When y = - x ^ 2 - (M-3) x + m is known, the distance between the two intersections of the parabola and X axis is equal to 3

When y = - x ^ 2 - (M-3) x + m is known, the distance between the two intersections of the parabola and X axis is equal to 3

Let two roots be x 1 x 2. As long as the absolute value of (x 1-x 2) is 3, it is equivalent to (x 1-x 2) = 9 (x 1 + x 2) - 4 x 1 x 2 = 9. By using Weida's theorem (3-m) - 4 M = 9 m = 10 or 0 △ 0 (M-3) + 4 m > 0, both of them hold, so m = 0 or 10

Given that the distance between the square of parabola y = x-mx-1 and the two intersections of x-axis is 4, find the value of M

The point of intersection with X axis means y = 0
So,
x^2-mx-1=0
Let W and y be two different roots of the equation, and W > = y
So w-y = 4
Here we can subtract two root formulas
It is concluded that [radical (b ^ 2-4ac)] / a = 4
Substituting coefficients into
[radical (m ^ 2-4 * - 1)] / 1 = 4
Root sign (m ^ 2 + 4) = 4
m^2+4=16
m^2=12
M = positive and negative root sign 12
So the value of M is the positive and negative root sign 12

Given that the distance between the image of parabola y = x square + MX + N and the two intersections of X axis is 7, find the value of M?

Let the coordinates of two intersections be (a, 0), (B, 0)
According to the meaning of the title, | A-B | = 7
According to Weida's theorem
a+b=-m
ab=n
(a-b)^2=m^2-4n
m^2-4n=49
m=±√(49+4n)

If the parabola y = 2x2 + 8x + m has only one common point with the X axis, then the value of M is______ ．

There is only one common point between ∵ parabola and x-axis, ∵ Δ = 0, ∵ b2-4ac = 82-4 × 2 × M = 0; ∵ M = 8