Two resistors connected in series have an equivalent resistance of 783.4 Ω. When they are connected in parallel, their equivalent resistance is 171.3 Ω. Find the resistance of each resistor.

1) small resistance (ohm)
2) large resistance (0hm)

Given:

Series Resistance: S = 783.4 Ω
Parallel Resistance: P = 171.3 Ω
Let the resistances be r and q.
S = r + q
P = rq/(r + q)

.: rq = SP
and S = r + SP/r

.: r^2 - Sr + SP = 0

By symmetry if r is one root of the quadratic then q is the other. Let r be the lesser.

r = (S-√(S^2-4SP))/2
q = (S+√(S^2-4SP))/2

I solved a problem exactly like this the other day, but I made a math mistake; I should have checked my math.

Equation 1.)

R1+R2=783.4

Equation 2.)

R1*R2/(R1+R2)=Req=783.4

Substitute equation 1 into 2:

R1=783.4-R2

and

R1*R2/(R1+R2)=Req=171.3

(783.4-R2)*R2/783.4-R2+R2=171.3

R2^2-783.4R2=-1.342 x 10^5

R2^2 -783.4R2+1.342 x 10^5=0

use the quadratic equation and solve for R

R=[-b + or - sqrt*(b^2-4ac)]/2a

a=1, b=-783.4, and c=1.342 x 10^5

R=[-(-783.4) + or - sqrt*((-783.4)^2-4(1)(1.342 x 10^5)]/2(1)

R=[783.4 + or - (6.1372 x 10^5-5.368 x 10^5)]/2

R=[783.4 + or - sqrt*(7.6916 x 10^4)]/2

R=(783.4-277.3/2) or (783.4+277.3/2)

R=253.05Ω or R=530.35Ω

To find the resistance of each resistor, we can set up a system of equations using the information given.

Let's denote the resistance of the first resistor as R1 and the resistance of the second resistor as R2.

When two resistors are connected in series, their equivalent resistance is given by the sum of their individual resistances:

R_series = R1 + R2 = 783.4 Ω ........(Equation 1)

When two resistors are connected in parallel, their equivalent resistance is given by the reciprocal of the sum of the reciprocals of their individual resistances:

1/R_parallel = 1/R1 + 1/R2 ........(Equation 2)

Substituting the given value of the equivalent resistance in parallel, we have:

1/171.3 Ω = 1/R1 + 1/R2

Now, we have a system of two equations (Equation 1 and Equation 2) with two unknowns (R1 and R2).

Let's solve this system to find the values of R1 and R2:

From Equation 1:
R1 + R2 = 783.4 Ω

From Equation 2:
1/R1 + 1/R2 = 1/171.3 Ω

To solve this system of equations, we can use the method of substitution or elimination. Let's use the substitution method for this example.

Rearrange Equation 1 to solve for R2:
R2 = 783.4 Ω - R1

Substitute this expression for R2 in Equation 2:

1/R1 + 1/(783.4 Ω - R1) = 1/171.3 Ω

Now, we can solve this equation to find the value of R1.