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

Revision as of 11:38, 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 $4 \cdot (-8)=\boxed{-32}$ .

Revision as of 11:38, 11 July 2021 (view source) Tigerzhang (talk \| contribs) (→‎Solution 4) ← Older edit		Revision as of 11:38, 11 July 2021 (view source) Tigerzhang (talk \| contribs) (→‎Solution 4) Newer edit →
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	==Solution 4==		==Solution 4==
−	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>4-8=\boxed{-4}</math>.	+	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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