PSAT Math Multiple-Choice Question 210: Answer and Explanation

Question: 210

Which of the following quadratic equations has no real solution?

• A. 0 = - 3(x + 1)(x - 8)
• B. 0 = 3(x + 1)(x - 8)
• C. 0 = - 3(x + 1)² + 8
• D. 0 = 3(x + 1)² + 8

Correct Answer: D

Explanation:

D

Difficulty: Medium

Category: Passport to Advanced Math/Quadratics

Getting to the Answer: The graph of every quadratic equation is a parabola, which may or may not cross the x-axis, depending on where its vertex is and which way it opens. When an equation has no solution, its graph does not cross the x-axis, so try to envision the graph of each of the answer choices (or you could graph each one in your graphing calculator, but this will probably take longer). Don't forget—if the equation is written in vertex form, y = a(x - h)² + k, then the vertex is (h, k) and the value of a tells you which way the parabola opens. When a quadratic equation is written in factored form, the factors tell you the x-intercepts, which means (A) and (B) (which are factored) must cross the x-axis, so eliminate them. Now, imagine the graph of the equation in (C): The vertex is (-1, 8) and a is negative, so the parabola opens downward and consequently must cross the x-axis. This means (D) must be correct. The vertex is also (-1, 8), but a is positive, so the graph opens up and does not cross the x-axis.