PSAT Math Multiple-Choice Question 217: Answer and Explanation

Question: 217

If the graph of the equation y= ax²+ bx+ c passes through the points (0, 2), (-6, -7), and (8, -14), what is the value of a+ b+ c ?

• A. - 19
• B. - 2
• C. 1.75
• D. 2.25

Correct Answer: C

Explanation:

C

Difficulty: Hard

Category: Passport to Advanced Math/Quadratics

Getting to the Answer: Writing quadratic equations can be tricky and time-consuming. If you know the roots, you can use factors to write the equation. If you don't know the roots, you need to create a system of equations to find the coefficients of the variable terms. You don't know the roots of this equation, so start with the point that has the nicest values (0, 2) and substitute them into the equation, y = ax² + bx + c, to get 2 = a(0)² + b(0) + c, or 2 = c. Now your equation looks like y = ax² + bx + 2. Next, use the other two points to create a system of two equations in two variables:

(-6, -7) → -7 = a(-6)² + b(-6) + 2 → -9 = 36 a - 6 b

(8, -14) → -14 = a(8)² + b(8) + 2 → -16 = 64 a + 8 b

You now have a system of equations to solve. If you multiply the top equation by 4 and the bottom equation by 3, and then add the equations, the b terms will eliminate each other:

Now, find b by substituting a = -0.25 into either of the original equations. Using the top equation, you get:

The value of a + b + c is (-0.25) + 0 + 2 = 1.75, so (C) is correct.