > Find Questions > Assessment: SAT + Test: Math + Domain: Advanced Math)

Comment: Focus on the discriminant.

Method 1: No real solutions means that the quadratic will have a negative value under the square root, known as the discriminant, within the quadratic formula. But first, set the equation to 0.

2x^2 - 4x - t = 0

b^2 - 4ac < 0

(-4)^2 - 4(2)(-t) < 0

16 + 8t < 0

8t < -16

t < -2

If t is less than -2, then the quadratic formula would produce no real solutions, so the correct answer must be (A), -3, the only value less than -2.

Method 2: Use Desmos and add a slider for t to test the answer choices.

When t = -3, the graph is above the x-axis, so there will be no real solutions, and (A) is the correct answer. Sliding the value to -1, 1, or 3 will reveal two solutions for each.

Another easy way to get the correct answer using a graphing calculator is to substitute each answer choice for x and create four parabolas. Only one will show that there are no real solutions.

The first graph, with a y-intercept at 3, must be the correct answer.

Method 3: For practice, you could complete the square and test the answer choices.

x^2 - 2x = t/2

x^2 - 2x + (-2/2)^2 = (t/2) + (-2/2)^2

(x - 1)^2 = (t/2) + 1

(x - 1)^2 = (t + 2)/2

To be honest, that is as far as you would need to take the algebraic solution. Any real number squared will be 0 at a minimum or positive, so to ensure that the equation would yield no real solutions, the numerator on the right-hand side would have to be negative.

t + 2 < 0

t < -2

Once again, among the answer choices in the lineup presented, the correct answer is (A).