For t>−1, let αt and βt be the roots of the equation ((t+2)71−1)x2+((t+2)61−1)x+((t+2)211−1)=0. If limt→−1+αt=a and limt→−1+βt=b, then 72(a+b)2 is equal to:
Get the complete, step-by-step math solution for: "For t > -1, let _t and _t be the roots of the equation (t + 2)^{(1)/(7)} - 1x² + (t + 2)^{(1)/(6)} - 1x + (t + 2)^{(1)/(21)} - 1 = 0. If _{t -1^+} _t ...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Simplify coefficients using substitution
To simplify the limit calculation, we introduce a substitution. Let y=t+1. As t approaches −1 from the right side (denoted as t→−1+), y will approach 0 from the positive side (denoted as y→0+). Substituting t+2=y+1 into the given quadratic equation transforms it into an expression in terms of y.
Step 2: Apply L'Hopital's Rule for coefficients
We need to find the limits of the coefficients as y→0+. Each coefficient is of the form limy→0+y(y+1)k−1. This is an indeterminate form 00, so we can apply L'Hopital's Rule. Alternatively, we can use the binomial approximation (1+y)k≈1+ky for small y. Applying this, we find the limits of the coefficients A, B, C.
Step 3: Determine the limiting quadratic equation
As y→0+, the original quadratic equation approaches a new quadratic equation whose coefficients are the limits we just calculated. We substitute the values of A, B, C into the general quadratic form Ax2+Bx+C=0. To clear the fractions, we multiply the entire equation by the least common multiple of the denominators, which is 42.
Step 4: Find the sum of the roots
For a quadratic equation in the standard form ax2+bx+c=0, the sum of its roots is given by the formula −ab. In our limiting equation, 6x2+7x+2=0, we identify a=6 and b=7. Therefore, the sum of the roots, which are a and b in the problem statement, is −67.
Step 5: Calculate the final expression
The problem asks for the value of 72(a+b)2. We have already found that a+b=−67. We substitute this value into the expression and perform the calculation. Squaring −67 gives 3649. Then, we multiply 72 by 3649, which simplifies to 2×49=98.