Given the equation x ^ 2 + 3 x + m = 0 at what value of m: 1) the difference between the roots of this equation

Given the equation x ^ 2 + 3 x + m = 0 at what value of m: 1) the difference between the roots of this equation will be equal to 6 2) one of the roots of the equation will be 2 times larger than the other?

x ^ 2 + 3 * x + m = 0. Let us write down the condition for the roots x1 and x2, (x1 – x2) = 6; Since according to the root theorem, x1 + x2 = -3; x1 * x2 = m; x1 = x2 + 6; x1 + x2 = -3; or x2 + 6 + x2 = -3; 2 * x2 + 6 = -3;, whence 2 * x2 = -9; x2 = -9/2 = -4.5. x1 = -4.5 + 6 = 1.5.

Then m = x1 * x2 = 1.5 * (-4.5) = -6.75.

Answer: m = -6.75.