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3 years ago

Find all real zeros of 4x^3-20x+16

Mathematics

2 answers:

Pani-rosa [81]3 years ago

5 0

Answer:

{1, (-1±√17)/2}

Step-by-step explanation:

There are formulas for the real and/or complex roots of a cubic, but they are so complicated that they are rarely used. Instead, various other strategies are employed. My favorite is the simplest--let a graphing calculator show you the zeros.

___

Descartes observed that the sign changes in the coefficients can tell you the number of real roots. This expression has two sign changes (+-+), so has 0 or 2 positive real roots. If the odd-degree terms have their signs changed, there is only one sign change (-++), so one negative real root.

It can also be informative to add the coefficients in both cases--as is, and with the odd-degree term signs changed. Here, the sum is zero in the first case, so we know immediately that x=1 is a zero of the expression. That is sufficient to help us reduce the problem to finding the zeros of the remaining quadratic factor.

Using synthetic division (or polynomial long division) to factor out x-1 (after removing the common factor of 4), we find the remaining quadratic factor to be x²+x-4.

The zeros of this quadratic factor can be found using the quadratic formula:

a=1, b=1, c=-4

x = (-b±√(b²-4ac))/(2a) = (-1±√1+16)/2

x = (-1 ±√17)2

The zeros are 1 and (-1±√17)/2.

_____

The graph shows the zeros of the expression. It also shows the quadratic after dividing out the factor (x-1). The vertex of that quadratic can be used to find the remaining solutions exactly: -0.5 ± √4.25.

The given expression factors as ...

4(x -1)(x² +x -4)

Rzqust [24]3 years ago

4 0

Answer:

Step-by-step explanation:

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Answer:

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Step-by-step explanation:

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3 years ago

Read 2 more answers

2 factors that add to 6 but multiply to 60

hodyreva [135]

Answer:

The short answer is there isn’t.

Start by writing each of these as an expression:

x * y = 60

x + y = 7

Next, solve each for the same variable; in this case, y:

(x * y) / x = 60 / x

.: y = 60 / x

(x + y) - x = 7 - x

.: y = 7 - x

Next, replace y of the second expression to the first

y = 60 / x & y = 7 - x

.: 7 - x = 60 / x

Now, solve for x:

(7 - x) * x = (60 / x) * x

.: x * 7 - x^2 = 60

This is quadratic, so write it in the form of ax2 + bx + x = 0

(-1)x^2 + (7)x + (-60) = 0

.: a = -1, b = 7, c = -60

Finally solve for b:

x = (-b +- sqrt(b^2 - 4*a*c)) / 2a

.: x = (-7 +- sqrt(7^2 - 4*-1*-60)) / (2 * -1)

.: x = (-7 +- sqrt(49 - 240)) / -2

.: x = (-7 +- sqrt(-191)) / -2

The square root of a negative value is imaginary and thus there’s no real answer to this problem.

8 0

2 years ago

Miriam is picking out some movies to rent, and she is primarily interested in comedies and foreign films. She has narrowed down

Keith_Richards [23]

Answer:

There are 795 combinations.

Step-by-step explanation:

The number of ways or combinations in which we can select k element from a group of n elements is given by:

$nCk=\frac{n!}{k!(n-k)!}$

So, if Miriam want to choose 3 movies with at least two comedies, she have two options: Choose 2 comedies and 1 foreign film or choose 3 comedies.

Then, the number of combinations for every case are:

1. Choose 2 Comedies from the 10 and choose 1 foreign film from 15. This is calculated as:

$10C2*15C1=\frac{10!}{2!(10-8)!}*\frac{15}{1!(15-14)!}$

$10C2*15C1=675$

2. Choose 3 Comedies from the 10. This is calculated as:

$10C3=\frac{10!}{3!(10-3)!}=120$

Therefore, there are 795 combinations and it is calculated as:

675 + 120 = 795

8 0

3 years ago

Help me out here pleaseeeeeee

Sveta_85 [38]

Answer:

(4, 80) Which is L

Step-by-step explanation:

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6 0

3 years ago

A. Do some research and find a city that has experienced population growth.

horrorfan [7]

A. The city we will use is Orlando, Florida, and we are going to examine its population growth from 2000 to 2010. According to the census the population of Orlando was 192,157 in 2000 and 238,300 in 2010. To examine this population growth period, we will use the standard population growth equation $N_{t} =N _{0}e^{rt}$
where:
$N(t)$ is the population after $t$ years
$N_{0}$ is the initial population
$t$ is the time in years
$r$ is the growth rate in decimal form
$e$ is the Euler's constant
We now for our investigation that $N(t)=238300$ , $N_{0} =192157$ , and $t=10$ ; lets replace those values in our equation to find $r$ :
$238300=192157e^{10r}$
$e^{10r} = \frac{238300}{192157}$
$ln(e^{10r} )=ln( \frac{238300}{192157} )$
$r= \frac{ln( \frac{238300}{192157}) }{10}$
$r=0.022$
Now lets multiply $r$ by 100% to obtain our growth rate as a percentage:
$(0.022)(100)$ =2.2%
We just show that Orlando's population has been growing at a rate of 2.2% from 2000 to 2010. Its population increased from 192,157 to 238,300 in ten years.

B. Here we will examine the population decline of Detroit, Michigan over a period of ten years: 2000 to 2010.
Population in 2000: 951,307
Population in 2010: 713,777
We know from our investigation that $N(t)=713777$ , $N_{0} =951307$ , and $t=10$ . Just like before, lets replace those values into our equation to find $r$ :
$713777=951307e^{10r}$
$e^{10r} = \frac{713777}{951307}$
$ln(e^{10r} )=ln( \frac{713777}{951307} )$
$r= \frac{ln( \frac{713777}{951307}) }{10}$
$r=-0.029$
$(-0.029)(100)$ = -2.9%.
We just show that Detroit's population has been declining at a rate of 2.2% from 2000 to 2010. Its population increased from 192,157 to 238,300 in ten years.

C. Final equation from point A: $N(t)=192157e^{0.022t}$ .
Final equation from point B: $N(t)=951307e^{-0.029t}$
Similarities: Both have an initial population and use the same Euler's constant.
Differences: In the equation from point A the exponent is positive, which means that the function is growing; whereas, in equation from point B the exponent is negative, which means that the functions is decaying.

D. To find the year in which the population of Orlando will exceed the population of Detroit, we are going equate both equations $N(t)=192157e^{0.022t}$ and $N(t)=951307e^{-0.029t}$ and solve for t:
$192157e^{0.022t} =951307e^{-0.029t}$
$\frac{192157e^{0.022t} }{951307e^{-0.029t} } =1$
$e^{0.051t} = \frac{951307}{192157}$
$ln(e^{0.051t})=ln( \frac{951307}{192157})$
$t= \frac{ln( \frac{951307}{192157}) }{0.051}$
$t=31.36$
We can conclude that if Orlando's population keeps growing at the same rate and Detroit's keeps declining at the same rate, after 31.36 years in May of 2031 Orlando's population will surpass Detroit's population.

E. Since we know that the population of Detroit as 2000 is 951307, twice that population will be $2(951307)=1902614$ . Now we can rewrite our equation as: $N(t)=1902614e^{-0.029t}$ . The last thing we need to do is equate our Orlando's population growth equation with this new one and solve for t:
$192157e^{0.022t} =1902614e^{-0.029t}$
$\frac{192157e^{0.022t} }{1902614e^{-0.029t} } =1$
$e^{0.051t} = \frac{1902614}{192157}$
$ln(e^{0.051t} )=ln( \frac{1902614}{192157} )$
$t= \frac{ln( \frac{1902614}{192157}) }{0.051}$
$t=44.95$
We can conclude that after 45 years in 2045 the population of Orlando will exceed twice the population of Detroit.

8 0

3 years ago