SAT Math Multiple Choice Question 373: Answer and Explanation

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Question: 373

13. Which of the following quadratic equations has no solution?

  • A. 0 = -2(x - 5)2 + 3
  • B. 0 = -2(x - 5)(x + 3)
  • C. 0 = 2(x - 5)2 + 3
  • D. 0 = 2(x + 5)(x + 3)

Correct Answer: C



Difficulty: Medium

Category: Passport to Advanced Math / Quadratics

Strategic Advice: Making connections between equations and their graphs will save valuable time on this question. 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. 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.

Getting to the Answer: The graph of an equation that has no solution does not cross the x-axis, so try to envision the graph of each of the answer choices. When a quadratic is written in factored form, the factors tell you the x-intercepts, which means every quadratic equation that can be written in factored form (over the set of real numbers) must have solutions. This means you can eliminate B and D. Now, imagine the graph of the equation in A: The vertex is (5, 3) and a is negative, so the parabola opens downward and consequently must cross the x-axis. This means you can eliminate A, and (C) must be correct. The graph of the equation in (C) has a vertex of (5, 3) and opens up, so it does not cross the x-axis and, therefore, has no solution.

You could also graph each of the answer choices in your graphing calculator, but this is not the most time-efficient way to answer the question.

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