PSAT Math Multiple-Choice Question 475: Answer and Explanation

Question: 475

Which of the following quadratics has only one real solution?

• A. 4x² = 3x - 8
• B. 10x = 2 - x²
• C. 7x² + 2x - 5 = 0
• D. 3x² - 6x + 3 = 0

Correct Answer: D

Explanation:

D

Difficulty: Hard

Category: Quadratic Formula

Strategic Advice: The discriminant is the part of the quadratic formula that determines whether a quadratic equation has 1 or 2 distinct real solutions or only imaginary solutions. Note that a quadratic will have only one distinct real solution when the discriminant equals 0.

Getting to the Answer: Convert the equations to standard quadratic form, if necessary, and calculate the discriminant for each choice to see which one equals 0:

(A) converts to 4x² - 3x + 8 = 0. The discriminant is (-3)² - 4(4)(8) = 9 - 128 = -119. This is not equal to 0, so eliminate (A).

(B) converts to x² + 10x - 2 = 0. The discriminant is (10)² - 4(1)(-2) = 100 + 8 = 108. Eliminate (B).

(C) is already in the proper form. The discriminant is (2)² - 4(7)(-5) = 4 + 140 = 144. Eliminate (C).

Only (D) is left, so it is correct. For the record, the discriminant for (D) does, in fact, equal 0. Divide through by the common factor of 3 to get x² - 2x + 1 = 0, so the discriminant is (-2)² -4(1)(1) = 0.