Algebra

Substituting x = 1 into the quadratic equation gives a + 2 - a - 2 = 0, meaning one root is always 1. Since the product of the roots is -2/a, the other root must be -2/a. For there to be at least four negative integers (which must be -1, -2, -3, and -4) between the roots, the second root must be strictly less than -4, which simplifies to 0 < a < 1/2.

For x^2 + kx + 81 = 0 to have real roots, the discriminant k^2 - 4(81) >= 0, so k^2 >= 324, implying k >= 18 (since k > 0). For x^2 - 6sqrt(2)x + k = 0 to have real roots, the discriminant (6sqrt(2))^2 - 4k >= 0, so 72 - 4k >= 0, implying k <= 18. Combining k >= 18 and k <= 18, the only solution is k = 18.

For distinct roots, the discriminant D = b^2 - 4ac > 0. Here, (-6)^2 - 4(1)(k) > 0 => 36 - 4k > 0 => 36 > 4k => k < 9.

For roots to be negatives of each other, the sum of roots must be zero. In ax^2 + bx + c = 0, sum of roots = -b/a. Here, -(-8m)/9 = 0, so 8m = 0, which means m = 0.

For coincident roots, the discriminant D = b^2 - 4ac must be 0. Here, a = k+3, b = -(5-k), c = 1. So, (5-k)^2 - 4(k+3) = 0. 25 - 10k + k^2 - 4k - 12 = 0. k^2 - 14k + 13 = 0. (k-1)(k-13) = 0. k = 1 or 13.

A quadratic equation with roots r1 and r2 is x^2 - (r1+r2)x + (r1*r2) = 0. Here, x^2 - (5+9)x + (5*9) = x^2 - 14x + 45 = 0.

For real and equal roots, the discriminant D = b^2 - 4ac = 0. Here a=k, b=4, c=1. 4^2 - 4*k*1 = 0. 16 - 4k = 0. 4k = 16, k = 4.

For real and equal roots, D = b^2 - 4ac = 0. Here a=2, b=-k, c=1. D = (-k)^2 - 4(2)(1) = k^2 - 8 = 0. k^2 = 8, so k = +/- 2*sqrt(2).

For equal roots, discriminant D = b^2 - 4ac = 0. k^2 - 4*3*3 = 0 => k^2 = 36 => k = +/- 6.

For real and equal roots, the discriminant D = b^2 - 4ac = 0. Here, (-2*sqrt(5))^2 - 4*k*4 = 0. 20 - 16k = 0, so 16k = 20, k = 20/16 = 5/4.

A quadratic equation with roots r1 and r2 is given by x^2 - (r1 + r2)x + (r1 * r2) = 0. Here, r1 = -3, r2 = 4. Sum = 1, Product = -12. Equation: x^2 - x - 12 = 0.

For a quadratic equation ax^2 + bx + c = 0 to have real and equal roots, the discriminant D = b^2 - 4ac must be zero. Here, (3k)^2 - 4(9)(4) = 0, so 9k^2 = 144, k^2 = 16, and k = +/- 4.

For equal roots, the discriminant b^2 - 4ac must be 0. Here, (-8)^2 - 4(p)(2p) = 0 => 64 - 8p^2 = 0 => p^2 = 8 => p = +/- 2*sqrt(2).

For a quadratic equation ax^2 + bx + c = 0 to have real equal roots, the discriminant D = b^2 - 4ac must be zero. Here, 3^2 - 4(2)(k) = 0, so 9 - 8k = 0, which means k = 9/8.