By concepts of polynomials and systems of linear equations, the constants c and d of the expression p(x) = x⁴ - 5 · x³ - 7 · x² + c · x + d are 29 and 30.
How to determine the missing coefficients of a quartic equation
A value x is a root of a polynomial if and only if p(x) = 0. We must replace the given equation with the given roots and solve the resulting system of linear equations:
(- 1)⁴ - 5 · (- 1)³ - 7 · (- 1)² + (- 1) · c + d = 0
- c + d = 1 (1)
3⁴ - 5 · 3³ - 7 · 3² + 3 · c + d = 0
3 · c + d = 117 (2)
The solution of this system is c = 29 and d = 30.
By concepts of polynomials and systems of linear equations, the constants c and d of the expression p(x) = x⁴ - 5 · x³ - 7 · x² + c · x + d are 29 and 30.
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A sequence is defined recursively by the formula f(n 1) = f(n) 3 . the first term of the sequence is –4. what is the next term in the sequence?
The next term in the sequence is -1.
Given a sequence is defined recursively by the formula f(n+1)=f(n)+3.
An ordered list of integers that frequently follows a pattern or function is known as a sequence. Sequences can have a finite or indefinite length. Terms refer to the distinct components of a sequence. The individual numbers or parts that make up a sequence are known as its terms.
The first term of the sequence is -4.
That means f(1)=-4 .......(1)
When compare f(1) with f(n+1), we get
n=0
So, when n=0 then it is the first term of the sequence.
When n=1 then substituting this in the given formula, we get
f(1+1)=f(1)+3
f(2)=f(1)+3
Now, Substitute the value of f(1) from equation (1), we get
f(2)=-4+3
f(2)=-1
Hence, the next term in the sequence when the sequence is defined recursively by the formula f(n+1) = f(n)+3 and the first term of the sequence is –4 is -1.
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Drag the tiles to the correct boxes to complete the pairs. Not all tiles will be used.
Find the average rate of change of each function over the interval [0, 3]. Match each representation with its respective average rate of change.
3
-2
5
-1
6
-3
The average rate of change is illustrated below.
How to illustrate the average rate?The average rate of change over interval [a,b]: r=[f(b)-f(a)]/(b-a).
In this case the interval is [0,2], then a=0, b=2
r=[f(2)-f(0)]/(2-0)
r=[f(2)-f(0)]/2
1) First function: h(x)
r=[h(2)-h(0)]/2
x=2→h(2)=(2)^2+2(2)-6
h(2)=4+4-6
h(2)=2
x=0→h(0)=(0)^2+2(0)-6
h(0)=0+0-6
h(0)=-6
r=[h(2)-h(0)]/2
r=[2-(-6)]/2
r=(2+6)/2
r=(8)/2
r=4
2) Second function: f(x)
A function, f, has an
x-intercept at (2,0)→x=2, f(2)=0
and a y-intercept at (0,-10)→x=0, f(0)=-10
r=[f(2)-f(0)]/2
r=[0-(-10)]/2
r=(0+10)/2
r=(10)/2
r=5
3) Third function: g(x)
r=[g(2)-g(0)]/2
From the graph:
g(2)=6
g(0)=2
r=(6-2)/2
r=(4)/2
r=2
4) Fourth function: j(x)
r=[j(2)-j(0)]/2
From the table:
x=2→j(2)=-8
x=0→j(0)=4
r=(-8-4)/2
r=(-12)/2
r=-6
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Lily read 3 books in 4 months. What was her rate of reading in books per month? Express your answer in simplest form.
HELP ASAP!!
Workers are required to pay 4½% of their salaries into an Educational Fund. A worker's salary is 120 Ghana cedis . How much does he pay into the Educational Fund?
Answer:
He will need to pay 5.4 Ghana cedis into his educational fund.
Step-by-step explanation:
We need to multiply 4 1/2 by 120 since it is a percentage. We convert 4 1/2 into a decimal which is 4.5. Now, we just need to multiply 120 by 4 1/2 to get an answer of 5.4.
17.
Lincoln stands on the top of a building and fires a gun upwards. The bullet travels
according to the equation h = -16t² + 384t + 50, where h is the height of the bullet off
the ground in metres at t seconds after it was fired.
a) How far is Lincoln above the ground when he shoots the gun?
b)How high does the bullet travel vertically relative to the ground?
c)How long does it take the bullet to reach its greatest height?
d)After how many seconds (round to nearest 100th) does the bullet hit the ground?
please help asap!
Answer:
a) 50 meters
b) 2354 meters
c) 12 seconds
d) 24.13 seconds
Step-by-step explanation:
Hello!
a) How far is Lincoln above the ground when he shoots the gun?We need to really understand what the variables represent in this question. Given that t is time, and h is height, the time at which he shoots the bullet will be 0, as the bullet hasn't started traveling yet. This means that we can solve for the original height of the bullet by plugging in 0 for time in the equation.
Equation: [tex]h = -16t^2 + 384t + 50[/tex]
Plug in 0 for t:
[tex]h = -16t^2 + 384t + 50[/tex][tex]h = -16(0)^2 + 384(0) + 50[/tex][tex]h = 50[/tex]The height at which Lincoln shot the bullet is 50 meters.
We can also look at this graphically. The height of origin will simply be when the x-value (in this case t-value) is 0. That means it is the point at which the graph intersects the y-axis, known as the y-intercept.
Standard form of a Parabola: [tex]y = ax^2 + bx + c[/tex]
The y-intercept is the "c" value.
Given our equation: [tex]h = -16t^2 + 384t + 50[/tex]
The c-value is 50. This proves that the y-intercept is 50.
b)How high does the bullet travel vertically relative to the ground?We want to find the highest point of the graph. To do that, we can find the vertex.
We can utilize the Axis of Symmetry (AOS) to find the Vertex.
First, find the AOS using the formula: [tex]AOS = -\frac{b}{2a}[/tex]
a = -16b = 384Plug it into the formula:
[tex]AOS = -\frac{b}{2a}[/tex][tex]AOS = -\frac{384}{2(-16)}[/tex][tex]AOS = \frac{384}{32}[/tex][tex]AOS = 12[/tex]Plug in 12 for t in the equation:
[tex]h = -16t^2 + 384t + 50[/tex][tex]h = -16(12)^2 + 384(12) + 50[/tex][tex]h = -2304 + 4608 + 50[/tex][tex]h = 2304+50[/tex][tex]h = 2354[/tex]Therefore, the highest point of the bullet is 2354 meters.
c)How long does it take the bullet to reach its greatest height?We answered this in the last question. The t-value when h is at its highest is 12.
d)After how many seconds (round to nearest 100th) does the bullet hit the ground?We have to solve for the values of t when h is 0 (when it touches the ground).
Set the equation to 0, and solve using the quadratic formula.
Quadratic formula: [tex]x = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a}[/tex]
[tex]x = \frac{-b\pm\sqrt{b^2 - 4ac}}{2a}[/tex][tex]x = \frac{-384\pm\sqrt{384^2 - 4(-16)(50)}}{2(-16)}[/tex][tex]x = \frac{-384\pm\sqrt{147456 +3200}}{-32}[/tex][tex]x = \frac{-384\pm\sqrt{150656}}{-32}[/tex][tex]x = \frac{-384\pm388.14}{-32}[/tex][tex]x = -\frac{4.14}{32}, x = \frac{772.14}{32}[/tex]We are only going to take the positive solution, as we can't have a negative time. The solution that doesn't work is called an extraneous solution.
[tex]x = \frac{772.14}{32}[/tex][tex]x = 24.13[/tex]It takes approximately 24.13 seconds for the bullet to hit the ground.
Which equation is equivalent to the given equation?
x² - 6x = 8
Find the solution to the equation z – 7 = 8.
Answer:
z=15
Step-by-step explanation:
z-7+7=8+7 ( add 7 on btoh sides to cancel out)
z=15 ( left with 7)
can someone help me with this i dont know what im doing
Answer:
ok sure will do the answer is
Step-by-step explanation:
Which equation matches the statement below?
Three times the difference of a number "n" and five is sixteen more than double the number "r
O 3(5-n)=2n+16
O 3(n-5)=16+2n
3(n-5)=2n+16
3n-5=2n+16
Answer:
3(n-5)=2n+16
Step-by-step explanation:
3 lots of 'n' and 5 means 3 times both values so these values are put into a bracket with a common multiplier (ie. the 3)
3(5 - n)
16 more than 2 lots of 'n' means that n has to be doubled. n + n = 2n. add 16 to this and we have
2n + 16
if these values are both equal (hence why they are in an equation) then all that is needed is for them to be set equal alongside each other.
3(n-5) = 2n+16
:)
In question 1, you found the squares of both positive and negative numbers. From the values you entered in the tables, what value or values can you substitute for x in x2 = 9 to make the equation true?
Answer:
x^2=9
so we find the sqare root of both sides
x=square root of 9
x=-3 or 3
Hope This Helps!!!
12x+10y=2
-x+4y= -5
Without actually solving, explain whether substitution or elimination would be a better method for solving the system.
Answer:
Elimination would be a better method for solving the system.
Given equation [tex] \sf12x+10y=2[/tex][tex] \sf-x+4y= -5 [/tex]Solution :-[tex]\Rrightarrow \sf12x+10y=2[/tex]
[tex]\Rrightarrow \sf6x+5y=1.......(i)[/tex]
[tex] \sf\Rrightarrow-x+4y= -5 ............(ii)[/tex]
[tex]\large \sf \red{ \blue{[ (ii)×6]}}[/tex]
[tex]\sf\Rrightarrow-6x+24y= -30 ............(iii)[/tex]
[tex] \large \sf \red{(i) + \blue{ (iii)}}[/tex]
[tex]\Rrightarrow \small\sf6x+5y=1+(-6x+24y= -30)[/tex]
[tex]\Rrightarrow \sf29y = - 29 \\ [/tex]
[tex]\Rrightarrow \blue{\sf \: y = - 1}[/tex]
putting the value in equation (i)
[tex]\Rrightarrow \sf6x+5 \times - 1=1[/tex]
[tex]\Rrightarrow \sf6x - 5=1[/tex]
[tex]\Rrightarrow \sf6x = 6[/tex]
[tex]\Rrightarrow \red{\sf \: x = 1}[/tex]
[tex]\overline{\underline{\rule{209pt}{2pt}}}[/tex]
What is the value of x for these problems? Pls help I’m so desperate
Answer:
First Question x = [tex]17\sqrt{2}[/tex] Second Question x = 36
Step-by-step explanation:
First Question:
Since The triangle is isosceles, [tex]17^2 +17^2=x^2[/tex] (Pythagoras)
[tex]x=17\sqrt{2}[/tex]
Second Question
This is half an equilateral triangle, so [tex](\frac{1}{2}x)^2+(18\sqrt{3})^2=x^2[/tex]
[tex]\frac{3}{4}x^2=972[/tex]
[tex]x^2=1296[/tex]
[tex]x=36[/tex] (cannot be negative)
Mike bought a photo frame which measured 8 inches in length and 7 inches in width. Determine the diagonal length of the photo frame
Answer:
10.63 inches
(b)2+(h)2=(H)2
(8)2+(7)2=(H)2
64+49=(H)2
113=(H)2
H=10.63
The diagonal length of the photo frame having measurements of 8 inches in length and 7 inches in width is 10.63 inches.
How is the diagonal of a rectangle determined?The diagonal of a rectangle having a length l and width w will be:
d = √(l² + w²).
How to solve the question?In the question, we are asked to determine the diagonal length of the photo frame bought by Mike, having measurements of 8 inches in length and 7 inches in width.
We know the photo frame is in the shape of a rectangle, as it has a length and a width.
We also know that the diagonal of a rectangle having a length l and width w will be:
d = √(l² + w²).
So, by substituting l = 8 and w = 7, in the above relation, we can calculate the diagonal of the photo frame as:
d = √(8² + 7²),
or, d = √(64 + 49),
or, d = √113,
or, d = 10.63.
Therefore, the diagonal length of the photo frame having measurements of 8 inches in length and 7 inches in width is 10.63 inches.
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Abhasra won 68 lollipops playing
basketball at her school's game night.
Later, she gave two to each of her friends.
She only has 8 remaining. How many
friends does she have?
Answer:
8
Step-by-step explanation:
because there is 8 in the question
The number of math classes offered online in the summer is 15 less than the number of math classes offered face-to face in the summer . Let Frepresent the number of face-to face summer classes . Which of the following expressions gives the number of online math classes ?
The expressions that gives the number of online math classes is as follows:
y = x - 15
How to express an equation?The number of math classes offered online in the summer is 15 less than the number of math classes offered face-to face in the summer.
Let
the number of math classes offered face-to face = x
the number of math classes offered online = y
Therefore, the expression that gives the number of online math classes is as follows:
y = x - 15
x = y + 15
Hence,
y = x - 15
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Jian Hui can do 14 pages of homework every 3 hours. How many pages can he finish if he works from 8pm to 11pm?
14 pages
-------------
3 hours
subtract 6x 2 −7x−2 from 6x^{2}-2x+6
Answer:
5x+8
Step-by-step explanation:
The from is the key word
That means the second term is first
6x^2 -2x +6 - (6x^2 -7x -2)
Distribute the minus sign
6x^2 -2x +6 - 6x^2 +7x +2
Combine like terms
5x+8
ANSWER ASAP PLS..........
The length of one leg of the right triangle is 5√10
How to find the leg of a right triangle?The leg of a right triangle can be found as follows;
Therefore, using trigonometric ratios,
sin 45 = opposite / hypotenuse
sin 45 = x / 10√5
1 / √2 = x / 10√5
cross multiply
10√5 = x√2
divide both sides by √2
x = 10√5 / √2
x = 10√5 / √2 × √2 / √2
x = 10√10 / 2 = 5√10
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find the slope of a line passing through the points A(-2,-7) and B(-2,2)
[tex]\Large\maltese\underline{\textsf{A. What is Asked}}[/tex]
Find the slope of a line passing through (-2,-7) and (-2,2)
[tex]\Large\maltese\underline{\textsf{B. This problem has been solved!}}[/tex]
[tex]\boxed{\begin{minipage}{6cm} To find the slope, \\ we use the formula below.\end{minipage}}[/tex]
[tex]\bf{\dfrac{y2-y1}{x2-x1}}[/tex] | put in the values
[tex]\bf{\dfrac{2-(-7)}{-2-(-2)}}[/tex] | look what happens when I simplify!
[tex]\bf{\dfrac{2+7}{-2+2}}[/tex] | uh-oh
[tex]\bf{\dfrac{9}{0}}[/tex] ...
The operation above is an operation we can't do.
[tex]\bf{Slope=\;Not\;D e f ined}[/tex]
[tex]\cline{1-2}[/tex]
[tex]\bf{Result:}[/tex]
[tex]\bf{=Slope=Not\;De fined}[/tex]
[tex]\LARGE\boxed{\bf{aesthetic\not1\theta\ell}}[/tex]
if you starting salary is $350000 and you receive a 6% increase at the end of every year, what is the total amount, in dollars. you will earn over the first 16 years that you work
The total amount will be $889123 after 16 years if the starting salary is $350000
What is an exponential function?It is defined as the function that rapidly increases and the value of the exponential function is always positive. It denotes with exponent [tex]y = a^x[/tex]
where a is a constant and a>1
We have:
Starting salary is $350000 and you receive a 6% increase at the end of every year.
We can find the total amount, from the exponential formula:
T = a(1 + r)ⁿ
T is the total amount
a is the starting amount
r is the rate
n is the total number of years
T = 350000(1 + 6%)¹⁶
T = 350000(1 + 0.06)¹⁶
T = 889123.08 ≈ $889123
Thus, the total amount will be $889123 after 16 years if the starting salary is $350000
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A triangle with side lengths of 5 meters, 7 meters and 9 meters.
Answer:
obtuse triangleStep-by-step explanation:
A triangle with side lengths of 5 meters, 7 meters and 9 meters is
an obtuse triangle
list both the negative and positive factors of -30 and sum is 1
Answer:
-5 and 6
Step-by-step explanation:
-5 * 6 = -30
-5 + 6 = 1
A kennel has 3 black puppies and 4 brown puppies. If someone is randomly choosing puppies, what is the probability they choose a black and a brown puppy?
A)0.2449
B)0.2857
C)0.7846
D)0.5
The probability they choose a black and a brown puppy is 0.2857 (option B)
What is the probability?Probability determines the chances that an event would happen. The probability the event occurs is 1 and the probability that the event does not occur is 0.
The probability they choose a black and a brown puppy = (number of black puppies / total number of puppies) x ( (number of brown puppies / total number of puppies - 1)
3/7 x 4/6 = 0.2857
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Match the concept with its definition and characteristics. i. A set of instructions (codes) that perform a specific task. ii. A calculation where the result is placed into a variable. iii. The computer programming equivalent of a recipe in cooking iv. A formula to calculate a new value based on other values and operations.
1. Algorithm 2. Expression 3. Equation
First definition matches with expression, second matches with equation ,third matches with algorithm and fourth matches with equation.
Given four definitions of equation, expression, algorithm but mixed.
We have to match the definitions with appropriate term.
We know that expression is a combination of numbers, symbols, fraction, coefficients, indeterminants mostly not found in equal to form. It exhibits a behaviour only.
Algorithm is a computer programming to do a specific task in a predetermined way.
Equation is a relationship between two variables expressed in equal to form. In this we have to put the value of variables and the equation gives us a value.
Hence First definition matches with expression, second matches with equation ,third matches with algorithm and fourth matches with equation.
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Follow the steps to find the
area of the shaded region.
First, use the formula
below to find the area
of the whole sector.
Sector Area
(angle of sector ) p2
360
Sector Area = [?] cm2
Round to four decimal places.
The area of the sector rounded to 4 decimal place is 78.6396 cm²
How to find the area of a sector?area of sector = ∅/ 360 × πr²
Therefore,
r = radius∅ = 46 degrees.Hence,
area of a sector = 46 / 360 × 3.14 × 14²
area of a sector = 46 / 360 × 3.14 × 196
area of a sector = 46 / 360 × 615.44
area of a sector = 28310.24 / 360
area of a sector = 78.6395555556
Hence,
area of a sector = 78.6396 cm²
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Find an equation perpendicular to y = (3/₂)x+ 1 and has a y-
intercept of 2.
Step-by-step explanation:
ATTACHED IS THE SOLUTION!!!!!!if nm = 8x - 14 and jk = x squared + 1, find JK
A parallelogram in which adjacent sides are perpendicular to each other is called a rectangle. The length of Jk can be either 10 or 26.
What is a rectangle?A parallelogram in which adjacent sides are perpendicular to each other is called a rectangle. A rectangle is always a parallelogram and a quadrilateral but the reverse statement may or may not be true.
Since in a rectangle the opposite sides are equal, therefore, the sides NM and JK will be equal.
JK = MN
8x - 14 = x² + 1
0 = x² - 8x + 15
x = 5, 3
Hence, the length of Jk can be either 10 or 26.
The complete questions are:
Quadrilateral JKMN is a rectangle, if NM= 8x - 14 and JK= x squared + 1, find JK.
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At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 16 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate (in ft/min) is the height of the pile changing when the pile is 2 feet high
The rate at which the height of the pile is changing when the height is 2 ft. is 0.5659 ft./min. Computed using differentiation.
The sand is falling onto a conical pile, thus we get the pile in the shape of a cone, whose volume is given as:
V = (1/3)πr²h,
where V represents the volume of the cone, r represents its radius, and h represents its height.
In the question, we are given that the rate of piling is 16 ft.³/min, which implies that:
dV/dt = 16 ... (i)
We are also given that the diameter is three times the altitude (height), which can be shown as:
2r = 3h,
or, r = (3/2)h.
Thus, the volume can now be shown as:
V = (1/3)π{(3/2)h}²h,
or, V = (3πh³)/4.
Differentiating this with respect to time, we get:
dV/dt = (3π/4)(3h²)dh/dt.
Now, we need to calculate the rate of change of height, when the height is 2 feet, thus we take h = 2
Using dV/dt = 16 from (i), we can write:
16 = (3π/4)(3(2)²)dh/dt.,
or, 16 = 9π(dh/dt),
or, dh/dt = 16/9π = 0.5659 ft./min.
Thus, the rate at which the height of the pile is changing when the height is 2 ft. is 0.5659 ft./min.
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What is the expression of g(x) when we perform the following sequence of transformations onto the parent function fx=1x:
a) stretch horizontally by a factor of 2
b) shift left 1 unit
c) shift up 3 units
The expression of function g(x) is [tex]\sqrt{\frac {x - 1}2} + 3[/tex]
How to determine the expression?The function is given as:
[tex]f(x) = \sqrt{x[/tex]
When the function is stretched horizontally by a factor of 2, the rule is:
f'(x) = f(x/2)
So, we have:
[tex]f'(x) = \sqrt{\frac x2[/tex]
When the function is shifted left by 1 unit, the rule is:
f"(x) = f(x + 1)
So, we have:
[tex]f"'(x) = \sqrt{\frac {x - 1}2[/tex]
When the function is shifted up by 3 units, the rule is:
g(x) = f"(x) + 3
So, we have:
[tex]g(x) = \sqrt{\frac {x - 1}2} + 3[/tex]
Hence, the expression of function g(x) is [tex]\sqrt{\frac {x - 1}2} + 3[/tex]
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Given: AC BD and AC and BD bisect each other.
Prove: BE EC
1) [tex]\overline{AC} \cong \overline{BD}[/tex] and [tex]\overline{AC}[/tex] and [tex]\overline{BD}[/tex] bisect each other (given)
2) [tex]ABCD[/tex] is a parallelogram (a quadrilateral with diagonals that bisect each other is a parallelogram)
3) [tex]ABCD[/tex] is a rhombus (a parallelgoram with congruent diagonals is a rhombus)
4) [tex]\overline{BE} \cong \overline{EC}[/tex] (halves of congruent segments are congruent)