Let f(x) = 3x^2 – 2x + n and g(x) = mx^2 – nx + 2. The functions are combined to form the new
functions h(x) = f(x) - g(x) and j(x) = f(x) + g(x). Point (6, 2) is in the function h(x), while the point
(-2, 10) is in the function j(x). Determine the exact values of m and n.
can you post a full solution?
So, doing the combinations,
h(x) = ( 3x^2 – 2x + n)-(mx^2 – nx + 2) = (3-m)x^2 - (n+2)x + (n+2)
Since h(6) = 2, that gives
(3-m)*6^2 - (n+2)*6 + (n+2) = 2
36m+5n = -96
Now do the same with j(x). That will give you two equations to solve for m and n.
To determine the exact values of m and n, we can use the information given about the points on the functions h(x) and j(x).
First, let's analyze the given point (6, 2) in the function h(x). We know that h(x) is equal to f(x) - g(x), so we substitute the x-coordinate of the point (6) into the equation:
h(6) = f(6) - g(6)
Next, we substitute the function expressions for f(x) and g(x):
2 = (3(6^2) - 2(6) + n) - (m(6^2) - n(6) + 2)
Simplifying further:
2 = (108 - 12 + n) - (36m - 6n + 2)
Now, let's analyze the given point (-2, 10) in the function j(x). We know that j(x) is equal to f(x) + g(x), so we substitute the x-coordinate of the point (-2) into the equation:
j(-2) = f(-2) + g(-2)
Substituting the function expressions for f(x) and g(x):
10 = (3(-2^2) - 2(-2) + n) + (m(-2^2) - n(-2) + 2)
Simplifying further:
10 = (12 + 4 + n) + (4m + 2n + 2)
We now have a system of two equations with two unknowns (m and n):
(1) 2 = 108 - 12 + n - 36m + 6n - 2
(2) 10 = 12 + 4 + n + 4m + 2n + 2
Let's simplify equation (1):
2 = 94 - 36m + 7n
And simplify equation (2):
10 = 18 + 4m + 3n
Now, we have a system of two linear equations:
(1) -36m + 7n = -92
(2) 4m + 3n = -8
We can solve this system of equations to find the exact values of m and n. For example, we can use the method of substitution or elimination.
Using the method of elimination:
Multiply equation (2) by 9 to cancel the n term:
36m + 27n = -72
Add equation (1) and the modified equation (2):
-36m + 7n + 36m + 27n = -92 - 72
34n = -164
n = -4
Substitute the value of n into equation (1):
-36m + 7(-4) = -92
-36m - 28 = -92
-36m = -64
m = 64/36
m = 8/9
Therefore, the exact values of m and n are m = 8/9 and n = -4.