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Difference between revisions of "2021 JMPSC Accuracy Problems/Problem 4"

(Created page with "==Problem== If <math>\frac{x+2}{6}</math> is its own reciprocal, find the product of all possible values of <math>x.</math> ==Solution==")
 
 
(11 intermediate revisions by 5 users not shown)
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==Solution==
 
==Solution==
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From the problem, we know that
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<cmath>\frac{x+2}{6} = \frac{6}{x+2}</cmath>
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<cmath>(x+2)^2 = 6^2</cmath>
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<cmath>x^2+ 4x + 4 = 36</cmath>
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<cmath>x^2 + 4x - 32 = 0</cmath>
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<cmath>(x+8)(x-4) = 0</cmath>
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Thus, <math>x = -8</math> or <math>x = 4</math>. Our answer is <math>(-8) \cdot 4=\boxed{-32}</math>
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~Bradygho
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==Solution 2==
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We have <math>\frac{x+2}{6} = \frac{6}{x+2}</math>, so <math>x^2+4x-32=0</math>. By Vieta's our roots <math>a</math> and <math>b</math> amount to <math>\frac{-32}{1}=\boxed{-32}</math>
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~Geometry285
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==Solution 3==
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<math>\frac{x+2}{6}=\frac{6}{x+2} \implies x^2+4x-32</math> Therefore, the product of the root is <math>-32</math>
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~kante314
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==Solution 4==
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The only numbers that are their own reciprocals are <math>1</math> and <math>-1</math>. The equation <math>\frac{x+2}{6}=1</math> has the solution <math>x=4</math>, while the equation <math>\frac{x+2}{6}=-1</math> has the solution <math>x=-8</math>. The answer is <math>4 \cdot (-8)=\boxed{-32}</math>.
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~tigerzhang
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==See also==
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#[[2021 JMPSC Accuracy Problems|Other 2021 JMPSC Accuracy Problems]]
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#[[2021 JMPSC Accuracy Answer Key|2021 JMPSC Accuracy Answer Key]]
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#[[JMPSC Problems and Solutions|All JMPSC Problems and Solutions]]
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{{JMPSC Notice}}

Latest revision as of 16:23, 11 July 2021

Problem

If $\frac{x+2}{6}$ is its own reciprocal, find the product of all possible values of $x.$

Solution

From the problem, we know that \[\frac{x+2}{6} = \frac{6}{x+2}\] \[(x+2)^2 = 6^2\] \[x^2+ 4x + 4 = 36\] \[x^2 + 4x - 32 = 0\] \[(x+8)(x-4) = 0\]

Thus, $x = -8$ or $x = 4$. Our answer is $(-8) \cdot 4=\boxed{-32}$

~Bradygho

Solution 2

We have $\frac{x+2}{6} = \frac{6}{x+2}$, so $x^2+4x-32=0$. By Vieta's our roots $a$ and $b$ amount to $\frac{-32}{1}=\boxed{-32}$

~Geometry285

Solution 3

$\frac{x+2}{6}=\frac{6}{x+2} \implies x^2+4x-32$ Therefore, the product of the root is $-32$

~kante314

Solution 4

The only numbers that are their own reciprocals are $1$ and $-1$. The equation $\frac{x+2}{6}=1$ has the solution $x=4$, while the equation $\frac{x+2}{6}=-1$ has the solution $x=-8$. The answer is $4 \cdot (-8)=\boxed{-32}$.

~tigerzhang


See also

  1. Other 2021 JMPSC Accuracy Problems
  2. 2021 JMPSC Accuracy Answer Key
  3. All JMPSC Problems and Solutions

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