(adsbygoogle = window.adsbygoogle || []).push({}); Q5 - Equations with an "additonal" restricted variable

1. The problem statement, all variables and given/known data

P1

Determine the parametermsuch that the sum of the squares of the roots of equation

x^{2}- (m-2)x + m - 3 = 0

is minimal.

P2

Determine the values of parameterp, so that the solution of equation

p - (1/x) = (p^{2}- 2x - 5)/((p+3)x)

is greather than 1.

2. Relevant equations

3. The attempt at a solution

I would like to know whether there is any specific way of calculating equations with an additonal "restricted" variable, like in the two problems above.

Is a rational reasoning followed by a trial and error used or is more specific calculation possible?

P1

x^{2}- (m-2)x + m - 3 = 0

The sum of the squares must be minimal, thus:

(ax - b)(cx - d) where ax^{2}+ cx^{2}must be minimal

roots are minimal when

(x - 1)^{2}or (x + 1)^{2}namely 1^{2}or (-1)^{2}which equals 1

This form

x^{2}- 2x +1

x^{2}+2x +1

Can be achieved when

x^{2}- (4-2)x + 4 - 3 = 0

Thus m = 4

Are my calculations here correct? Is this the correct way to solve this problem?

P2

p - (1/x) = (p^{2}- 2x - 5)/((p+3)x)

I find this problem a lot more difficult. Reasoning here is a lot trickier (for me at least). I can only see that p can not equal -3

When either working the equation out of fractions, ie

(px - 1)(p +3) = p^{2}- 2x - 5

or moving everything to one side, I am still left with one equation and "two" unknowns. Another approach I tried when anaylizing this problem was simply putting in a value for x, x=1, x=2 and x=3 all return values of p=-2

How should I approach such a problem?

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# Q5 - Equations with an additonal restricted variable

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