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

I squared the first equation, and then substituted for y^2:

x^4 - 16x^2 + 64 = -5x + 44 --> x^4 - 16x^2 + 5x + 20 = 0 --> x^2*(x^2 - 4) + 5*(x+4) --> x^2*(x - 4)(x + 4) + 5*(x + 4) -->

(x^2(x - 4) + 5)(x + 4) = 0. Then....umm...not sure. One solution is x = -4, but not sure how to find the other one.

Jun 22, 2018

#1
+22343
+2

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

$$\begin{array}{|lrcll|} \hline (1) & y &=& x^2-8 \\\\ (2) & y^2 &=& -5x+44 \quad | \quad y^2 = (x^2-8)^2 \\ & (x^2-8)^2 &=& -5x+44 \\ & x^4-16x^2+84 &=& -5x+44 \\ & x^4-16x^2+5x+20 &=& 0 \\ & x^2(x^2-16)+5(x+4) &=& 0 \quad | \quad x^2-16 = (x-4)(x+4) \\ & x^2(x-4)(x+4)+5(x+4) &=& 0 \\ & (x+4)\left(x^2(x-4)+5\right) &=& 0 \\ \\ 1. & x_1+4 &=& 0 \\ & \mathbf{x_1} &\mathbf{=}& \mathbf{-4} \\ & y_1 &=& x_1^2-8 \\ & y_1 &=& (-4)^2-8 \\ & \mathbf{y_1} &\mathbf{=}& \mathbf{8} \\ & \text{solution (-4,8) } \\\\ 2. & x^2(x-4)+5 &=& 0 \\ & x^3-4x^2 + 5 &=& 0 \\ \text{try } x=-1 & (-1)^3-4(-1)^2 + 5 &\overset{?}{=}& 0 \\ & -1-4 + 5 & = & 0\ \checkmark \\ & \mathbf{x_2} &\mathbf{=}& \mathbf{-1} \\ & y_2 &=& x_2^2-8 \\ & y_2 &=& (-1)^2-8 \\ & \mathbf{y_2} &\mathbf{=}& \mathbf{-7} \\ & \text{solution (-1,-7) } \\\\ \hline \end{array}$$

Polynomial long division

$$\begin{array}{|rcll|} \hline x^2-5x+5 &=& 0 \\ x &=& \dfrac{5\pm\sqrt{25-4\cdot 5 }}{2}\\ x &=& \dfrac{5\pm\sqrt{5}}{2} \\\\ \mathbf{x_3} &\mathbf{=}& \mathbf{\dfrac{5+\sqrt{5}}{2}=3.618 } \\ y_3 &=& x_3^2-8 \\ y_3 &=& \left(\dfrac{5+\sqrt{5}}{2} \right)^2-8 \\ \mathbf{y_3} &\mathbf{=}& \mathbf{\dfrac{5\sqrt{5}-1}{2} = 5.09 } \\ \text{solution (\dfrac{5+\sqrt{5}}{2},\dfrac{5\sqrt{5}-1}{2}) \\ =(3.618,5.09) } \\\\ \mathbf{x_4} &\mathbf{=}& \mathbf{\dfrac{5-\sqrt{5}}{2}=1.382 } \\ y_4 &=& x_4^2-8 \\ y_4 &=& \left(\dfrac{5-\sqrt{5}}{2} \right)^2-8 \\ \mathbf{y_4} &\mathbf{=}& \mathbf{-\left(\dfrac{5\sqrt{5}+1}{2}\right) = -6.09 } \\ \text{solution (\dfrac{5-\sqrt{5}}{2},-\left(\dfrac{5\sqrt{5}+1}{2}\right)) \\ =(1.382,-6.09) } \\ \hline \end{array}$$

Find the product of the y-coordinates of all the distinct solutions (x,y):

$$\begin{array}{|rcll|} \hline && y_1y_2y_3y_4 \\ &=& 8\times(-7)\times \left(\dfrac{5\sqrt{5}-1}{2}\right) \times \left(-\left(\dfrac{5\sqrt{5}+1}{2}\right)\right) \\ &=& 8\times(-7)\times (-31) \\ &\mathbf{=}& \mathbf{1736} \\ \hline \end{array}$$

Jun 22, 2018

#1
+22343
+2

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

$$\begin{array}{|lrcll|} \hline (1) & y &=& x^2-8 \\\\ (2) & y^2 &=& -5x+44 \quad | \quad y^2 = (x^2-8)^2 \\ & (x^2-8)^2 &=& -5x+44 \\ & x^4-16x^2+84 &=& -5x+44 \\ & x^4-16x^2+5x+20 &=& 0 \\ & x^2(x^2-16)+5(x+4) &=& 0 \quad | \quad x^2-16 = (x-4)(x+4) \\ & x^2(x-4)(x+4)+5(x+4) &=& 0 \\ & (x+4)\left(x^2(x-4)+5\right) &=& 0 \\ \\ 1. & x_1+4 &=& 0 \\ & \mathbf{x_1} &\mathbf{=}& \mathbf{-4} \\ & y_1 &=& x_1^2-8 \\ & y_1 &=& (-4)^2-8 \\ & \mathbf{y_1} &\mathbf{=}& \mathbf{8} \\ & \text{solution (-4,8) } \\\\ 2. & x^2(x-4)+5 &=& 0 \\ & x^3-4x^2 + 5 &=& 0 \\ \text{try } x=-1 & (-1)^3-4(-1)^2 + 5 &\overset{?}{=}& 0 \\ & -1-4 + 5 & = & 0\ \checkmark \\ & \mathbf{x_2} &\mathbf{=}& \mathbf{-1} \\ & y_2 &=& x_2^2-8 \\ & y_2 &=& (-1)^2-8 \\ & \mathbf{y_2} &\mathbf{=}& \mathbf{-7} \\ & \text{solution (-1,-7) } \\\\ \hline \end{array}$$

Polynomial long division

$$\begin{array}{|rcll|} \hline x^2-5x+5 &=& 0 \\ x &=& \dfrac{5\pm\sqrt{25-4\cdot 5 }}{2}\\ x &=& \dfrac{5\pm\sqrt{5}}{2} \\\\ \mathbf{x_3} &\mathbf{=}& \mathbf{\dfrac{5+\sqrt{5}}{2}=3.618 } \\ y_3 &=& x_3^2-8 \\ y_3 &=& \left(\dfrac{5+\sqrt{5}}{2} \right)^2-8 \\ \mathbf{y_3} &\mathbf{=}& \mathbf{\dfrac{5\sqrt{5}-1}{2} = 5.09 } \\ \text{solution (\dfrac{5+\sqrt{5}}{2},\dfrac{5\sqrt{5}-1}{2}) \\ =(3.618,5.09) } \\\\ \mathbf{x_4} &\mathbf{=}& \mathbf{\dfrac{5-\sqrt{5}}{2}=1.382 } \\ y_4 &=& x_4^2-8 \\ y_4 &=& \left(\dfrac{5-\sqrt{5}}{2} \right)^2-8 \\ \mathbf{y_4} &\mathbf{=}& \mathbf{-\left(\dfrac{5\sqrt{5}+1}{2}\right) = -6.09 } \\ \text{solution (\dfrac{5-\sqrt{5}}{2},-\left(\dfrac{5\sqrt{5}+1}{2}\right)) \\ =(1.382,-6.09) } \\ \hline \end{array}$$

Find the product of the y-coordinates of all the distinct solutions (x,y):

$$\begin{array}{|rcll|} \hline && y_1y_2y_3y_4 \\ &=& 8\times(-7)\times \left(\dfrac{5\sqrt{5}-1}{2}\right) \times \left(-\left(\dfrac{5\sqrt{5}+1}{2}\right)\right) \\ &=& 8\times(-7)\times (-31) \\ &\mathbf{=}& \mathbf{1736} \\ \hline \end{array}$$

heureka Jun 22, 2018