Question

Find the quadratic equation in x whose roots are 2a + 1/b and 2b + 1/a, given that ab = 3.5, a + b = -2.

Answer 1

\[Ax^2+Bx+C=0\] sum of the roots is \(-\frac{B}{A}=-2\) product of the roots is \(\frac{C}{A}=3.5\)
perhaps that will help
is it \(2a+\frac{1}{b}\) or \(\frac{2a+1}{b}\)?

Answer 2

the former

Answer 3

hmm

Answer 4

how do I continue?

Answer 5

not sure

Answer 6

but what i wrote is wrong anyway
a and b are not the roots, so that is a mistake
maybe another tactic is needed

Answer 7

ok i have an idea
lets multiply the two roots and see what we get

Answer 8

No you are right, the roots are 2alpha + 1/beta, and 2beta + 1/alpha, I couldn't find the symbols for alpha and beta so i used a and b

Answer 9

\[(2a+\frac{1}{b})(2b+\frac{1}{a})=4ab+2a+2b+\frac{1}{ab}\] if my algebra is right

Answer 10

no my algebra is bad tonight, let me try it again

Answer 11

\[(2\alpha+\frac{1}{\beta})(2\beta+\frac{1}{\alpha})=4\alpha \beta+4+\frac{1}{\alpha\beta}\]
i think that is better

Answer 12

Ok got it. Thanks!

Answer 13

we know all those numbers because \(\alpha \beta=\frac{7}{2}\)
if my arithmetic is correct, that number is \(\frac{128}{7}\) but maybe my arithmetic is as bad as my algebra

Answer 14

you good from there? you still have to add them

Answer 15

i get the sum of the roots is \(-\frac{32}{7}\) but you should check it

Answer 16

i like this question, it is cute