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(D) For equation $I$: $x^{2}-8 \sqrt{3} x+45=0$We need two numbers whose product is $45$ and sum is $8 \sqrt{3}$.These numbers are $5 \sqrt{3}$ and $3

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if $x = y$ or relationship between $x$ and $y$ cannot be established.

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(D) For equation $I$: $x^{2}-8 \sqrt{3} x+45=0$We need two numbers whose product is $45$ and sum is $8 \sqrt{3}$.These numbers are $5 \sqrt{3}$ and $3 \sqrt{3}$.$x^{2}-5 \sqrt{3} x-3 \sqrt{3} x+45=0$$x(x-5 \sqrt{3})-3 \sqrt{3}(x-5 \sqrt{3})=0$$(x-5 \sqrt{3})(x-3 \sqrt{3})=0$So,$x = 5 \sqrt{3} \approx 5 	imes 1.732 = 8.66$ and $x = 3 \sqrt{3} \approx 3 	imes 1.732 = 5.196$.For equation $II$: $y^{2}-\sqrt{2} y-24=0$We need two numbers whose product is $-24$ and sum is $-\sqrt{2}$.These numbers are $-4 \sqrt{2}$ and $3 \sqrt{2}$.$y^{2}-4 \sqrt{2} y+3 \sqrt{2} y-24=0$$y(y-4 \sqrt{2})+3 \sqrt{2}(y-4 \sqrt{2})=0$$(y-4 \sqrt{2})(y+3 \sqrt{2})=0$So,$y = 4 \sqrt{2} \approx 4 	imes 1.414 = 5.656$ and $y = -3 \sqrt{2} \approx -4.242$.Comparing the values:$x_1 = 8.66, x_2 = 5.196$$y_1 = 5.656, y_2 = -4.242$Since $x_1 > y_1$ but $x_2 < y_1$,the relationship between $x$ and $y$ cannot be established.

Solve the given two equations and select the correct option.
$I.$ $x^{2}-8 \sqrt{3} x+45=0$
$II.$ $y^{2}-\sqrt{2} y-24=0$

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Solve the given two equations and select the correct option.$I.$ $x^{2}-8 \sqrt{3} x+45=0$$II.$ $y^{2}-\sqrt{2} y-24=0$