The statement "R is numerically equal to the square of the reactance X" simply means `R = X^2`.

Recall (from Application of Complex Numbers) that

`|Z|=sqrt(R^2+(X_L-X_C)^2`

In this case, from the definition, and to make life easier, we assume that XL XC = X.

So

`|Z|=sqrt(R^2+X^2)=2Omega`

Now, on squaring both sides, we have

R2 + X2 = 4

But R = X2 (since R is equal to the square of X) so

R2 + R = 4

Then

R2 + R − 4 = 0

Using quadratic formula gives

`R=(-1+-sqrt(1+16))/2` `=(-1+-sqrt17)/2`

Only the positive root has meaning (since we cannot have negative resistance), so

R = 1.56 Ω. and therefore X = √1.56 = 1.25 Ω.

In this question, the 2 interssecting functions are a parabola (`X = R^2 + R - 4`) and a straight line (`X=4`). In the graph, we can see the 2 solutions we obtained, one is negative (`R=-2.56`) and the other one is positive, at (`R=1.56`).

12345-1-2-3-4-5123456-1-2-3RXOpen image in a new page

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