If in the expansion of (1+x)p(1−x)q, the coefficients of x and x2 are 1 and -2, respectively, then p2+q2 is equal to:
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Step-by-Step Solution
Step 1: Expand the given expression using binomial theorem
We use the binomial theorem to expand each term. The binomial theorem states that (1+y)n=1+ny+2n(n−1)!y2+…. We apply this to (1+x)p and (1−x)q, noting that for (1−x)q, the terms alternate in sign.
Step 2: Multiply the expansions and find coefficients of x and x2
Now we multiply the two expanded series. To find the coefficient of x, we multiply the constant term from the first expansion by the x term from the second, and the x term from the first by the constant term from the second. Similarly, for the coefficient of x2, we consider all combinations that result in an x2 term: constant with x2, x with x, and x2 with constant.
Step 3: Formulate equations from given coefficients
We are given that the coefficient of x is 1 and the coefficient of x2 is -2. We set up two equations based on these given values and the coefficients we derived in the previous step.
Step 4: Simplify and solve the system of equations
We simplify Equation 2 by multiplying by 2 and rearranging terms. We recognize that p2−2pq+q2 is (p−q)2. Substituting p−q=1 from Equation 1, we can solve for p+q.
Step 5: Solve for p and q
Now we have a simple system of two linear equations with two variables (p and q). We can add the two equations to find p, and subtract them to find q.
Step 6: Calculate p2+q2
Finally, we substitute the values of p and q we found into the expression p2+q2 to get the final answer.