FP1 roots of polynomial equations

The equation x^2+2x+5=0 has roots alpha and beta. Find the equation which has the roots
(alpha +2(beta)) and (beta + 2(alpha))

Could some one briefly run through to a solution, as I can't spot the trick on this one...

Reply 1

Kevin De Bruyne

21

Original post by Mathematicus65

The equation x^2+2x+5=0 has roots alpha and beta. Find the equation which has the roots
(alpha +2(beta)) and (beta + 2(alpha))

Could some one briefly run through to a solution, as I can't spot the trick on this one...

I would suggest calculating what alpha and beta are, then calculate the value of those two roots and go from there.

Reply 2

Harvey.Specter

3

Original post by Mathematicus65

The equation x^2+2x+5=0 has roots alpha and beta. Find the equation which has the roots
(alpha +2(beta)) and (beta + 2(alpha))

Could some one briefly run through to a solution, as I can't spot the trick on this one...

What did you get for your alpha and beta? Where is it that you're getting stuck?

Reply 3

username1732133

15

Original post by Mathematicus65

The equation x^2+2x+5=0 has roots alpha and beta. Find the equation which has the roots
(alpha +2(beta)) and (beta + 2(alpha))

Could some one briefly run through to a solution, as I can't spot the trick on this one...

\alpha+ 2 \beta = (\alpha+\beta)+\beta

Spoiler

Reply 4

iylianuraiman

3

x^2 - (sum of roots)x + (product of roots) = 0 find the sums and products of the new roots

Reply 5

Muttley79

20

Original post by Mathematicus65

The equation x^2+2x+5=0 has roots alpha and beta. Find the equation which has the roots
(alpha +2(beta)) and (beta + 2(alpha))

Could some one briefly run through to a solution, as I can't spot the trick on this one...

These questions are NOT about finding the roots.

sum of roots = -b/a

product = c/a

Find the sum of the new roots and express in terms of these expressions and then ditto for the product of the new roots.

Then find the new equation.

Reply 6

username1732133

15

Original post by Muttley79

These questions are NOT about finding the roots.

sum of roots = -b/a

product = c/a

Find the sum of the new roots and express in terms of these expressions and then ditto for the product of the new roots.

Then find the new equation.

Easier with the substitution I gave above. Pretty much a write down..

Reply 7

Muttley79

20

Original post by BuryMathsTutor

Easier with the substitution I gave above. Pretty much a write down..

Wrong technique though for FP1 - surely a tutor knows this?

Reply 8

Mathematicus65

OP

9

Original post by SeanFM

I would suggest calculating what alpha and beta are, then calculate the value of those two roots and go from there.

Thank you this worked. I was too busy trying to find a more algebraic method for this one! Thank you again!

Reply 9

username1732133

15

Original post by Muttley79

Wrong technique though for FP1 - surely a tutor knows this?

To clarify:

The observation that the new roots are

\alpha -2

and

\beta - 2

allows you to write down

(x+2)^2+2(x+2)+5=0

which is easy to simplify.

Using linear transformations is mentioned in specifications as something students should be able to do.

Sorry if my first post caused any offence. That was not my intention.