Rom

we're told that both points lie on the line

\(a z + b \bar{z} = 38\)

\(\text{So just plug both of these points into the formula there and we solve for }a \text{ and }b\\ \text{I use the complex conjugate because that's how they've specified the line}\)

\(\text{The polynomial evaluated at 1 is the sum of the coefficients. This must be a rational number}\\ \text{Thus each root that is not a rational number must have it's conjugate also as a root}\\ \text{This will be all the roots of the form }a + \sqrt{b} \text{ where }\sqrt{b} \not \in \{2,3,\dots,1001\}\)

\(\text{There are 30 numbers in 2-1001 that are perfect squares}\\ \text{So }1000-30=970 \text{ of the roots must also have their conjugate as a root}\\ \text{Thus we have a total of }\\ 30 + 2\cdot 970 = 1970 \text{ is the smallest possible degree of the polynomial}\)

I'm pretty sure this is correct but I might be missing something.  Someone else should take a look.

\(\text{Both points are on the line so plug them in}\\ a(-5+4i) + b(-5-4i)=38\\ a(7+2i)+b(7-2i) = 38\\ (-5a-5b)+i(4a-4b)=38\\ (7a+7b)+i(2a-2b)=38\)

\(-5(a+b)-14(a+b)=-38\\ -19(a+b)=-38\\ a+b=2\)

\(-5(2) + i(4a-4(2-a))=38\\ i(8a-8)=48\\ i(a-1)=6\\ a-1 = -6i\\ a=1-6i\\ b=2-(1-6i) = 1+6i\)

\((a,b) = (1-6i,1+6i)\)

retired systems engineer who works as a ranch hand now

\(x^2+ax+5a = (x-i_1)(x-i_2)=x^2 - (i_1+i_2)x+i_1i_2 \\ 5a=i_1 i_2\\ i_1+i_2=a\\ 5a = i_1(a-i_1)\\ i_1^2-i_1a+5a=0\\ i_1 = \dfrac{-a\pm\sqrt{a^2-20a}}{2}\\ \sqrt{a^2-20a} \in \mathbb{N}+\{0\}\)

\(a=20\text{ and }a=36 \text{ seem to be the only solutions}\)

watts to kilowatts

\(\text{It hopefully is pretty obvious that it uses}\\ \dfrac{23}{1000}\cdot 336~kw\cdot hr\)

\(1~kw\cdot hr = 3.6 \cdot 10^6~J\\ E=\dfrac{23}{1000}\cdot 336 \cdot 3.6 \cdot 10^6 ~J\)

you can do all the calculator work

d) is the correct choice.

By the Central Limit theorem the sample average of i.i.d samples from any underlying distribution

will converge in distribution to that of a a normal random variable with mean equal to the mean of the underlying distribution

and variance equal to the variance of the underlying distribution divided by the number of samples taken.

\(3^3\sqrt{2a}-6^3\sqrt{2a}=\\ \sqrt{2a}\left(3^3-(3\cdot 2)^3\right)=\\ \sqrt{2a}3^3(1-2^3) = \\ 27\sqrt{2a}(-7) = \\ -189\sqrt{2a}\)

\(3^5(x+2)^3+3=27\\ 3^5(x+2)^3=24\\ (x+2)^3 = \dfrac{24}{3^5} = \dfrac{8}{81}\\ x+2 = \dfrac{2}{3\sqrt[3]{3}}\\ x = \dfrac{2}{3\sqrt[3]{3}} -2 = \\2\left( \dfrac{1}{3\sqrt[3]{3}} -1\right)\)