Problem 5: Symmetrical Properties of Roots

Algebra Level 4

The roots of the quadratic equation $3x^2-2kx+k+4=0$ are $\alpha$ and $\beta$ . If $\alpha^2+\beta^2=\frac{16}{9}$ , find the sum of all possible values of $k$ .

The answer is 1.5.

2 solutions

Alan Enrique Ontiveros Salazar
Jul 22, 2014

Rewrite the given expression:

$(\alpha+\beta)^2-2\alpha \beta = \dfrac{16}{9}$

Now, by Vieta's formulas:

$\left(\dfrac{2k}{3}\right)^2-2\left(\dfrac{k+4}{3}\right)=\dfrac{16}{9}$

Rearrange and simplify:

$4k^2-6k-40=0$

Again, by Vieta's formulas, the sum of all values of $k$ is $\dfrac{3}{2}=\boxed{1.5}$

done by the same method

Ayush Garg - 6 years, 10 months ago

Hey the equation is coming out to be: 12k^2 - 2k - 24=0. Whose solutions are -4/3 and 3/2. So the sum is 1/6 or 0.167. Where am I wrong??

Sarthak Tanwani - 6 years, 10 months ago

You are wrong with the equation you obtained. How did you obtain it?

Alan Enrique Ontiveros Salazar - 6 years, 10 months ago

Is ABC formula wrong Vietas formula wrong?

-b(+/-)√(b^2-4a.c)/2.a

I got k 4 and 2.5 which is the sum gone to 6.5 .

Hafizh Ahsan Permana - 6 years, 10 months ago

@Hafizh Ahsan Permana – bro, u must have got -2.5, check it!

devesh golwalkar - 6 years, 9 months ago

The roots of the equation what u got are 8 and 9

Kandarp Kakkad - 5 years, 11 months ago

same method!!!!

Kartik Sharma - 6 years, 10 months ago

haha!!! guys we all have same approach towards this, i.e the veita function :)

devesh golwalkar - 6 years, 9 months ago

Fox To-ong
Jan 21, 2015

using To-ong's little theorem

Can you please tell what is that theorem?

Swapnil Das - 5 years, 10 months ago

Problem 5: Symmetrical Properties of Roots

The answer is 1.5.

2 solutions

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