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Julli [10]
3 years ago
5

Find an m > 0 such that the the equation x^4−(3m+2)x^2+m^2=0 has four real solutions that form an arithmetic sequence.

Mathematics
1 answer:
Aleonysh [2.5K]3 years ago
4 0

Answer:

The value of m is 6.

Step-by-step explanation:

Here, the given equation,

x^4-(3m+2)x^2+m^2=0

x^4+0x^3-(3m+2)x^2+0x+m^2=0

Let the roots of the equation are a-3b, a-b, a+b and a + 3b, ( they must be form an AP )

Thus, we can write,

a-3b+a-b+a+b+a+3b=\frac{\text{coefficient of }x^3}{\text{coefficient of }x^4}

=\frac{0}{1}=0

\implies a=0----(1)

(-3b)(-b)+(-b)(b)+(b)(3b)+(3b)(-3b)+(-b)(3b)+(-3b)(b)=\frac{\text{coefficient of }x^2}{\text{coefficient of }x^4}}

=\frac{-3m-2}{1}

3b^2-b^2+3b^2-9b^2-3b^2-3b^2=-3m-2

-10b^2=-3m-2

\implies b^2=\frac{3m+2}{10}-----(2)

(-3b)(-b)(b)(3b)=\frac{\text{Constant term}}{\text{coefficient of}x^4}= m^2

9b^4=m^2

9(\frac{3m+2}{10})^2=m^2

9(\frac{9m^2+4+12m}{100})=m^2

81m^2+36+108m=100m^2

-19m^2+108m+36=0

19m^2-108m-36=0

19m^2-114m+6m-36=0

19m(m-6)+6(m-6)=0

(19m+6)(m-6)=0

\implies m=-\frac{6}{19}\text{ or }m=6

But m > 0,

Hence, the value of m is 6.

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2. Lorena is trying to choose between two different savings accounts at H-Town Bank. She will deposit $3,500 dollars and will le
Ivahew [28]

Answer:

First of all, this question is not complete. So, we will be hypothetically create two accounts and try to choose the one with the greater profit.

Choice 1 = X = Final Amount after 12 years = $5600

Choice 2 = Y = Final Amount after 12 years = $6285.50

Hence, Lorena should choose choice 2, as it gives more profit over than choice 1.

Step-by-step explanation:

First of all, this question is not complete. So, we will be hypothetically create two accounts and try to choose the one with the greater profit.

Initial Deposit Amount = $3500

Time period = 12 years

Let's Suppose Lorena has two X and Y choices of accounts at H-town Bank.

X = simple interest of 5% over the initial amount for 12 years.

Y = Compound interest 5% over the initial amount for 12 years.

Solution:

Let's calculate the profit for choice 1 = X

Simple interest = 5%

Initial amount = $3500

Time period = 12 years

Formula = A = P (1 + rt)

where, A = Final amount after 12 years

t = time period in years  = 12

P = initial amount to be deposited = $3500

r = annual interest rate = 5%

Plug in the values to get the total amount.

A = 3500(1+0.05x12)

A = $5600 = final amount Lorena will get after 12 years.

Let's calculate the profit for choice 2 = Y

Compound interest = 5%

Initial amount = $3500

Time period = 12 years

Formula = A = P (\frac{r}{n} + 1)^{nt}

where,

A = Final amount after 12 years

t = time period in years  = 12

P = initial amount to be deposited = $3500

r = annual interest rate = 5%

n = number of times interest rate is applied or compounded. = 12

Plug in the values to get the total amount.

A = P (\frac{r}{n} + 1)^{nt}

A = $6285.50

Hence, Lorena should choose choice 2, as it gives more profit over than choice 1.

4 0
2 years ago
Which of the following graphs represents the function f(x) = x2 + x − 6?
Darya [45]

The answer is: D


Step-by-step explanation:

Direction: Opens Up

Vertex: ( −1/2, −25/4)

Focus: (−1/2, −6)

Axis of Symmetry: x = -12

Directrix:    y = -132                        


3 0
3 years ago
Read 2 more answers
Values of k for which the quadratic equation 2x - kx + k 0 has equal roots is
WARRIOR [948]

Answer:

The answer is D.

Step-by-step explanation:

We have to apply Discriminant Law. When a quadratic equation, ax² + bx + c = 0 has equal roots so the discriminant will be 0. Then, you have to substitute the values into the formula :

D =  {b}^{2}  - 4ac

let   \: D = 0 ,a = 2,b = k,c = k

0 =  {k}^{2}  - 4(2)(k)

{k}^{2}  - 8k = 0

k(k - 8) = 0

k = 0

k - 8 = 0

k = 8

4 0
3 years ago
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Answer:

$780

Step-by-step explanation:

15x60 = 900   900 - 120 = $780

6 0
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Answer: 1/6th

Step-by-step explanation:idk fr

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