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
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
3 answers
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
-
36 m + 27 n = - 72
_______________
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
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
-
36 m + 27 n = - 72
_______________
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.
I forgot to write nick.