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Difference between revisions of "1970 AHSME Problems/Problem 14"

(Created page with "== Problem == Consider <math>x^2+px+q=0</math>, where <math>p</math> and <math>q</math> are positive numbers. If the roots of this equation differ by 1, then <math>p</math> equa...")
 
 
(5 intermediate revisions by 3 users not shown)
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<math>\text{(A) } \sqrt{4q+1}\quad
 
<math>\text{(A) } \sqrt{4q+1}\quad
−
\text{(B) } \q-1\quad
+
\text{(B) } q-1\quad
 
\text{(C) } -\sqrt{4q+1}\quad
 
\text{(C) } -\sqrt{4q+1}\quad
 
\text{(D) } q+1\quad
 
\text{(D) } q+1\quad
Line 10: Line 10:
  
 
== Solution ==
 
== Solution ==
−
<math>\fbox{A}</math>
+
From the quadratic equation, the two roots of the equation are <math>\frac{-p\pm\sqrt{p^2-4q}}{2}</math>. The positive difference between these roots is <math>\sqrt{p^2 - 4q}</math>. Setting <math>\sqrt{p^2-4q}=1</math> and isolating <math>p</math> gives <math>\sqrt{4q+1}</math>, or choice <math>\boxed{\text{(A)}}</math>.
  
 
== See also ==
 
== See also ==
−
{{AHSME box|year=1970|num-b=3|num-a=14}}   
+
{{AHSME 35p box|year=1970|num-b=13|num-a=15}}   
  
 
[[Category: Introductory Algebra Problems]]
 
[[Category: Introductory Algebra Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 00:30, 24 February 2023

Problem

Consider $x^2+px+q=0$, where $p$ and $q$ are positive numbers. If the roots of this equation differ by 1, then $p$ equals

$\text{(A) } \sqrt{4q+1}\quad \text{(B) } q-1\quad \text{(C) } -\sqrt{4q+1}\quad \text{(D) } q+1\quad \text{(E) } \sqrt{4q-1}$

Solution

From the quadratic equation, the two roots of the equation are $\frac{-p\pm\sqrt{p^2-4q}}{2}$. The positive difference between these roots is $\sqrt{p^2 - 4q}$. Setting $\sqrt{p^2-4q}=1$ and isolating $p$ gives $\sqrt{4q+1}$, or choice $\boxed{\text{(A)}}$.

See also

1970 AHSC (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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35
All AHSME Problems and Solutions

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