Please answer this question

masha68 [24]

Answer:

$\displaystyle f(x) = 3x^2 + 2x + 5\text{ and } g(x) =2x^2 - 4x -2\text{ or } \\ \\ f(x) = 3x^2 + 5 \text{ and } g(x) = x^2 - 4x -2$

Step-by-step explanation:

We are given the two functions:

$\displaystyle f(x) = 3x^2 + mx +5 \text{ and } g(x) = nx^2 - 4x -2$

And that:

$\displaystyle h(x) = f(x)\cdot g(x)$

With the given conditions that (1, -40) and (-1, 24) satisfy the new function, we want to determine functions f and g.

First, find h:

$\displaystyle \begin{aligned} h(x) & = f(x)\cdot g(x) \\ \\ & = (3x^2 + mx +5)(nx^2 - 4x -2) \end{aligned}$

Because (1, -40) and (-1, 24) are points on the graph of h, we have that h(-1) = 40 and h(-1) = 24. In other words:

$\displaystyle \begin{aligned} h(1) = -40 & = (3(1)^2 + m(1) +5)(n(1)^2 - 4(1) -2) \\ \\ & = (3 + m +5)(n-4 -2) \\ \\ & = (m+8)(n-6) \\ \\ -40 &= mn-6m+8n-48 \end{aligned}$

And:

$\displaystyle \begin{aligned} h(-1) = 24 & = (3(-1)^2 + m(-1) +5)(n(-1)^2 -4(-1) -2) \\ \\ & = (3 - m +5)(n + 4 -2) \\ \\ & = (-m+8)(n+2) \\ \\ 24 & = -mn -2m + 8n +16 \end{aligned}$

Solve the system of equations. Adding the two equations together yield:

$\displaystyle -16 = -8m+16n - 32$

Solve for either m or n:

$\displaystyle \begin{aligned} -16 & = -8m + 16n - 32 \\ \\ 16 & = -8m + 16n \\ \\ 8m & = 16n - 16 \\ \\ m & = 2n -2\end{aligned}$

Substitute this into one of the two equations above and solve:

$\displaystyle \begin{aligned} -40 & = mn - 6m + 8n - 48 \\ \\ 0 & = (2n-2)n -6 (2n-2) + 8n -8 \\ \\ &= (2n^2 - 2n) + (-12n + 12) +8 n - 8 \\ \\ & = 2n^2 -6n + 4 \\ \\ & = n^2 - 3n + 2 \\ \\ &= (n-2)(n-1) \\ \\ & \end{aligned}$

Therefore:

$\displaystyle n = 2 \text{ or } n = 1$

Solve for m:

$\displaystyle \begin{aligned}m &= 2n-2 & \text{ or } m & = 2n-2 \\ \\ & = 2(2) - 1 &\text{ or } & =2(1) -2 \\ \\ &= 2 &\text{ or } & = 0 \end{aligned}$

Hence, the values of n and m are either: 2 and 2, respectively; or 1 and 0, respectively.

In conclusion, functions f and g are:

$\displaystyle f(x) = 3x^2 + 2x + 5\text{ and } g(x) =2x^2 - 4x -2\text{ or } \\ \\ f(x) = 3x^2 + 5 \text{ and } g(x) = x^2 - 4x -2$

7 0

2 years ago

What is the equation of a line, in general form, with a slope of -2 and a y-intercept of 8?

Amanda [17]

Slope of -2, y int of 8....

y = -2x + 8
2x + y - 8 = 0 <== general form

8 0

4 years ago

Read 2 more answers

24. The farmer's market has strawberries that sell for $5 per pound and cherries that sell for $3 a pound. Write an

mylen [45]

Answer:

5s+3c=30

Step-by-step explanation:

6 0

3 years ago

The half-life of Potassium-15 is 1,250 years

mel-nik [20]

Half life = 1250 years
Start amount = 36
End amount = 15
Time = ?

15 = 36 * (1/2)^(x/1250)
0.417 = (1/2)^(x/1250)
ln(0.417) = x/1250 * ln(1/2)
x = 1577.95 years

8 0

2 years ago

Without graphing, compare the slopes and the intercepts of the

pishuonlain [190]

Answer:

graph f(x) passes through the y-axis at 1:00 whereas g(x) passes through the origin.

4 0

3 years ago

Please answer this question ​

Please answer this question