Difference between revisions of "2002 AMC 12A Problems/Problem 1"
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| + | {{duplicate|[[2002 AMC 12A Problems|2002 AMC 12A #1]] and [[2002 AMC 10A Problems|2002 AMC 10A #10]]}} | ||
| + | |||
== Problem == | == Problem == | ||
| + | Compute the sum of all the roots of | ||
| + | <math>(2x+3)(x-4)+(2x+3)(x-6)=0 </math> | ||
| + | |||
| + | <math> \textbf{(A) } \frac{7}{2}\qquad \textbf{(B) } 4\qquad \textbf{(C) } 5\qquad \textbf{(D) } 7\qquad \textbf{(E) } 13 </math> | ||
| − | == Solution == | + | ==Solution 1== |
| − | {{ | + | We expand to get <math>2x^2-8x+3x-12+2x^2-12x+3x-18=0</math> which is <math>4x^2-14x-30=0</math> after combining like terms. Using the quadratic part of [[Vieta's Formulas]], we find the sum of the roots is <math>\frac{14}4 = \boxed{\textbf{(A) }7/2}</math>. |
== See also == | == See also == | ||
{{AMC12 box|year=2002|ab=A|before=First Question|num-a=2}} | {{AMC12 box|year=2002|ab=A|before=First Question|num-a=2}} | ||
| + | {{AMC10 box|year=2002|ab=A|num-b=9|num-a=11}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 15:44, 25 July 2024
- The following problem is from both the 2002 AMC 12A #1 and 2002 AMC 10A #10, so both problems redirect to this page.
Problem
Compute the sum of all the roots of
Solution 1
We expand to get 
which is 
after combining like terms. Using the quadratic part of Vieta's Formulas, we find the sum of the roots is 
.
See also
| 2002 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by First Question |
Followed by Problem 2 |
| 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 | |
| 2002 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 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 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
