x-2-2-y-2-2-97-1-x-2-3-y-2-3-793-2-solve-the-simultaneous-equation-

Question Number 44783 by Necxx last updated on 04/Oct/18

{(x^{2})}^{2} + {(y^{2})}^{2} = 97 \dots \dots . 1

{(x^{2})}^{3} + {(y^{2})}^{3} = 793 \dots \dots . 2

s o l v e t h e s i m u l t a n e o u s e q u a t i o n

Answered by ajfour last updated on 05/Oct/18

a^{2} + b^{2} = 97

a^{3} + b^{3} = 793

\Rightarrow {(a + b)}^{2} - 2 a b = 97

& {(a + b)}^{3} - 3 a b (a + b) = 793

l e t a + b = r & a b = s

\Rightarrow r^{2} - 2 s = 97 &

r^{3} - 3 r s = 793

\Rightarrow 2 r^{3} - 3 r (r^{3} - 97) = 1586

\Rightarrow r^{3} - 291 r + 1586 = 0

s o l u t i o n o f a c u b i c o f t h e f o r m

x^{3} + p x + q = 0 i s

x = {(- \frac{q}{2} + \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}})}^{1 / 3}

- {(\frac{q}{2} + \sqrt{\frac{q^{2}}{4} + \frac{p^{3}}{27}})}^{1 / 3}

(i f i t h a s o n e r e a l r o o t)

H e r e w e h a v e

r^{3} - 291 r + 1586 = 0

c a l c u l a t o r p r o v i d e s t h e r e a l r o o t,

r = 13, s = \frac{r^{2} - 97}{2} = \frac{169 - 97}{2}

\Rightarrow s = 36

\Rightarrow a + b = 13, a b = 36

x = \pm 9, y = \pm 4 o r

x = \pm 4, y = \pm 9 .

Commented by Necxx last updated on 05/Oct/18

p l e a s e s o l v e h o w r = 13 . T h a n k s

Commented by Necxx last updated on 05/Oct/18

O h \dots . I r e a l l y d i d n t r e m e m b e r

t h e c u b i c f o r m s y o u a n d M r M J S

u s e d . T h a n k s s o m u c h

Facebook Tweet Pin