f(x)=x^2-mx+5 and g(x)=nx^2+x-3. The functions are combined to form the new function h(x)=f(x)+g(x). Points (1,3) and (-2,18) satisfy the new function. Determine the values of m and n.
To determine the values of m and n, we need to substitute the given points into the function h(x) and solve the resulting equations.
Let's substitute the point (1,3) into the function h(x):
h(x) = f(x) + g(x)
3 = f(1) + g(1)
Now, let's substitute the point (-2,18) into the function h(x):
h(x) = f(x) + g(x)
18 = f(-2) + g(-2)
To solve these equations, we need to find the values of f(1), g(1), f(-2), and g(-2).
First, let's find f(1):
f(x) = x^2 - mx + 5
f(1) = (1)^2 - m(1) + 5
f(1) = 1 - m + 5
f(1) = 6 - m
Similarly, let's find g(1):
g(x) = nx^2 + x - 3
g(1) = n(1)^2 + (1) - 3
g(1) = n + 1 - 3
g(1) = n - 2
Now, let's find f(-2):
f(x) = x^2 - mx + 5
f(-2) = (-2)^2 - m(-2) + 5
f(-2) = 4 + 2m + 5
f(-2) = 9 + 2m
Similarly, let's find g(-2):
g(x) = nx^2 + x - 3
g(-2) = n(-2)^2 + (-2) - 3
g(-2) = 4n - 2 - 3
g(-2) = 4n - 5
Now, substitute these values back into the equations:
3 = f(1) + g(1)
3 = (6 - m) + (n - 2)
18 = f(-2) + g(-2)
18 = (9 + 2m) + (4n - 5)
Simplify these equations:
3 = 6 - m + n - 2
18 = 9 + 2m + 4n - 5
Now, group the constants together and the variables together:
3 = 4 + n - m
18 = 4n + 2m + 4
Simplify further:
n - m = -1
2m + 4n = 14
Now, we have a system of linear equations. We can solve this system using any method, such as substitution or elimination.
Let's solve it using the substitution method:
From the first equation, n - m = -1, we can solve for n:
n = m - 1
Substitute n = m - 1 into the second equation:
2m + 4(m - 1) = 14
2m + 4m - 4 = 14
6m - 4 = 14
6m = 18
m = 3
Now, substitute m = 3 back into n = m - 1:
n = 3 - 1
n = 2
Therefore, the values of m and n are m = 3 and n = 2.