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.
Could someone please show me the full solutions, I've tried so many times so I want to see where I went wrong.
Did you not check with this?
https://www.jiskha.com/questions/1829690/let-f-x-3x2-2x-n-and-g-x-mx2-nx-2-the-functions-are-combined-to-form-the
so why don't you show us your work?
h(x) = (3x^2 – 2x + n)-(mx^2 – nx + 2) = (3-m)x^2 + (-2+n)x + (n-2)
j(x) = (3x^2 – 2x + n)+(mx^2 – nx + 2) = (3+m)x^2 + (-2-n)x + (n+2)
You know that h(6) = 2 and j(-2) = 10, so plug them in and you have
(3-m)*36 + (-2+n)*6 + (n-2) = 2
(3+m)*4 + (2+n)*2 + (n+2) = 10
or, more usefully,
-36m+7n = -92
4m+3n = -8
solve these (multiply #2 by 9 and add to eliminate x) and you get
m = 55/34
n = -82/17
To determine the exact values of m and n, we need to use the given points and the new functions h(x) and j(x). Let's start with the equation h(x) = f(x) - g(x).
We are given that the point (6, 2) is in the function h(x), which means that when x = 6, h(x) = 2. Substituting these values into the equation, we get:
2 = f(6) - g(6)
Next, let's find the values of f(6) and g(6):
f(6) = 3(6)^2 - 2(6) + n (Substituting x = 6 into f(x))
g(6) = m(6)^2 - n(6) + 2 (Substituting x = 6 into g(x))
Simplifying these equations, we have:
f(6) = 108 - 12 + n
g(6) = 36m - 6n + 2
Now let's substitute these values back into the equation h(x) = f(x) - g(x):
2 = 108 - 12 + n - (36m - 6n + 2)
Simplifying further, we get:
2 = 108 - 12 + n - 36m + 6n - 2
Combining like terms, we have:
2 = 94 - 30m + 7n
This is our first equation.
Now let's move on to the equation j(x) = f(x) + g(x).
We are given that the point (-2, 10) is in the function j(x), which means that when x = -2, j(x) = 10. Substituting these values into the equation, we get:
10 = f(-2) + g(-2)
Next, let's find the values of f(-2) and g(-2):
f(-2) = 3(-2)^2 - 2(-2) + n (Substituting x = -2 into f(x))
g(-2) = m(-2)^2 - n(-2) + 2 (Substituting x = -2 into g(x))
Simplifying these equations, we have:
f(-2) = 12 + 4 + n
g(-2) = 4m + 2n + 2
Now let's substitute these values back into the equation j(x) = f(x) + g(x):
10 = 12 + 4 + n + 4m + 2n + 2
Simplifying further, we get:
10 = 18 + 3n + 4m + 2n
Combining like terms, we have:
10 = 18 + 5n + 4m
This is our second equation.
Now we have a system of equations:
Equation 1: 2 = 94 - 30m + 7n
Equation 2: 10 = 18 + 5n + 4m
To solve this system, we can use substitution or elimination method.
By rearranging Equation 1, we have:
7n - 30m = -92 (Equation 3)
Now we can multiply Equation 2 by 15 to eliminate the n term:
150 = 270 + 75n + 60m
Simplifying further, we get:
15n + 12m = -120 (Equation 4)
Now we can solve Equations 3 and 4 as a system of linear equations.
Multiplying Equation 3 by 5 and Equation 4 by 7, we get:
35n - 150m = -460 (Equation 5)
105n + 84m = -840 (Equation 6)
Now we can use either substitution or elimination method to solve this system of linear equations. I will use the elimination method.
Multiplying Equation 5 by 3 and Equation 6 by 1, we get:
105n - 450m = -1380 (Equation 7)
105n + 84m = -840 (Equation 6)
Subtracting Equation 6 from Equation 7, we get:
(^1) 105n - 450m - (105n + 84m) = -1380 - (-840)
Simplifying further, we get:
(^2) - 450m - 84m = -1380 + 840
Combining like terms, we have:
-534m = -540
Dividing both sides by -534, we get:
m ≈ 1.0112
Now, substituting this approximate value of m back into Equation 6:
105n + 84(1.0112) = -840
Simplifying further, we get:
105n + 84.9341 = -840
Subtracting 84.9341 from both sides, we get:
105n ≈ -924.9341
Dividing both sides by 105, we get:
n ≈ -8.8088
Therefore, the approximate values for m and n are m ≈ 1.0112 and n ≈ -8.8088.
Please note that these values are approximate due to rounding during calculations. To obtain the exact values, it would be necessary to work with the original equations without approximation.