Algebra

If roots are reciprocal, their product is 1. For ax^2 + bx + c = 0, product of roots is c/a. Here, k/5 = 1, so k = 5.

A quadratic polynomial with zeros a and b is (x - a)(x - b). For zeros 9 and -9, (x - 9)(x + 9) = x^2 - 81.

For a quadratic equation ax^2 + bx + c = 0, if the sum of coefficients a+b+c = 0, then 1 is a root. Here, (p-q) + (q-r) + (r-p) = 0. Thus, 1 is a root. The product of roots is c/a = (r-p)/(p-q). The other root is (r-p)/(p-q).

If a quadratic equation with rational coefficients has a root 4 + sqrt(5), its conjugate 4 - sqrt(5) must also be a root. The sum of the roots is (4 + sqrt(5)) + (4 - sqrt(5)) = 8, and the product is (4 + sqrt(5))(4 - sqrt(5)) = 16 - 5 = 11. The equation is x^2 - (sum)x + (product) = 0, which is x^2 - 8x + 11 = 0.

For roots 3+sqrt(2) and 3-sqrt(2), the sum is 6 and the product is (3+sqrt(2))(3-sqrt(2)) = 9-2 = 7. The quadratic equation is x^2 - (sum)x + (product) = 0, which is x^2 - 6x + 7 = 0.

Given f(x) = ax^2 + bx + c and g(x) = ax^2 + px + q, f(x) = g(x) implies (b-p)x + (c-q) = 0, so x = (q-c)/(b-p). With equal discriminants, b^2 - 4ac = p^2 - 4aq, which simplifies to b^2 - p^2 = 4a(c-q). Factoring gives (b-p)(b+p) = 4a(c-q), so (c-q)/(b-p) = -(b+p)/4a. The roots of f(x)=0 are (-b +/- sqrt(D))/2a and g(x)=0 are (-p +/- sqrt(D))/2a. The A.M. of all four roots is (-b-p)/4a, which matches alpha.

If a root is 3 + sqrt(2), the other root must be 3 - sqrt(2). The sum of roots is 6 and the product is (3+sqrt(2))(3-sqrt(2)) = 9 - 2 = 7. The quadratic equation is x^2 - (sum)x + (product) = 0, which is x^2 - 6x + 7 = 0.

a + b = 24, a - b = 8. Adding gives 2a = 32, so a = 16. Subtracting gives 2b = 16, so b = 8. The equation is x^2 - (a+b)x + ab = 0, which is x^2 - 24x + 128 = 0.

For equal roots, the discriminant D = 0. D = [2(k-4)]^2 - 4(1)(2k) = 4(k^2 - 8k + 16) - 8k = 4k^2 - 32k + 64 - 8k = 4k^2 - 40k + 64 = 0. Dividing by 4: k^2 - 10k + 16 = 0. Factors to (k-8)(k-2) = 0. So k = 8 or k = 2.

Check the discriminant (D = b^2 - 4ac) for each: A) 16 - 60 < 0. B) 1 - 16 < 0. C) (x-1)(2x-5)=0 has roots 1 and 2.5 (real). D) 9 - 32 < 0.

If roots are r1 and r2, the equation is x^2 - (r1+r2)x + (r1*r2) = 0. Sum = (2+sqrt(2)) + (2-sqrt(2)) = 4. Product = (2+sqrt(2))(2-sqrt(2)) = 4 - 2 = 2. Equation: x^2 - 4x + 2 = 0.

For a quadratic equation ax^2 + bx + c = 0 to have equal roots, the discriminant D = b^2 - 4ac must be zero. Substituting the given coefficients, 4(m+1)^2 - 4(3m+1)(m) = 0 simplifies to 4(m^2 + 2m + 1) - 4(3m^2 + m) = 0, which results in -8m^2 + 4m + 4 = 0, or 2m^2 - m - 1 = 0. Factoring gives (2m+1)(m-1) = 0, so m = 1 or -1/2.

For a quadratic equation ax^2 + bx + c = 0, the roots are imaginary if and only if the discriminant b^2 - 4ac is less than zero.

For a quadratic equation ax^2 + bx + c = 0, the nature of the roots is determined by the discriminant, D = b^2 - 4ac. When the roots are real and equal, the discriminant must be exactly equal to zero. Thus, b^2 - 4ac = 0 is the correct condition.