Algebra

For a quadratic equation x^2 - (sum of roots)x + (product of roots) = 0. Given sum = 3, product = 2. Equation is x^2 - 3x + 2 = 0.

If 1/alpha is a root, then 2(1/alpha)^2 - 5(1/alpha) + 7 = 0. Multiplying by alpha^2 gives 2 - 5alpha + 7alpha^2 = 0. Rearranging, 7alpha^2 - 5alpha = -2.

For x^2 - 6x + 6 = 0, a + b = 6 and ab = 6. a^2 + b^2 = (a + b)^2 - 2ab = 6^2 - 2 * 6 = 36 - 12 = 24.

For -1 < a < 1, the coefficient a - 1 is negative and sqrt(1 - a^2) is positive. Thus the discriminant a^2 - 4(a - 1)sqrt(1 - a^2) is positive, so both roots are real.

The equation x^2 - 1 = 0 has solutions x = 1 and x = -1. The equation x^2 = -1 has solutions x = i and x = -i. Since the solution sets are different, the statement is false.

x^2 + x = 0 factors to x(x+1) = 0. The solutions are x = 0 and x = -1. Neither solution is greater than zero.

Let u = |x|. The equation is u^2 + u - 6 = 0, which factors to (u+3)(u-2) = 0. Since u = |x| >= 0, u = 2. Thus |x| = 2, so x = 2 or x = -2. The roots are 2 and -2, and their sum is 0.

The equation 4(x-p)(x-q) - r^2 = 0 can be written as 4x^2 - 4(p+q)x + 4pq - r^2 = 0. The discriminant D = [4(p+q)]^2 - 4(4)(4pq - r^2) = 16(p^2 + 2pq + q^2) - 16(4pq - r^2) = 16(p^2 - 2pq + q^2 + r^2) = 16[(p-q)^2 + r^2]. Since (p-q)^2 >= 0 and r^2 >= 0, D >= 0, so roots are real. If p=q and r=0, D=0, so roots are equal.

For equal roots, the discriminant D = b^2 - 4ac = 0. Here, [b(c-a)]^2 - 4[a(b-c)][c(a-b)] = 0. This simplifies to b^2(c-a)^2 - 4ac(ab - b^2 - ac + bc) = 0. This is a known condition for a, b, c in HP.

For x^2 - 30x + 221 = 0, p+q = 30 and pq = 221. p^3 + q^3 = (p+q)(p^2 - pq + q^2) = (p+q)((p+q)^2 - 3pq) = 30 * (30^2 - 3*221) = 30 * (900 - 663) = 30 * 237 = 7110.

n^2 + 4n = 0 => n(n + 4) = 0. Roots are n = 0 and n = -4.

4m^2 - 9 = 0 implies 4m^2 = 9, so m^2 = 9/4. Taking the square root of both sides gives m = +/- 3/2.

A quadratic polynomial with sum S and product P is given by x^2 - Sx + P. Here S=5, P=-50. So x^2 - 5x - 50.

A quadratic polynomial with sum of zeroes S and product P is given by x^2 - Sx + P. Here, S = -11 and P = 10, so the polynomial is x^2 - (-11)x + 10 = x^2 + 11x + 10.