Algebra Foundation Quiz #2 Q9

GRE Quant Problem Solving
1

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Can someone help me understand how to get to -4? When solving for the roots, it does not result in whole numbers. When I apply the quadratic formula, it results in a rather time consuming and difficult product of (-5 +/- root (41)) / 2).

Is there a simpler way to solve it that I am missing?

Thank you in advance for the help!

2

The product of the roots will be of the form:
(a+b) * (a-b) = a² - b², where a = -5/2 and b = √41/2
This gives (25/4) - (41/4) = -16/4 = -4

Additionally, if you want to remember a shortcut, if we are given an equation of the form ax2 + bx + c = 0,

  1. The sum of roots = -b/a
  2. The product of roots = c/a

Here, a=1, c = -4 and c/a gives -4

3

The roots would be

\frac{-5 + \sqrt{41}}{2}

and

\frac{-5 - \sqrt{41}}{2}

Finding the product isn’t all that hard manually:

\frac{-5 + \sqrt{41}}{2} \times \frac{-5 - \sqrt{41}}{2}
=\frac{1}{4} (-5 + \sqrt{41})(-5 - \sqrt{41})
=\frac{1}{4}(41 - 25)
=4

Note that knowledge of Vieta’s formulas (as @vision_35 said with his shortcuts) is not required for the GRE.

4

Thank you both so much!