Question 2: Consider the function Q(x) = ax^2 +bx + c for x belongs to all real numbers where a belongs to all real numbers, a doesn't equal to 0 and b, c belong to all real numbers. Let r_1, r_2 belong to all real numbers and be the roots of Q(x) = 0. d(i) write down an expression for the sum of roots r_1 + r_2 in terms of a and b. d(ii) write down an expression for the product of roots r_1 + r_2 in terms of a and c. If a, r_1, b, r_2, and c in that order are in arithmetic sequence, then Q(x) is said to be an AS quadratic function. e(i) Given that Q(x) is an AS-quadratic function, write down an expression for r_2 - r_1 in terms of a and b. Use your answers to parts d(i) and e(i) to show that r_1 = (a^2 - ab - b)/2a. e(iii) Use the result from e(ii) to show that b=0 or a=-(1/2). (f) Consider the case where b=0. Determine the two AS-quadratic functions that staisfy this condition. g(i) Now consider the case where a=-(1/2). Find an expression for r_1 in terms of b. g(ii) Hence or otherwise, determine the exact values of b and c, such that AS-quadratic functions are formed. Give your answers in the form (-p±q√s)/2 where p, q, s all belong to all positive integers.
Question 2: Consider the function Q(x) = ax^2 +bx + c for x belongs to all real numbers where a belongs to all real numbers, a doesn't equal to 0 and b, c belong to all real numbers. Let r_1, r_2 belong to all real numbers and be the roots of Q(x) = 0. d(i) write down an expression for the sum of roots r_1 + r_2 in terms of a and b. d(ii) write down an expression for the product of roots r_1 + r_2 in terms of a and c. If a, r_1, b, r_2, and c in that order are in arithmetic sequence, then Q(x) is said to be an AS quadratic function. e(i) Given that Q(x) is an AS-quadratic function, write down an expression for r_2 - r_1 in terms of a and b. Use your answers to parts d(i) and e(i) to show that r_1 = (a^2 - ab - b)/2a. e(iii) Use the result from e(ii) to show that b=0 or a=-(1/2). (f) Consider the case where b=0. Determine the two AS-quadratic functions that staisfy this condition. g(i) Now consider the case where a=-(1/2). Find an expression for r_1 in terms of b. g(ii) Hence or otherwise, determine the exact values of b and c, such that AS-quadratic functions are formed. Give your answers in the form (-p±q√s)/2 where p, q, s all belong to all positive integers.
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