2. The equation x² + px + q = 0, where p and q are constants, has roots -3 and 5.
(i) Find the values of p and q.

(ii) Using these values of p and q, find the value of the constant r for which the equation
x² + px + q + r = 0 has equal roots.
[3]

Answers

Answer 1

The result are-

p = -2 and q = -15.constant r = 64.What is defined as the roots of the quadratic equation?

The values of x that correspond to the given quadratic equation ax² + bx + c = 0 are known as its roots.

They are, in other words, the values of the variable (x) that satisfy the equation. The roots of such a quadratic function are indeed the x-coordinates of the function's x-intercepts.The discriminant value indicates how often roots f(x) has: - The quadratic function does have two distinct real roots if b² - 4ac > 0. The quadratic function contains a single repeated real root if b² - 4ac = 0. If b² - 4ac = 0, the quadratic function has really no true roots.

For, the given expression;

x² + px + q = 0.      ....... (equation 1)

where p and q are constants, has roots -3 and 5.

Part 1: solve for p and q.

Put x = -3 in equation 1.

p(-3) = (-3)² + p(-3) + q = 0

9 - 3p + q = 0         .........(equation 2)

Now, put x = 5 in equation 1.

p(5) = (5)² + p(5) + q = 0

25 + 5p + q = 0     ..........(equation 2)

Solve equation 2 and 3 by elimination method.

p = -2 and q = -15.

Part 2: Estimate the constant r.

The equation is; x² + px + q + r = 0.

Substitute the values of p = -2 and q = -15.

x² + (-2)x + (-15) + r = 0.

As, the equation given real equal roots; then,

discriminant value: b² - 4ac = 0.

Substitute the values;

(-2)² - 4×1×(-15 + r) = 0

On simplification;

4 + 60 - r = 0

r = 64.

Therefore, the value of the constant r is estimated as 64.

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Related Questions

Monica has a goal to complete a triathlon. The race consists of a 2.4-mile swim, a 112-mile bike ride, and a 26.2-mile run. In training, Monica can swim at an average pace of 1.8mph. She rides her bike at an average of 19.2mph and runs at 4.8mph. Estimate how long it will take her to complete the triathlon.

Answers

Taking into account the definition of speed, Monica will complete the triathlon in 12.62 hours.

Definition of speed

Speed can be defined as the amount of space traveled per unit of time with which a body moves and can be calculated using the expression:

speed= distance traveled÷ time

Time to complete the triathlon

In this case, you know

Swim:speed= 1.8 mphdistance traveled= 2.4 milesRide a bikespeed= 19.2 mphdistance traveled= 112 milesRun:speed= 4.8 mphdistance traveled= 26.2 miles

Replacing these values ​​in the speed definition:

Swim: 1.8 mph= 2.4 miles÷ time to swimRide a bike: 19.2 mph= 112 miles÷ time to ride a bikeRun: 4.8 mph= 26.2 miles÷ time to run

Solving:

Swim: 1.8 mph×time to swim= 2.4 miles

time to swim= 2.4 miles÷ 1.8 mph

time to swim= 1.33 h

Ride a bike: 19.2 mph× time to ride a bike = 112 miles

time to ride a bike= 112 miles÷ 19.2 mph

time to ride a bike=5.83 h

Run: 4.8 mph× time to run= 26.2 miles

time tu run= 26.2 miles÷ 4.8 mph

time to run= 5.46 h

The time to complete the triathlon is the sum of each of the stages:

Total time= time to swim + time to ride a bike + time to run

Total time= 1.33 h + 5.83 h + 5.46 h

Total time= 12.62 h

In summary, Monica will complete the triathlon in 12.62 hours.

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Satisfactory internal control ______ the probability of errors or frauds in the accounts.

Answers

Satisfactory internal control reduces the probability of errors or frauds in the accounts.

What functions may a strong internal control system perform?

To ensure that goals and objectives are met, effective internal controls are necessary. For management choices, they offer trustworthy financial reports. To reduce the possibility of public scandals, they make sure that applicable laws and regulations are followed.

Why are internal controls necessary?

Internal controls are primarily used to protect organizations and advance their goals. Internal controls serve to reduce risks, safeguard assets, maintain record accuracy, enhance operational effectiveness, and promote conformity to laws, rules, and regulations.

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∠ABC and ∠DEF are complementary. If m∠ABC = (2x+4)∘ and m∠DEF = (3x−5)∘, find the measure of each angle.

m∠ABC =

m∠DEF =

Answers

The measure of each angle are m∠ABC =  40.4 and m∠DEF =  49.6

How to determine the measure of each angle?

The given parameters are

m∠ABC = (2x+4)∘ and m∠DEF = (3x−5)∘,

Complementary angles add up to 90 degrees

So, we have

2x + 4 + 3x - 5 = 90

Evaluate the like terms

5x = 91

Divide by 5

x = 18.2

Substitute x = 18.2 in m∠ABC = (2x+4)∘ and m∠DEF = (3x−5)∘,

m∠ABC = (2 x 18.2 + 4) = 40.4

m∠DEF = (3 x 18.2 − 5)∘ = 49.6

Hence, the measure of each angle are m∠ABC =  40.4 and m∠DEF =  49.6

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Without multiplying, tell whether the value of the expression is positive or negative.
4/7 • (-3 1/2)
Explain your reasoning.

Answers

Answer:

Negative

Step-by-step explanation:

Because knowing that a positive number times a negative number equals negative.

Which driver's race times were the most variable if Driver A has a mean of 4.01 seconds and a standard deviation of 0.05 seconds and Driver B has a mean of 3.96 seconds and a standard deviation of 0.12 seconds and Driver C has a mean of 3.99 seconds and a standard deviation of 0.19 seconds?

Answers

The driver B has the fastest typical race time.

It is required to find which driver's race times were the most variable.

What is a statement?

A statement is a declarative sentence that is either true or false but not both. A statement is sometimes called a proposition. The key is that there must be no ambiguity. To be a statement, a sentence must be true or false, and it cannot be both.

Given:

From the options presented, driver B has the fastest typical race time because he recorded the smallest time taken to complete an individual race.

Also from the standard deviation given 3.96  seconds  for driver B, it shows that the individual time spread from the standard deviation is minimal and that the driver maintained a fairly consistent time of race throughout the racing period.

Therefore, the driver B has the fastest typical race time.

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Question 2 of 10
Which situation shows a constant rate of change?
A. The amount raised by a booster club compared with the number
of raffle tickets sold
B. The weight of a kitten compared with its age in months
C. The number of goals scored in a soccer game compared with the
minutes played
D. The height of a paper airplane over time

Answers

The amount raised by a booster club compared with the number of  raffle tickets sold shows a constant rate of change;

A constant rate of change is realized when a change in one situation always results in a change in another. In the scenario of a booster club that regularly sells raffle tickets, the more tickets it sells, the higher the club gets.

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Which best describes the triangle?

Answers

It being a pyramid need more information
i cant see the question???

write 26/50 as a decimal fraction​

Answers

Answer:

13/25

Step-by-step explanation:

hope this helps :)

Which of the following numbers can be expressed as decimals that terminate? (1 point) 5 over 2 , 4 over 5 , 2 over 7 , 4 over 3 5 over 2 and 2 over 7 4 over 5 and 4 over 3 5 over 2 and 4 over 5 2 over 7 and 4 over 3

Answers

Answer:

all of them are terminating decimals

Step-by-step explanation:

5/2=2.5

4/5=1.25

2/7=3.5

4/5=1.25

4/3=0.75

5/2=2.5

4/5=1.25

2/7=3.5

4/3=0.75

they are terminating decimals because they have a finite end

a²-7a²b² + b² factor​

Answers

You can’t factor that any further…

an amusement park charges 18 dollars for 4 rides. how much will 7 rides cost

Answers

Answer: $31.5
Explanation: 18 divided by 4 = 4.5
4.5 X 7 rides = $31.5

Help!!!

6 divided by 2(1+2)

Answers

Answer:

9

Step-by-step explanation:

6 / 2(1+2)

6 / 2 (3)

6 / 2 * 3

3 * 3

9

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About 70% of people are right-handed. Of 352students in a school. Which is a reasonable number of right-handed students ?

Answers

Answer: 246 right-handed students

Step-by-step explanation:

70%=

70/100=0.7

352*0.7=246.4

246.4=246 right-handed students   ==> I rounded the answer since you can have 246 students and an additional 0.4 students. You can't have part of a student.

Eric rented a truck for one day. There was a base fee of 19.95 and there was an additional charge of 97 cents for each mile driven. Eric had to pay 190.67 when he returned to the truck. For how many miles did he drive the truck?

Answers

Answer: 176 miles

Step-by-step explanation:

m=mile.

97 cents=

97/100 dollars=$0.97

190.67=19.95+0.97m

170.72=0.97m

m=176 miles

Apply the rule for negative exponents to select the equivalent expression. 8-1 08 8X -8​

Answers

The equivalent expression of 8^-1 is 1/8

How to apply the rule for negative exponents?

The expression is given as

8^-1

The above represents 8 exponent -1

The rule for negative exponents states that

a^-1 = 1/a

This means that

8^-1 = 1/8

Hence, the equivalent expression of 8^-1 is 1/8

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1. 60 high school seniors taking AP classes (advanced classes for college credit) were polled (asked a question) to see if they were taking European History or US History. A total Of 29 students said they were taking AP European History and a total of 50 students sald they were taking AP US History. How many students take both AP European History and AP US History? (Hint: Draw a venn diagram.)

Answers

The number of students take both AP European History and AP US History is 19 students


Venn Diagram

A Venn diagram is an illustration of the relationships between and among sets, groups of objects that share something in common.

Let the students take both AP European History and AP US History be x such that;

n(U) = 29 - x + 50 - x

If 60 high school seniors taking AP classes, then n(U) = 60. Substitute to have:

60 = 29 - x + x + 50 - x

60 = 79 - x

-x = 60 - 79

-x =-19

x = 19

Therefore the number of students take both AP European History and AP US History is 19 students

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A taxi ride costs p40.00 for the first 500 meters and each additional 300 meters or a fraction there of adds p3.50 to the fare.use a piecewise function to represent the taxi fare in terms of distance d in meter?

Answers

The function which represents the taxi fare in terms of distance d in meter is f(d) = 3.50 ×[tex]\frac{d-500}{300}[/tex] + 40 .

According to the question,

For first 500 meters cost is  40.00 and For each additional 300 meters cost is 3.50 .

Let, the total distance of taxi ride = d meters

Now, representing the equation of taxi fare in terms of distance when distance is less than 500

taxi fare = 3.50 ×  [tex]\frac{d-500}{300}[/tex]  + 40

∴  f(d) = 3.50 ×  [tex]\frac{d-500}{300}[/tex] + 40

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what is the unit rate for 1/4 kilometer in 1/3 hours

Answers

Answer: 0.75km

Step-by-step explanation:

To find a baseball pitcher's earned run average (era), you can use the formula ei = 9r, in which e represents era, i represents the number of innings pitched, and r represents the number of earned runs allowed. solve the equation for e. what is a pitcher's era if he allows 33 earned runs in 45 innings pitched? if necessary, round your answer to the nearest hundredth.

Answers

The value of e is 33/5 or 6.6 era.

It is given in the question that:-

e*i = 9r

Where,

e represents era

i represents the number of innings pitched, and

r represents the number of earned runs allowed

Also, we are given that:-

i = 45 innings, and

r = 33 earned runs allowed

We have to find the value of e for the given data.

Hence, applying the formula given in the question, we get

e*45 = 9*33

Now, as only one variable is left in the equation, it has become a linear equation in one variable.

Hence, solving the linear equation, we get

e = (9*33)/45

e = 33/5

e = 6.6 era

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What is the solution to this equation?
5x+15+ 2x = 24 + 4x
O A. x = 13
OB. x = 3
O C. x=39/11
O D. x =9/11​

Answers

Answer:

X=3

Step-by-step explanation:

Remember to get all your variables on one side by doing the "Reverse function" etc...

Once youve got your variables on one side (and only 1 is remaining) you should get rid of whats BESIDE the variable by dividing it by *whatever it is thats beside the variable!* AND REMEMBER TO DO THIS ON BOTH SIDES OF THE EQUATION SO THAT THEYRE EQUAL!!

use the diagram to find bc. (help please!!)

Answers

Answer:6

Step-by-step explanation:

The length of ac is 20 and the length of ab is 14 so all you have to do is 20-14=6

Diego worked the SAME number of hours for FIVE days at his job at the cheese factory plus an extra 7 hours on Saturday. He worked a total 52-hours last week.

Using H as a variable for the number of hours he worked each normal day, model (create an equation) for Diego’s work week.

How many hours did he work each day for his normal work week?

Answers

The total hours worked on each day of his normal work week is 7.5 hours.

How many hours did Diego work on each normal day ?

The total hours worked is a function of the total hours worked during the week and the hours worked on Saturday.

The equation that represents the total hours that Diego worked is:

Total hours = total hours worked during the five days + hours worked on Saturday

52 = (5 X H) + 7 + H

52 = 5H + 7 + H

In order to determine the value of H, take the following steps:

Combine similar terms together: 52 - 7 = 5H + H

Add similar terms together: 45 = 6h

Divide both sides of the equation by 6 : H = 45 /6

H = 7.5 hours

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what is 3-5b=-32 in algebra

Answers

B = 7
Because well it is

Answer:

b=7

Step-by-step explanation:

3-5b=-32

-5b=-32-3

-5b=-35

Divide both sides by -5

b=7

( I also had trouble with these kind of expressions)

100 Points Please Help Quick
Which function has a constant additive rate of change of –1/4?

A coordinate plane with a straight line with a negative slope. The line passes through (negative 2, 2) and (2, 1).

A coordinate plane with a curved line passing through (negative 1, 2), (0, negative 1), the minimum (2, negative 2), and (4, negative 1).

A two-column table with five rows. The first column, x, has the entries, 20, 21, 22, and 23. The second column, y, has the entries negative 1, negative 1.5, negative 2, and negative 2.5.

A two-column table with five rows. The first column, x, has the entries, negative 12, negative 11, negative 10, and negative 9. The second column, y, has the entries, 7, 11, 14, and 17.

Answers

Answer:

(A) is correct.

Step-by-step explanation:

As we know,

The ratio that y increase as x increases is the slope of a line. The slope of a line reflects how steep it is, but how much y increases as x increases. Anywhere on the line, the slope stays unchanged (the same).

From the first graph:

The line goes through the points:

(-2, 2) and (2, 1)

y - 1 = (1-2)/(2+2)[x -2]

y - 1 = (-1/4)[x -2]

y = -x/4 + 1/2 + 1

y = -x/4 + 3/2

The constant additive rate of change = -1/4

Thus, the function shown in graph (A) has a constant additive rate of change is -1/4 option (A) is correct.

A car and an emergency are heading toward each other. The car is traveling at a speed of 30
mph or 44 feet per second. The emergency vehicle is traveling at a speed of 50 mph or about
74 feet per second. If the vehicles are 1000 feet apart and the conditions are ideal, in how
many seconds will the drive of the car first hear the siren?

Answers

If the speed of car is 44 feet per second and the speed of emergency vehicle is 74 feet per second,then they the driver of the car will hear the siren after 8.48 seconds after cover the distance of 1000 feet.

Given that the speed of car be 30 mph or 44 feet per second, the speed of emergency vehicle be 50 mph or 74 feet per second and the distance between both is 1000 feet apart.

We are required to find the seconds after which the driver of the car first hear the siren.

Speed is basically the distance that a vehicle covers in a particular period of time.

Speed=Distance/Time

Time=Distance/Speed

Total speed in feet per second=44+74=118 feet per second

Time=1000/118

=8.48 seconds

Hence if the speed of car is 44 feet per second and the speed of emergency vehicle is 74 feet per second,then they the driver of the car will hear the siren after 8.48 seconds.

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Open the file to answer the question

Answers

1) 14 radical 3
2) 11 radical 2 by 7
3) -300
4) 27 radical 5
5) -4 radical 7
6) -8 radical 7
7) 7 radical 6
8) 72 radical 3

Arrange the students in order from shortest to tallest. kim 56.03 alexa 56.3 roy 56.14 tom 57.1

Answers

Arrangement of students from shortest to tallest is

Kim 56.03 < Roy 56.14 < Alexa 56.3 < Tom 57.1.

As given in the question,

Number of students =  4

Height of the students

Kim = 56.03 units

Alexa = 56.3 units

Roy = 56.14 units

Tom = 57.1 units

Arrange the students from shortest to tallest

56.03 < 56.14 < 56.3 < 57.1

Kim < Roy < Alexa < Tom

Therefore, arrangement of students from shortest to tallest is

Kim 56.03 < Roy 56.14 < Alexa 56.3 < Tom 57.1.

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What is a first step to solve the equation 0.3n - 15 = 0.2n - 5?​

Answers

Answer:

100

Step-by-step explanation:

0.3n - 15 = 0.2n - 5

0.3n - 0.2n = - 5 + 15

0.1n = 10

10/0.1 = n

n = 100

PLEASE HELP, URGENT, POINTS GIVEN
A sheet of paper contains a list of 5 double-digit prime numbers.
Their mean is 31, while their median and mode is 29.
The largest number is 41.
Show how you can discover the 5 prime numbers written on the sheet.

Answers

Answer:

19, 29, 29, 37, 41

Step-by-step explanation:

1. What we know: x y 29 z 41

(x+y+29+z+41)/5 =31

x y and z each >9 and prime

Possibile numbers: 11, 13, 17, 19, 23, 29, 31, 37, 41

2. (x+y+29+z+41)/5 =31

x+y+29+z+41 = (31×5)

x+y+z = (31×5) -29 -41

x+y+z = 85

z>=29

x<=29

y<=29

x and y can be 11, 13, 17, 19, 23, 29

z could be 29, 31, 37, 41

At least one if them must equal 29 because it's the mode.

So two of them must equal 56

11+29=40 so both numbers cannot be less than or equal to 40

This means y is 29 because x + z has to equal 56 and that wouldn't be possible if x equals 29.

Now we know x+z=56

Possible numbers: 19+37

I found these by doing 56- each possible number for x and seeing if that equaled a possible number for z. For example 56-11=45 but z cannot be 45 therefore x cannot be 11 and so on.

Then you just plug in the numbers!

19, 29, 29, 37, 41

This was so much fun to solve! I hope this is helpful, let me know if you have any questions! You got this :)

Verify the identity.

secx-secxsin^2x= cos x

Answers

The identity is verified, yes. secx-secxsin^2x=cos x

What is an identity?

An identity in mathematics is an equality connecting two mathematical expressions, A and B, such that both expressions provide the same result for any value of the variables falling within a given range of validity.

An identity is an equation that, regardless of the values used, is always true. 2 x + 3 x = 5 x is an identity because, according to the preceding equation, 2 x + 3 x will always equal 1 regardless of the value;

According to given information;

secx = 1/cosx

cos²x+sin²x= 1

**sin²x = 1-cos²x

by (*) :

secx - secx sin²x=(1/cosx) -(1/cosx) sin²x

                       = (1/cosx) (1-  sin²x)....factory :(1/cosx)

by(**)

                     sec x-sec x sin²x  =  (1/cos x)cos²x

                     sec x-sec x sin²x  = cos x

( simplify by : cos x)

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