To find the resistance [tex]\( R_2 \)[/tex] in a circuit with parallel resistors, we start with the equation for the total resistance in a parallel circuit. The formula for the total resistance [tex]\( R_T \)[/tex] in parallel is given by:
[tex]\[ \frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2} \][/tex]
Given the total resistance [tex]\( R_T \)[/tex] is 2 ohms and [tex]\( R_1 \)[/tex] is 6 ohms, we can rearrange the equation to solve for [tex]\( R_2 \)[/tex]:
First, we isolate [tex]\(\frac{1}{R_2}\)[/tex] by subtracting [tex]\(\frac{1}{R_1}\)[/tex] from both sides:
[tex]\[ \frac{1}{R_2} = \frac{1}{R_T} - \frac{1}{R_1} \][/tex]
Next, plug in the values of [tex]\( R_T \)[/tex] and [tex]\( R_1 \)[/tex]:
[tex]\[ \frac{1}{R_2} = \frac{1}{2} - \frac{1}{6} \][/tex]
To subtract these fractions, we need a common denominator, which in this case is 6. So, we rewrite the fractions:
[tex]\[ \frac{1}{2} = \frac{3}{6} \][/tex]
Now our equation looks like:
[tex]\[ \frac{1}{R_2} = \frac{3}{6} - \frac{1}{6} \][/tex]
Subtract the fractions:
[tex]\[ \frac{1}{R_2} = \frac{2}{6} \][/tex]
This simplifies to:
[tex]\[ \frac{1}{R_2} = \frac{1}{3} \][/tex]
Now, to find [tex]\( R_2 \)[/tex], we take the reciprocal of both sides:
[tex]\[ R_2 = 3 \][/tex]
Therefore, [tex]\( R_2 \)[/tex] is approximately 3 ohms.
