Difference between revisions of "2006 AMC 10B Problems/Problem 14"

Revision as of 14:17, 26 January 2022

Problem

Let $a$ and $b$ be the roots of the equation $x^2-mx+2=0$ . Suppose that $a+\frac1b$ and $b+\frac1a$ are the roots of the equation $x^2-px+q=0$ . What is $q$ ?

$\textbf{(A) } \frac{5}{2}\qquad \textbf{(B) } \frac{7}{2}\qquad \textbf{(C) } 4\qquad \textbf{(D) } \frac{9}{2}\qquad \textbf{(E) } 8$

Video Solution

https://youtu.be/3dfbWzOfJAI?t=457

~ pi_is_3.14

Solution

In a quadratic equation of the form $x^2 + bx + c = 0$ , the product of the roots is $c$ (Vieta's Formulas).

Using this property, we have that $ab=2$ and

$q = (a+\frac{1}{b})\cdot(b+\frac{1}{a}) = \frac{ab+1}{b} \cdot \frac{ab+1}{a} = \frac{(ab+1)^2}{ab} = \frac{(2+1)^2}{2} = \boxed{\textbf{(D) }\frac{9}{2}}$ .

Notice the fact that we never actually found the roots.

Solution 2

Assume without loss of generality that $m=3$ . We can factor the equation $x^2-3x+2=0$ into $(x-1)(x-2)=0$ . Therefore, $a=1$ and $b=2$ . Using these values, we find $a+\frac1b=\frac32$ and $b+\frac1a=3$ . By Vieta's formulas, $q$ is the product of the roots of $x^2-px+q=0$ , which are $a+\frac1b$ and $b+\frac1a$ . Therefore, $q=\left(a+\frac1b\right)\left(b+\frac1a\right)=\frac32\cdot3=\boxed{\textbf{(D) }\frac92}$ .

2006 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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Vieta's formulas

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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 == Solution 2 ==
-Assume [[without loss of generality]] that <math>m=3</math>. We can factor the equation <math>x^2-3x+2=0</math> into <math>(x-1)(x-2)=0</math>. Therefore, <math>a=1</math> and <math>b=2</math>. Using these values, we find <math>a+\frac1b=\frac32</math> and <math>b+\frac1a=3</math>. By [[Vieta's formulas]], <math>q</math> is the product of the roots of <math>x^2-px+q=0</math>, which are <math>a+\frac1b</math> and <math>b+\frac1a</math>. Therefore, <math>q=\left(a+\frac1b\right)\left(b+\frac1a\right)=\frac32\cdot3=\frac92\Rightarrow \boxed{D}</math>
+Assume [[without loss of generality]] that <math>m=3</math>. We can factor the equation <math>x^2-3x+2=0</math> into <math>(x-1)(x-2)=0</math>. Therefore, <math>a=1</math> and <math>b=2</math>. Using these values, we find <math>a+\frac1b=\frac32</math> and <math>b+\frac1a=3</math>. By [[Vieta's formulas]], <math>q</math> is the product of the roots of <math>x^2-px+q=0</math>, which are <math>a+\frac1b</math> and <math>b+\frac1a</math>. Therefore, <math>q=\left(a+\frac1b\right)\left(b+\frac1a\right)=\frac32\cdot3=\boxed{\textbf{(D) }\frac92}</math>.
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