Answers
Answer:
(x+1+i)(x+1-i) goes with x^2+2x+2
(x+2i)(x-2i) goes with x^2+4
(x-2+2i)(x-2-2i) goes with x^2-4x+8
Step-by-step explanation:
(x+1+i)(x+1-i)
(x+[1+i])(x+[1-i])
Use foil.
First: x(x)=x^2
Outer: x(1+i)=x+ix
Inner: x(1-i)=x-ix
Last: (1+i)(1-i)=1-i^2 since 1+i and 1-i are conjugates
__Add together to get: x^2+2x+1-i^2
We can actually simplify this because i^2=-1
So x^2+2x+1-i^2=x^2+2x+1-(-1)=x^2+2x+2
(x+2i)(x-2i)
These are conjugates so just do first and last of foil.
First: x(x)=x^2
Last: 2i(-2i)=-4i^2=-4(-1)=4
==Adding together gives x^2+4
(x-2+2i)(x-2-2i)
(x+[-2+2i])(x+[-2-2i])
This is similar to first.
Foil time!
First: x(x)=x^2
Outer: x(-2-2i)=-2x-2ix
Inner: x(-2+2i)=-2x+2ix
Last: (-2-2i)(-2+2i)=4-4i^2 (multiplying conjugates again)
==Add together giving us x^2-4x+4-4i^2
This can be simplified since i^2=-1.
So applying this gives us x^2-4x+4-4(-1)
=x^2-4x+4+4
=x^2-4x+8
Answer:
1. The first expression is equivalent to [tex]x^2+2x+2[/tex].
2. The second expression is equivalent to [tex]x^2+4[/tex].
3. The third expression is equivalent to [tex]x^2-4x+8[/tex].
Step-by-step explanation:
(1).
The given expression is
[tex](x+1+i)(x+1-i)[/tex]
[tex][(x+1)+i][(x+1)-i][/tex]
Using the algebraic properties, we get
[tex](x+1)^2-(i)^2[/tex] [tex][\because a^2-b^2=(a-b)(a+b)][/tex]
[tex]x^2+2x+1-(i)^2[/tex] [tex][\because (a+b)^2=a^2+2ab+b^2][/tex]
[tex]x^2+2x+1-(-1)[/tex] [tex][\because i^2=-1][/tex]
[tex]x^2+2x+2[/tex]
Therefore the first expression is equivalent to [tex]x^2+2x+2[/tex].
(2).
The given expression is
[tex](x+2i)(x-2i)[/tex]
Using the algebraic properties, we get
[tex](x)^2-(2i)^2[/tex] [tex][\because a^2-b^2=(a-b)(a+b)][/tex]
[tex]x^2-4i^2[/tex]
[tex]x^2-4(-1)[/tex] [tex][\because i^2=-1][/tex
[tex]x^2+4[/tex]
Therefore the second expression is equivalent to
[tex]x^2+4[/tex].
(3)
The given expression is
[tex](x-2+2i)(x-2-2i)[/tex]
[tex][(x-2)+2i][(x-2)-2i][/tex]
Using the algebraic properties, we get
[tex](x-2)^2-(2i)^2[/tex] [tex][\because a^2-b^2=(a-b)(a+b)][/tex]
[tex]x^2-4x+4-4i^2[/tex] [tex][\because (a-b)^2=a^2-2ab+b^2][/tex]
[tex]x^2-4x+4-4(-1)[/tex] [tex][\because i^2=-1][/tex]
[tex]x^2-4x+4+4[/tex]
[tex]x^2-4x+8[/tex]
Therefore the third expression is equivalent to [tex]x^2-4x+8[/tex].
Related Questions
briana has earned the following test grades: 82%, 75% and 90%. She wants to maintain an 85% average for her test scores. what must she receive on the final test in order to acieve this goal?
Answers
Answer:
93%
Step-by-step explanation:
Formula for calculating average = sum of data/ total number of data
Step 1: Add all percentages
Let the percentage on the last test be x
82% + 75% + 90% + x = 247% + x
Step 2: Apply the formula
Average = sum of percentages/ total number of tests
85 = 247 + x /4
340 = 247 + x
x = 340 - 247
x = 93%
Therefore, Briana must receive 93% on her final test to maintain an average of 85%.
!!
which two quantities form a proportional relationship?
A.1/4 and 3/8
B.2/15 and 3/30
C.10/18 and 45/81
D.5/6 and 5/24
Answers
c.10/18 and 45/81
10/18=.5555555555555555555555
45/81=.5555555555555555555555
Answer:
D 5/6 and 5/24
step by step explanation :
when you multiply 6 times 4 it equals 24
How to use substitution? (With picture) thanks!
Answers
Answer: x = 10 y = 6
Step-by-step explanation:
0.1X + 5 = y
0.2X + 4 = y
0.1X + 5 = 0.2X + 4
-0.1X - 4 = -0.1X - 4
1 = .1X
x = 10
0.1(10) + 5 = y
1 +5 = y
6 = y
Question 7
Answers:
Jane's Uber business has the function f(x) = 0.10x+5
Joe's Uber business has the function f(x) = 0.20x+4
x is the number of miles, f(x) is the total cost for the customer
-------------------------
Explanation:
Jane charges $5 no matter how many miles are driven. This is the base fee. On top of this base fee, an additional price is charged based on the number of miles driven. If Jane drives 1 mile, then an additional 0.10*1 = 0.10 dollars is added on. If 2 miles are driven, then 0.10*2 = 0.20 dollars is added on. And so on.
In general, an additional 0.10*x dollars is added on the base fee to get the total cost to be 5 + 0.10x which rearranges to 0.10x + 5. That's how I ended up with f(x) = 0.10x + 5
Joe's equation will be constructed in a similar way. His base fee is $4 and you add on 0.20x dollars (since it costs $0.20 per mile for Joe's company). Overall, we get f(x) = 0.20x + 4 for Joe's company.
Note how your teacher is not asking you to solve for x or f(x). If you want, you can replace f(x) with y; however, this isn't in function notation.
=============================================================
Question 8
Answers:
Function for Joseph's plan is: f(x) = 0.10x + 20
Function for Micki's plan is: f(x) = 0.15x + 10
The plans cost the same when 200 text messages are sent
-------------------------
Explanation:
We'll set up the equations in a similar way done in problem 7. We start with $20 as the base fee for Joseph's plan and then add on 0.10x dollars because $0.10 is charged per text message. We have x represent the number of text messages. Joseph's function is therefore f(x) = 0.10x + 20. Similarly, Micki's function is f(x) = 0.15x + 10
Let's replace f(x) with y and we have these two equations: y = 0.10x + 20 and y = 0.15x + 10
Now use substitution to solve for x and y
y = 0.10x + 20
0.15x + 10 = 0.10x + 20 ... y replaced with 0.15x+10
0.15x-0.10x = 20-10
0.05x = 10
x = 10/0.05
x = 200
This means that the two plans cost the same when 200 messages are sent.
Note plugging x = 200 into each f(x) function leads to
f(x) = 0.10x + 20
f(200) = 0.10*200 + 20
f(200) = 20 + 20
f(200) = 40 <<--- it costs Joseph $40 to send 200 messages
and
f(x) = 0.15x + 10
f(200) = 0.15*200 + 10
f(200) = 30 + 10
f(200) = 40 <<--- it costs Micki $40 to send 200 messages
What is LCM of 30 and 62 using Division Method
Answers
Answer:
930
Step-by-step explanation:
To use the division method
Write the given numbers in a horizontal row separated by commas.
Divide them by a suitable prime number
Place the quotient under the numbers in the next row. If a number is not divided exactly then bring it down to the next row.
Repeat the process until only co prime numbers are left in the last row.
Multiply all the prime numbers used as divisors.
The product is the LCM
Given 30 and 62, then
2 | 30, 62
3 | 15, 31
5 | 5, 31
31 | 1, 31
| 1, 1 ← co primes left
Hence LCM = 2 × 3 × 5 × 31 = 930
Sarah was 50 1/4 in tall and she was 12 years old she was 4 and 1/2 inches tall when she was 11 years old how much did she grow during the year
Answers
If p and q prime number greater than 2which of the following is not even integer ? a.p+q b.p×q c.p^2-q^2 d.p-q
Answers
B, because every prime number greater than 2 is odd, and the product of two odd numbers is odd.
Answer: I'm pretty sure its B
Step-by-step explanation:
..
Jim Smith is a salesman who receives a $1,100 draw per week. He receives a 12% commission on all sales. Sales for Jim were $205,000 for the month. Assuming a four-week month. Jim's commission after the draw is
Answers
Answer:
$29,000
Step-by-step explanation:
Given:
Receivable draw per week= $11,00
Commission on all sales= 12%
Sale for the month= $205,000
weeks in a month= 4
Jim's commission after the draw=?
total draws= 11,00 (4)
= 44,00
Commission on the sale= 205000(0.12)
=24600
Jim's commission after the draw= 4400 +24600
=29,000 !
Jim Smith's commission after the draw is calculated by determining his total commission from sales and then subtracting the draw he receives weekly. For a month where he made $205,000 in sales, his commission after the draw is $20,200.
The subject of this question is Jim Smith's commission after the draw, who works as a salesman and receives a draw of $1,100 per week plus a 12% commission on all sales. Considering that Jim made $205,000 in sales for the month, we would first calculate his total commission by multiplying the sales amount by the commission percentage. Next, we deduct the total draw for the month from the total commission to find Jim's commission after the draw.
To calculate the total commission earned, you would use the formula: Commission = Sales x Commission Rate. Therefore, for a sales amount of $205,000 and a commission rate of 12%, the total commission would be $205,000 x 0.12, which equals $24,600. Since a month is considered to have four weeks, Jim's total draw for the month would be $1,100 x 4, totaling $4,400.
To find out how much commission Jim has after accounting for the draw, we subtract the draw from the total commission: Commission after Draw = Total Commission - Total Draw, or $24,600 - $4,400, which equals $20,200.
76degrees faherheit = what in Celsius?
Answers
Answer:
24.4444
Step-by-step explanation:
76 degrees faherheit = 24.4444 degrees Celsius
(76°F − 32) × 5/9 = 24.444°C
Answer:
rjjj3krktkkrkrkektk4kgk4mgk3mrl3lfkwjkeoc
Van was asked to add 46 to 37 and round the result to the nearest ten. He figured that since he was going to have to round anyway, he might as well do it before adding the numbers. He rounded 46 to 50 and 37 to 40, thus getting an answer of 50+40=90. This answer is not correct. What is the correct answer?
Answers
Answer:
80
Explanation:
46 + 37 = 83 so 83 rounded to the nearest 10 is 80
Answer:
80
Step-by-step explanation:
46+37 = 83
Rounding to the nearest ten
80
The area of two rectangles is given by he functions: area of a rectangle A:f(x)= 4x2+6x area of rectangle B:g(x)=3x2-x Which function represents the difference? A.h(x)=7x2-5x B.h(x)=x2-7x C.h(x)=x2+5x D.h(x)=x2+7x
Answers
Answer:
x^2+7x if you are asked to find the difference of a function f and function g
Step-by-step explanation:
We are asked to A-B or f(x)-g(x).
(4x^2+6x)-(3x^2-x)
4x^2+6x-3x^2+x
The like terms I'm going to pair up.
4x^2-3x^2+6x+x
1x^2 +7x
x^2 +7x
The answer is x^2+7x
Answer:
OPTION D: [tex]h(x)=x^2+7x[/tex]
Step-by-step explanation:
You know that the area of Rectangle A is given by :
[tex]f(x)= 4x^2+6x[/tex]
And the area of Rectangle B is given by:
[tex]g(x)=3x^2-x[/tex]
Therefore, in otder to find the function that represents the difference, you need to subtract the functions f(x) and g(x).
Then, you get this function h(x):
[tex]h(x)=f(x)-g(x)\\\\h(x)= (4x^2+6x)-(3x^2-x)\\\\h(x)=4x^2+6x-3x^2+x\\\\h(x)=x^2+7x[/tex]
Nathan wanted to make a safe investment with decent returns. His financial advisor suggested a financial service that provided a higher rate of interest. However, he warned Nathan that he won’t be able to withdraw the money before the maturity date. Which financial service has Nathan’s financial advisor suggested?
A.
checking account
B.
certificates of deposit
C.
savings account
D.
stocks and bonds
Answers
Nathan was advised to get a saving account
Find the midpoint between (4,-1) and (3,2)
Answers
Midpoint has form of,
[tex]M(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2})[/tex]
We can insert the data,
[tex]M(\dfrac{4+3}{2},\dfrac{-1+2}{2})[/tex]
Which simplifies to,
[tex]\boxed{M(3.5, 0.5)}[/tex]
Hope this helps.
r3t40
Nikita knows the following information about her food club that has 11 members: 3 members like neither fruit nor vegetables. 4 members like fruit but not vegetables. 5 members in total like fruit. Can you help Nikita organize the results into a two-way frequency table?
Answers
Answer:
3 like vegetables but do not like fruit
Step-by-step explanation:
We have 11 total members
There are 5 members that like fruit
11-5=6
That means there are 6 members that do not like fruit
Of those 6 members ,3 do not like vegetables
6-3 =3
That means we have 3 members who do not like fruit who like vegetables
Answer:
3
Step-by-step explanation:
Assemble the proof by dragging tiles to the statements HELLPPPPPP PLEASEEEE
Answers
Answer:
See explanation
Step-by-step explanation:
1. [tex]JK\cong LM[/tex] - given
2. [tex]JL\cong LN[/tex] - definition of midpoint (the midpoint of the segment divides the segment into two congruent segments)
3. [tex]\angle LJK\cong \angle NLM[/tex] - corresponding angles theorem (corresponding angles theorem states: If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.)
4. [tex]\triangle JLK\cong \triangle LNM[/tex] - SAS (SAS postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then these two triangles are congruent).
What is the value of f(6) in the function below?
f(x) = 2x
Answers
Answer:
see below
Step-by-step explanation:
f(x) = 2x
Let x = 6
f(6) = 2*6 = 12
If you meant f(x) = 2^x
Let x = 6
f(6) = 2^6 = 64
what is the slope of the line for y = 1/2x + 2 and what is the y intercept of the line
Answers
Remember that the slope intercept formula is:
y = mx + b
m is the slope
b is the y-intercept
In this case m is 1/2 and b is 2.
This means that the slope of this line is 1/2 and the y-intercept is (0, 2) or simply 2
Hope this helped!
~Just a girl in love with Shawn Mendes
Answer:
see explanation
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
y = [tex]\frac{1}{2}[/tex] x + 2 ← is in slope- intercept form
with slope m = [tex]\frac{1}{2}[/tex] and y- intercept c = 2
What is the value of x?
Answers
Answer:
x = 8
Step-by-step explanation:
The sum of the exterior angles of a polygon = 360°
Sum the exterior angles and equate to 360
62 + 66 + 77 + 59 + 12x = 360, that is
264 + 12x = 360 ( subtract 264 from both sides )
12x = 96 ( divide both sides by 12 )
x = 8
31. A driver
can be charged with Driving While Intoxicated (DWI).
A. under 21 years old only
B. over 21 years old only
c. under 18 years old only
D. of any age
Answers
Answer:
D. of any age
Step-by-step explanation:
A driver can be charged with Driving While Intoxicated of any age.
It doesn't matter how old you are because a crime is a crime. If you are younger there will most likely be less of a punishment as adults.
-2.5y⁰ for y= -1⅔
find the value of the expression
Answers
[2 -6] - [-2 4] please help
Answers
The answer is 20. The explanation is shown in the picture.
Answer:
[tex]\Huge \boxed{20}[/tex]
Step-by-step explanation:
This question is about order of operations. It stands for parenthesis, exponent, multiply, divide, add, and subtract from left to right.
First thing you do is parenthesis.
[tex]\displaystyle [2-6]=-4[/tex]
[tex]\displaystyle -4-[-24][/tex]
Then, add and subtract numbers from left to right to find the answer.
[tex]\displaystyle -4-[-24][/tex]
[tex]\displaystyle -4-[-24]=-4+24=20[/tex]
20, which is our answer.
Hope this helps!
Kelly has 11 markers in a backpack. One of them is pink and one is blue. Find the probability Kelly will reach into the backpack without looking and grab the pink marker and then reach in a second time and grab the blue marker. Express your answer as a fraction in simplest form.
Answers
Answer:
1/110
Step-by-step explanation:
Her chance of originally grabbing a pink marker is 1/11. Her chance of grabbing a blue marker after the pink marker is taken out is 1/10. When you multiply 1/11 by 1/10, you get a 1/110 chance of getting the colored markers in this series
Answer:
The probability of her picking both out in a sequence would be [tex]\frac{1}{110}[/tex]
Step-by-step explanation:
Hello, this is a great question and one that many people struggle with in school. Hopefully I can help you understand it more clearly. Kelly has 11 markers in total within her backpack and needs to randomly pick out the pink marker. since there is only 1 that would mean her probability of picking out the pink marker is 1/11 .
Now there are 10 markers inside the backpack and she needs to randomly pick out the blue marker. since there is only 1 blue marker that would mean her probability of picking that one out is 1/10.
So now we have the following probabilities
1/11 for the pink marker 1/10 for the blue markerNow if we want to find the probability of her getting the pink marker and the blue marker one after another we would need to multiply both fractions together
[tex]\frac{1}{11} * \frac{1}{10} = \frac{1}{110}[/tex]
So the probability of her picking both out in a sequence would be [tex]\frac{1}{110}[/tex]
One angle in a triangle has a measure that is three times as large as the smallest angle. The measure of the third angle is 20 degrees more than that of the smallest angle. Find the measure of the LARGEST angle.
The LARGEST angle has a measure of ____ degrees.
Answers
Answer:
LARGEST ANGLE= 96 DEGREES
Step-by-step explanation:
MEASURE OF LARGEST ANGLE= 3A
SMALLEST ANGLE= A
THIRD ANGLE= 20+A
3A+A+20+A=180(SUM OF INTERIOR ANGLES OF TRAINGLE IS 180 DEGREES)
TRANSPOSE 20 TO RHS.
5A=180-20=160
A=160/5
=32
MEASURE OF SMALLEST ANGLE=32
LARGEST ANGLE=3A=3*32=96 DEGREES
THIRD ANGLE=32+20=52 DEGREES
Answer:
The largest angle has a measure of 96 degrees.
Step-by-step explanation:
In order to solve this problem, we have to know the fact that the sumatory of the internal angles of any triangle is 180 degrees.
With this statement in mind, and looking at the image, we can say:
180 = A + B + C (eq. 1)
Before continue any further, let's affirmt that B is the smallest angle.
Now the enunciate says "One angle in a triangle has a measure that is three times as large as the smallest angle"; This can be express as:
A = 3B (eq. 2)
The other enunciate is "The measure of the third angle is 20 degrees more than that of the smallest angle" This can be express as:
C = B + 20 (eq. 3)
Now, replacing equations 2 and 3 into 1:
(eq. 1) 180 = 3B + B + B + 20
And clearing B:
B = 32.
By knowing B, we can clear A and C from equations 2 and 3 respectively:
(eq. 2) A = 3B, so A= 96
(eq. 3) C = B + 20, so C =52
Which set of polar coordinates are plotted in the graph below
Answers
Answer:
C. (-2,2π/3)
Step-by-step explanation:
Given:
In the given plane, each line represents angles and the circles shows the radii. Now as full circle is of 360 degrees or 2π and there are 12 straight lines that means
every angle is π/6 apart (30 degrees).
As the given red point is on circle 2, this means the radii, r=2
and Ф=300 degrees or 5π/3
r=(2,5π/3)
but above is not given in the options.
Polar coordinates can have positive or negative theta and also have positive or negative radii. In case of negative radii the polar coordinate moves in opposite direction of the pole.
if we consider r=-2 then we have to move in opposite direction of pole of Ф=300 degrees or 5π/3 is 120 degrees or 2π/3.
hence r=-2 andФ=2π/3
Correct option is C:(-2,2π/3)!
-
[tex] - \sqrt{5 } - 3 \sqrt{5} [/tex]
Answers
Step-by-step explanation:
[tex]-\sqrt5-3\sqrt5=-1\sqrt5-3\sqrt5=(-1-3)\sqrt5=-4\sqrt5[/tex]
can someone please help me with this?
Answers
Answer:
A
Step-by-step explanation:
The table of values is linear so we can express the function as
y = mx + c ( m is the slope and c the y- intercept )
Calculate m using the slope formula
m = (y₂ - y₁ ) / (x₂ - x₁ )
with (x₁, y₁ ) = (- 6, - 4) and (x₂, y₂ ) = (- 3, - 2) ← ordered pairs from the table
m = [tex]\frac{-2+4}{-3+6}[/tex] = [tex]\frac{2}{3}[/tex]
Note the ordered pair (0, 0) ⇒ c = 0
y = [tex]\frac{2}{3}[/tex] x → A
A certain type of cell can double itself every hour, because it has a growth factor of 2. A laboratory has 15 of these cells.
Which function best models the amount of cells in the laboratory after x hours?
f(x)=2^x+15
f(x)=15⋅2^x
f(x)=2⋅15^x
f(x)=15^x+2
Answers
Answer:
Second one
Step-by-step explanation:
Second choice.
Second one is growing by a factor of 2 because the exponent on 2 tells me that 2 is is being repeated as a factor.
If x=0 then 15*2^0=15(1)=15 which is what we have initially.
Final answer:
The correct option is f(x)=15⋅2^x. The function that best models the cell growth after x hours is f(x) = 15⋅ 2^x, illustrating the exponential nature of cell division starting with 15 cells.
Explanation:
The amount of cells in the laboratory after x hours, when the number of cells doubles every hour, is best represented by the function f(x) = 15⋅ 2^x. This is because the cell population increases exponentially, starting with 15 cells and doubling each hour. The formula represents exponential growth where the original amount of cells (15) is multiplied by two raised to the power of the number of hours elapsed (x).
What is the equation of the following line? Be sure to scroll down first to see
all answer options.
Answers
Answer:
I'm pretty sure its A
Step-by-step explanation:
To find the slope, you divide the difference of the y-coordinates of 2 points on a line by the difference of the x-coordinates of those same 2 points .
Option: A is the correct answer:
[tex]A.\ y=-3x[/tex]
Step-by-step explanation:The equation of a line passing through two points (a,b) and (c,d) is calculated with the help of a two-point formula:
[tex]y-b=\dfrac{d-b}{c-a}\times (x-a)[/tex]
Based on the graph we observe that the line passes through:
(0,0) and (3,-9)
i.e. we have: (a,b)=(0,0) and (c,d)=(3,-9)
i.e. the equation of line is given by:
[tex]y-0=\dfrac{-9-0}{3-0}\times (x-0)\\\\i.e.\\\\y=\dfrac{-9}{3}\times x\\\\i.e.\\\\y=-3x[/tex]
The answer is: Option: A
Find the area the sector
Answers
Answer:
A= 706.85m^2
Step-by-step explanation:
A= 4• Pi•r^2
= 4•Pi•15^2
=2827.43/4
=706.8575m^2
For this case we have that by definition, the area of a circular sector is given by:
[tex]A = \frac {a * r ^ 2} {2}[/tex]
Where:
r: It's the radio
a: It is the angle of the sector
We have to:
a = 90 degrees
[tex]90\ degrees = \frac {\pi} {2}[/tex]
Then, replacing:
[tex]A = \frac {\frac {\pi} {2} * 15 ^ 2} {2} = \frac {225 \pi} {4}[/tex]
Answer:
Option A
The first term of a sequence is 4 and the common ratio is –2. What is the formula for the sequence based on term number
Answers
Answer:
its B and the next one is C
Step-by-step explanation:
Formula for the sequence based on term number is
[tex]a_{n} =(4)(2)^{n-1}[/tex].
What is geometric sequence?In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
Given
First term [tex]a[/tex] = 4
Common ratio r = -2
It is a geometric sequence because we have common ratio
Formula for the sequence based on term number is
[tex]a_{n} =ar^{n-1}[/tex]
[tex]a_{n} =(4)(2)^{n-1}[/tex]
Hence, Formula for the sequence based on term number is
[tex]a_{n} =(4)(2)^{n-1}[/tex].
Find out more information about geometric sequence here
brainly.com/question/11266123
#SPJ2
Explain the steps necessary to convert a quadratic function in standard form to vertex form
Answers
Answer:
Sample Response: A quadratic function in standard form is converted to vertex form by completing the square. The first two terms are used to create a perfect square trinomial after a zero pair is added. The zero pair is found by taking half of the x-term coefficient and squaring it. The original constant term and the negative value of the zero pair are then combined.
Step-by-step explanation:
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To prove a projectile's trajectory is parabolic, we start with the horizontal motion equation, solve for time, and substitute into the vertical motion equation, ending up with a quadratic equation in the form of y = ax + bx², showing parabolic motion.
Explanation:To prove that the trajectory of a projectile is parabolic, we start with the two separate equations for horizontal and vertical motion. The horizontal motion equation is given by x = Voxt, where Vox is the horizontal velocity component. We solve this for time t to get t = x / Vox. We then substitute this expression for t into the vertical motion equation y = Voyt - 1/2gt2, where Voy is the vertical velocity component and g is the acceleration due to gravity.
Substituting t = x / Vox into the vertical equation, we get y = Voy(x / Vox) - (1/2)g(x / Vox)2. This simplifies to y = (Voy/ Vox)x - (g / 2Vox2)x2, which is of the form y = ax + bx2, proving that the trajectory is parabolic with a and b as constants.
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