Let f(x) = 3x2– 2x + n and g(x) = mx2 – 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
Reiny 's calculation is correct to the step:
12 + 4 m + 4 + 2 n + n + 2 = 10
It should be calculated as follows:
12 + 4 m + 4 + 3 n + 2 = 10
12 + 4 + 2 + 4 m + 3 n = 10
18 + 4 m + 3 n = 10
4 m + 3 n = 10 - 18
4 m + 3 n = - 8
Now you must solve system of two equation:
36 m - 7 n = 92 , 4 m + 3 n = - 8
4 m + 3 n = - 8
Multiply both sides by 9
36 m + 27 n = - 72
36 m - 7 n = 92
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36 m + 27 n = - 72
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36 m - 36 m - 7 n - 27 n = 92 - ( - 72 )
- 34 n = 92 + 72
- 34 n = 164
n = 164 / - 34
n = - 2 ∙ 82 / 2 ∙ 17
n = - 82 / 17
Replace n with - 82 / 17 in equation:
4 m + 3 n = - 8
4 m + 3 ∙ ( - 82 / 17 ) = - 8
4 m - 246 / 17 = - 8
Add 246 / 17 to both sides
4 m = - 8 + 246 / 17
4 m = - 136 / 17 + 246 / 17
4 m = 110 / 17
Divide both sides by 4
m = 110 / 68
m = 2 ∙ 55 / 2 ∙ 34
m = 55 / 34
The solutions are:
m = 55 / 34 , n = - 82 / 17
Check of result:
h(x) = x² ∙ ( 3 - m ) + ( n - 2 ) ∙ ( x + 1 )
for x = 6
h(6) = 6² ∙ ( 3 - 55 / 34 ) + ( - 82 / 17 - 2 ) ∙ ( 6 + 1 ) =
36 ∙ ( 102 / 34 - 55 / 34 ) + ( - 82 / 17 - 34 / 17 ) ∙ 7 =
36 ∙ 47 / 34 + ( - 116 / 17 ) ∙ 7 =
1692 / 34 - ( 116 / 17 ) ∙ 7 =
2 ∙ 846 / 2 ∙ 17 - 812 / 17 = 846 / 17 - 812 / 17 =
34 / 17 = 2
j(x) = x² ∙ ( 3 + m ) + ( n + 2 ) ∙ ( 1 - x )
for x = - 2
j( - 2 ) = ( - 2 )² ∙ ( 3 + 55 / 34 ) + ( - 82 / 17 + 2 ) ∙ [ ( 1 - ( - 2 ) ] =
4 ∙ ( 102 / 34 + 55 / 34 ) + ( - 82 / 17 + 34 / 17 ) ∙ ( 1 + 2 ) =
4 ∙ 157 / 34 + ( - 48 / 17 ) ∙ 3 =
628 / 34 - 3 ∙ 48 / 17 =
2 ∙ 314 / 2 ∙ 17 - 144/ 17 =
314 / 17 - 144/ 17 =
170 / 17 = 10
By the way my nickname is Bosnian.
I forgot to write nick.
To determine the values of m and n, we will use the given information about the points (6, 2) and (-2, 10) that lie on the respective functions h(x) and j(x).
1. Point (6, 2) lies on the function h(x), which is defined as h(x) = f(x) - g(x).
Substituting the x-coordinate (6) and the y-coordinate (2) into the equation, we get:
2 = f(6) - g(6)
To find the value of f(6), substitute x = 6 into the equation f(x):
f(6) = 3(6)^2 - 2(6) + n
Simplifying,
f(6) = 108 - 12 + n
f(6) = 96 + n
Similarly, to find the value of g(6), substitute x = 6 into the equation g(x):
g(6) = m(6)^2 - n(6) + 2
g(6) = 36m - 6n + 2
Substituting the values of f(6) and g(6) back into the equation for h(x), we have:
2 = 96 + n - (36m - 6n + 2)
Simplifying,
2 = 98 + 7n - 36m
This gives us equation (1): 7n - 36m = -96
2. Point (-2, 10) lies on the function j(x), which is defined as j(x) = f(x) + g(x).
Substituting the x-coordinate (-2) and the y-coordinate (10) into the equation, we get:
10 = f(-2) + g(-2)
To find the value of f(-2), substitute x = -2 into the equation f(x):
f(-2) = 3(-2)^2 - 2(-2) + n
Simplifying,
f(-2) = 12 + 4 + n
f(-2) = 16 + n
Similarly, to find the value of g(-2), substitute x = -2 into the equation g(x):
g(-2) = m(-2)^2 - n(-2) + 2
Simplifying,
g(-2) = 4m + 2n + 2
Substituting the values of f(-2) and g(-2) back into the equation for j(x), we have:
10 = 16 + n + (4m + 2n + 2)
Simplifying,
10 = 18 + 3n + 4m
This gives us equation (2): 3n + 4m = -8
To solve for the values of m and n, we can solve equations (1) and (2) simultaneously.
By multiplying equation (1) by 3 and equation (2) by 7, we obtain a system of two equations:
21n - 108m = -288 (equation 1 multiplied by 3)
21n + 28m = -56 (equation 2 multiplied by 7)
Now, subtract equation (2) from equation (1):
-136m = -232
Dividing both sides by -136, we find:
m ≈ 1.7059
Substituting this value of m into equation (2), we have:
3n + 4(1.7059) = -8
3n + 6.8236 = -8
3n = -14.8236
n ≈ -4.9412
Therefore, the approximate values of m and n are m ≈ 1.7059 and n ≈ -4.9412, respectively.
h(x) = f(x) - g(x)
= 3x^2 - 2x + n -(mx^2 - nx + 2) = x^2(3 - m) + x(n - 2) + n - 2
(6,2) lies on it, so
36(3-m) + 6(n-2) + n-2 = 2
108 - 36m + 6n - 12 + n - 2 = 2
36m -7n = 92 **
j(x) = 3x^2 - 2x + n + (mx^2 - nx + 2) = x^2(3+m) - x(2 + n) + n + 2
for (-2,10)
4(3+m) + 2(2+n) + n + 2 = 10
12 + 4m + 4 + 2n + n + 2 = 10
4m + 7n = -8 ***
add ** and ***
40m = 84
m = 21/10
in ** 36(21/10) - 7n = 92
n = -82/35
check my algebra and my arithmetic, was expecting nicer numbers