For t-1, let _t and _t be the roots of the equation ( (t+2 )^1/7-1)x^2+( (t+2 )^1/6-1)x+( (t+2 )^1/21-1)=0. if limt -1^+ 0.05em _t=a and limt -1^+ 0.05em _t=b then 72 (a+b )^2 is equal to ______.

Question

For t>-1, let _t and _t be the roots of the equation ( (t+2 )^1/7-1)x^2+( (t+2 )^1/6-1)x+( (t+2 )^1/21-1)=0. if limt -1^+ 0.05em _t=a and limt -1^+ 0.05em _t=b then 72 (a+b )^2 is equal to ______.

Accepted Answer

Correct answer: 98. (98) a+b= lim_t -1^+ ( + )= lim_t -1^+- (t+2 )^ 16-1 (t+2 )^ 17-1 lett+2=y a+b= lim_y 1^+ 0.05em y^1/6-1y^1/7-1= 76 72 (a+b )^2=72 4936=98

$For t > - 1, let α_{t} and β_{t} be the roots of the equation ((t + 2)^{1/7} - 1) x^{2} + ((t + 2)^{1/6} - 1) x + ((t + 2)^{1/21} - 1) = 0. if lim t \to - 1^{+} α_{t} = a and lim t \to - 1^{+} β_{t} = b then 72 (a + b)^{2} is equal to ______.$

Solution

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