Answer:
m = 3/4, n = -1/3
m = 4, n = -32.
Step by step calculation:
If the equations mx² + 5x + 2 = 0 and 3x² + 10x + n = 0 have a common root, then that means there exists some value of x that satisfies both equations simultaneously.
Let's call the common root of these equations "r". Then we can write:
mx² + 5x + 2 = 0 (1)
3x² + 10x + n = 0 (2)
r is a common root of (1) and (2), so we have:
mr² + 5r + 2 = 0 (3)
3r² + 10r + n = 0 (4)
We want to find the values of m and n that satisfy both equations (1) and (2), which means we want to find the value of r that satisfies both equations (3) and (4).
To do this, we can use the fact that r is a common root of both equations (3) and (4), which means we can set them equal to each other:
mr² + 5r + 2 = 3r² + 10r + n
Simplifying this equation, we get:
(m - 3)r² + (5 - 10)r + (2 - n) = 0
(m - 3)r² - 5r + (n - 2) = 0
Since r is a common root of both equations (3) and (4), this equation must have a solution for r. That means the discriminant of the quadratic equation must be zero:
(-5)² - 4(m - 3)(n - 2) = 0
Simplifying this equation, we get:
25 - 4mn + 8m + 12n - 24 = 0
4mn - 8m - 12n + 1 = 0
We now have two equations in two variables (m and n). We can solve for one variable in terms of the other and substitute into the other equation to obtain a quadratic equation in one variable.
Let's solve for n in terms of m:
4mn - 8m - 12n + 1 = 0
n = (4m - 1)/(3m - 4)
Substituting this into the other equation, we get:
25 - 4mn + 8m + 12n - 24 = 0
25 - 4m(4m - 1)/(3m - 4) + 8m + 12(4m - 1)/(3m - 4) - 24 = 0
Simplifying this equation, we get:
-4m² + 49m - 12 = 0
Using the quadratic formula to solve for m, we get:
m = 3/4 or m = 4
Substituting these values back into the equation we found for n in terms of m, we get:
If m = 3/4, then n = -1/3
If m = 4, then n = -32
Therefore, the possible values of m and n that make the equations mx² + 5x + 2 = 0 and 3x² + 10x + n = 0 have a common root are:
m = 3/4, n = -1/3
m = 4, n = -32.
This is quite confusing question