1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
gulaghasi [49]
2 years ago
9

What is the value of k in the equation, so that one root exceeds the other by 4

Mathematics
1 answer:
tatiyna2 years ago
4 0
<h3>Answer:   k = -15</h3>

===========================================================

Explanation:

Let p(x) = 4x^2+4x+k be the polynomial function.

Also, let r and s be the two roots of the polynomial p(x).

By definition of what it means to be a root, we know that

p(r) = 0

p(s) = 0

So this means p(r) = p(s).

Because one root exceeds another by 4, we can say s = r+4.

So the equation p(r) = p(s) updates to p(r) = p(r+4).

----------------------------

Let's compute p(r) and p(r+4)

So,

p(x) = 4x^2+4x+k

p(r) = 4r^2+4r+k

and

p(x) = 4x^2+4x+k

p(r+4) = 4(r+4)^2+4(r+4)+k

p(r+4) = 4(r^2+8r+16)+4(r+4)+k

p(r+4) = 4r^2+32r+64+4r+16+k

p(r+4) = 4r^2+36r+80+k

------------------------------

Now equate those results

p(r) = p(r+4)

4r^2+4r+k = 4r^2+36r+80+k

4r+k = 36r+80+k        ...... the 4r^2 terms cancel

4r = 36r+80                ..... the k terms cancel as well

4r-36r = 80

-32r = 80

r = 80/(-32)

r = (16*5)/(-16*2)

r = -5/2 = -2.5 is one of the roots

s = r+4

s = -2.5+4

s = 1.5 = 3/2 is the other root.

------------------------------

With this in mind, we can use either r or s to find the value of k

p(x) = 4x^2 + 4x + k

p(r) = 4r^2 + 4r + k

p(r) = 4(-2.5)^2 + 4(-2.5) + k

p(r) = 15+k

0 = 15+k

k+15 = 0

k = -15

------------------------------

To confirm this answer, you can use the quadratic formula to solve 4x^2+4x-15 = 0. You should get the two roots r = -5/2 = -2.5 and s = 3/2 = 1.5

Then note how s-r = 4 which is the same as saying s = r+4.

You might be interested in
Joe, working alone, takes 8 hours to mow a lawn. Sally, working alone, can mow the same lawn in 6 hours. How long does it take t
tamaranim1 [39]

Answer: 3 3/7 hours

Step-by-step explanation:

you need to find out how much work they can do in an hour.

let x represent the total collaborative work

1/8 + 1/6 = 1/x

3/24 + 4/24 = 1/x

7x = 24

x = 24/7 or 3 3/7 hours (in decimal form this is 3.4286)

I cannot guarantee this is correct but hope it helped

6 0
3 years ago
Solve this system by elimination <br> 10x+6y=14<br> -x-6y=-23
ludmilkaskok [199]

Answer:

(-1, 4)

General Formulas and Concepts:

<u>Pre-Algebra</u>

  • Order of Operations: BPEMDAS
  • Equality Properties

<u>Algebra I</u>

  • Solving systems of equations using substitution/elimination

Step-by-step explanation:

<u>Step 1: Define systems</u>

10x + 6y = 14

-x - 6y = -23

<u>Step 2: Solve for </u><em><u>x</u></em>

<em>Elimination</em>

  1. Add 2 equations together:                    9x = -9
  2. Divide 9 on both sides:                          x = -1

<u>Step 3: Solve for </u><em><u>y</u></em>

  1. Define original equation:                    -x - 6y = -23
  2. Substitute in <em>x</em>:                                     -(-1) - 6y = -23
  3. Multiply:                                                1 - 6y = -23
  4. Subtract 1 on both sides:                     -6y = -24
  5. Divide -6 on both sides:                      y = 4
8 0
2 years ago
What is 7/1 dividend by 14/3
katrin2010 [14]
7/1 x 3/14 = 21/14 = 3/2 = 1 1/2

The answer is 1 1/12 or 1.5, but I recommend the fraction way since the problem was in fractions!
4 0
3 years ago
Read 2 more answers
Please help with these two questions, will give thanks
AURORKA [14]

Answer:

Step-by-step explanation:

1. True

2. C

3 0
2 years ago
Dan earns $9.50 per hour as a dishwasher. Determine the fewest number of hours he must work to earn
Oksana_A [137]

Answer:

43 hours

Step-by-step explanation:

\frac{y}{1} :\frac{408}{9.5}

y × 9.5 = 408 × 1

9.5y = 408

9.5y ÷ 9.5 = 408 ÷ 9.5

y=42\frac{18}{19}

43 hours

8 0
3 years ago
Read 2 more answers
Other questions:
  • Mariah took 76 pictures on vacation.Her sister took 29 more pictures than she did.How many pictures did Mariah’s sister take
    7·1 answer
  • Apply the Gram Schmidt orthogonalization process for decomposition for the matrix = (1 −2 3 5)
    13·1 answer
  • Alexander is going to an amusement park. The price of admission into the park is $15, and once he is inside the park, he will ha
    12·2 answers
  • the mean raduis of earth is 6371.0 kilometers and the mean radius of earth moon is 1737.5 kilometers what is the approximate dif
    14·1 answer
  • There it is. Help me pls
    11·1 answer
  • What decimal number is halfway between 20 and 21?
    10·1 answer
  • I'm slow I need help :(​
    11·1 answer
  • If 5(3x – 7 ) = 20, then what is 6x – 8? *<br> 8<br> 12<br> 14<br> 16
    9·1 answer
  • For each item, would a milliliter or a liter be the more reasonable unit of measure?
    9·2 answers
  • Find Q2 of the following data set.3, 9, 12, 4, 6, 40, 25, 14, 10
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!