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Find the product of the $y$-coordinates of all the distinct solutions $(x,y)$ for the two equations $y=x^2-8$ and $y^2=-5x+44$.

Jul 25, 2019

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Find the product of the $y$-coordinates of all the distinct solutions $(x,y)$ for the two equations $y=x^2-8$ and $y^2=-5x+44$.

Hello Guest!

$$y=x^2-8\\ y^2=-5x+44$$.

$$y^2=x^4-16x^2+64$$

$$x^4-16x^2+64=-5x+44\\ x^4-16x^2+5x+20=0$$

http://www.arndt-bruenner.de/mathe

$$x_4=3,618\\ x_3=1,382\\ x_2=-1\\ x_1=-4\\ y_{1,2,3,4}=0$$

The product of the y-coordinates is $$y_1\cdot y_2\cdot y_3\cdot y_4\cdot =0\cdot 0\cdot 0\cdot 0=0$$

!

That was a wrong consideration. Sorry.
I correct.

$$f(x)=y=x^2-8\\ g(x)=y =\pm\sqrt{-5x+44}$$

http://www.arndt-bruenner.de/mathe/

$$P_1(-4/8)\\ P_2(3,618/5,0902)\\ P_3(-1/-7)\\ P_4(1,382/-6,0902)$$

The product of the y-coordinates is $$8\cdot 5,0902\cdot (-7)\cdot (-6,0902)=\color{blue}1736,02$$

!

Jul 25, 2019
edited by asinus  Jul 25, 2019