Difference between revisions of "2023 AMC 12B Problems/Problem 14"

Revision as of 19:29, 15 November 2023

Denote three roots as $r_1 < r_2 < r_3$ . Following from Vieta's formula, $r_1r_2r_3 = -6$ .

Case 1: All roots are negative.

We have the following solution: $\left( -3, -2, -1 \right)$ .

Case 2: One root is negative and two roots are positive.

We have the following solutions: $\left( -3, 1, 2 \right)$ , $\left( -2, 1, 3 \right)$ , $\left( -1, 2, 3 \right)$ , $\left( -1, 1, 6 \right)$ .

Putting all cases together, the total number of solutions is $\boxed{\textbf{(A) 5}}$ .

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

2023 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

@@ Line 13: / Line 13: @@
 Putting all cases together, the total number of solutions is
-\boxed{\textbf{(A) 5}}.
+<math>\boxed{\textbf{(A) 5}}</math>.
 ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
+==See Also==
+{{AMC12 box|year=2023|ab=B|num-b=13|num-a=15}}
+{{MAA Notice}}