find-all-values-for-k-such-that-the-eq-x-3-13x-k-0-has-three-integer-roots-

Question Number 200388 by mr W last updated on 18/Nov/23

f i n d a l l v a l u e s f o r k s u c h t h a t t h e e q .

x^{3} - 13 x + k = 0 h a s t h r e e i n t e g e r r o o t s .

Answered by Frix last updated on 18/Nov/23

(x - p) (x + \frac{p}{2} - q) (x + \frac{p}{2} + q) = 0

x^{3} - \frac{3 p^{2} + 4 q^{2}}{4} x - \frac{p (p^{2} - 4 q^{2})}{4} = 0

\Rightarrow

q = \frac{\sqrt{52 - 3 p^{2}}}{2} \land k = - p (p^{2} - 13)

\Rightarrow p = \pm {1, 3, 4} \land k = \pm 12

Commented by mr W last updated on 18/Nov/23

t h a n k s s i r!

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