John has 7 five-dollar bills, 5 ten-dollar bills and 3 twenty-dollar bills.
Total number of bills with John = 15
Total money with John = $145
John has a mix of $5, $10 and $20 bills
Let the number of $5 bills be x, $10 bills are y and $20 bills be z
Formulating the equations:
5x + 10y + 20z = 145-------(1)
x + y + z = 15--------(2)
As per the question, the number of ten dollar bills is one less than twice the number of twenty dollar bills:
y = 2z - 1
Putting y = 2z-1 in equating (1) and (2) we get:
5x + 10(2z-1) + 20z = 145
5x + 20z - 10 + 20z = 145
5x + 40z = 155--------(3)
x + y + z = 15
x + 2z - 1 + z = 15
x + 3z = 16--------(4)
Multiplying equation (4) with 5 and using Gaussian Elimination to equate (3) and (4)
5x + 40z = 155
5x + 15z = 80
25z = 75
z = 3
Putting z=3 in (4) we get
x = 7 and y = 5
So, John has 7 five-dollar bills, 5 ten-dollar bills and 3 twenty-dollar bills.
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Which property of addition is shown in the equation below?
a + bi + 0 + 0i = a + bi
The property of addition that represents this is the zero property of addition
How to determine the property of addition?The equation is given as:
a + bi + 0 + 0i = a + bi
In the above equation, the number is added by 0 and the result is the initial number
The property of addition that represents this is the zero property of addition
Because it states that
a + 0 = 0
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a multiplication problem that will have the answer of -48.
[tex]{\sf{12 \times ( - 4) = -48}}[/tex]
Which of the following is not a function for all values of x?
Answer:
f(X) = ( - X) ½
is the answer
pls brainliest
Angel owns 10 more DVDs than Philip. Choose an equation
that represents the number of DVDs Philip owns (p) as a
function of the number of DVDs Angel owns (a).
The function of the number of DVDs Angel owns (a) will be a = p + 10
How to calculate the value?From the information, Angel owns 10 more DVDs than Philip.
The number of DVD that Philip owns was represented asp.
Therefore, the function of the number of DVDs Angel owns (a) will be:
a = p + 10
Therefore the function is a = p + 10
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Identify at least two hidden assumptions in the following argument.
I recommend giving to the Charity A because it supports so many worthwhile causes.
Which of the following are hidden assumptions for this argument? Select all that apply.
A. Giving to Charity A is better than giving to Charity B.
B. The money you give is spent on worthwhile causes.
C. Charity A does not support unworthy causes.
D. Charity A supports worthwhile causes.
Answer:
A, B, C yes
D no
Step-by-step explanation:
A. is a no-brainer. yes, that is indirectly implied.
B. this is compared to D a hidden assumption, that the money you give is guaranteed to be spent on worthwhile causes (not to mention what THAT actually means).
C. this goes combined with B. it is suggested that charity A is not engaging in unworthy causes (again, whatever that means).
D. is not hidden. that is openly said.
The Henderson family and the Ramirez family each used their sprinklers last summer. The water output rate for the Henderson family's sprinkler was 40 L per hour. The water output rate for the Ramirez family's sprinkler was 35L per hour. The families used their sprinklers for a combined total of 55 hours, resulting in a total water output of 2125 L. How long was each sprinkler used?
Answer:
Henderson family spent 40 hrs, while Ramirez spent 15 hrs.
Step-by-step explanation:
Let t = time in hrs the Henderson family used their sprinkler
Let 55 - t = time in hrs the Ramirez family used their sprinkler.
40t+35*(55-t)=2125
40t+1925-35t=2125
calculcate this and you will get t=40.
then 55-40=15.
So, Henderson family spent 40 hrs, while Ramirez spent 15 hrs.
Which may be a cost of inflation?
Higher prices
Consumer panic
Higher pay for workers
Increased business profits
Answer:
consumer panic
Step-by-step explanation:
Exit Ticket
A plumber charges a one-time surcharge fee of $50 for
emergency calls plus a rate of $65 per hour. Write a function
using function notation to determine the total amount charged
for an emergency service call. Then find f(4.5), and explain the
real-world interpretation of f(4.5).
Answer:
f(x) = $65x + $50
f(4.5) = $342.5
The real-world interpretation of f(4.5) would be an emergency service call where they have to work for 4.5 hours.
Step-by-step explanation:
We will write a function where x is the hours. Since you pay $65 for every hour, we will multiply $65 by x. Next, there is a fee of $50 so we will add that onto the end.
f(x) = $65x + $50
Now, we will plug 4.5 into the function for x and solve.
f(x) = $65x + $50
f(4.5) = $65(4.5) + $50
f(4.5) = $292.5 + $50
f(4.5) = $342.5
The real-world interpretation of f(4.5) would be an emergency service call where they have to work for 4.5 hours.
A local company has K2,500 to set up an annuity to be paid quarterly for 5 years. The payments of that annuity are to be increased in line with the current interest rate. The account pays interest at 10% p.a. compounded quarterly,
a) Calculate the size of each payment during the first year.
b) What is the size of the final payment (payment at the end)?
Answer:
Step-by-step explanation:
(a).
Within 1 year, there would be 4 payments to be made.
therefore, PMT= [tex]\frac{P(1+i)}{n} = \frac{2500(1+0.025)}{4*5} =\frac{2562.5}{20} = $128.125[/tex]
(i) 1st payment will be :PMT $128.125
(ii) 2nd payment will be :128.125*(1.025)^1= $131.33
(iii) 3rd payment will be :128.125*(1.025)^2=$134.61
(iv) 4th payment will be :128.125*(1.025)^3=$137.98
Hence the payments are: $128.125, $131.33, $134.61, and $137.98.
(b).
The size of the final payment [tex]PMT(1+i)^{m-1}[/tex] where m 4 *5= 20
[tex]=128.125(1+0.025)^{20-1} \\=128.125(1.025)^{19} \\=$204.83[/tex]
4/5 divided 1/10 math problem trying to figure out
Answer:
8
Step-by-step explanation:
(4/5)/(1/10)
oreja
take the top most number and bottom most number, multiply the two. Then multiply the two middle numbers.
(4*10)/(5*1)
Simplify and you get:
40/5
=8
19. 9a2b = c + 4a, for a
Answer:x/5 + 2b/5
Step-by-step explanation:
9a-2b=c+4a solve for a
9a-2b=c+4a
add -4a to both sides
9a-4a-2b=x+4a-4a
5a-2b=x
add 2b to both sides
5a-2b+2b=x+2b
5a= x+2b
divide by 5
5a/5 = x/5 +2b/5
a= x/5 + 2b/5
Tommy pays $550 dollars for a new cell phone plus $121-dollar monthly fee (f) for data. write an expression for the total expense of his phone?
Answer: C=550+121m
Step-by-step explanation:
help pleaseeeeeeeeee its difficult I will slide brainiest
Answer: hope this help
step-step explanation:
A pole 11 feet tall is used to support a guy wire for a tower, which runs from the tower
to a metal stake in the ground. After placing the pole, Violet measures the distance
from the pole to the stake and from the pole to the tower, as shown in the diagram
below. Find the length of the guy wire, to the nearest foot.
The length of the guy wire, to the nearest foot is 48 ft.
What is mean by foot of Maths?A unit of length (or distance) in US units equal to 12 inches. The abbreviation is: ft. Or sometimes the single quote mark: ' Example: 10 feet can be written 10 ft, or just 10' A foot is exactly 0.3048 meters in the Metric System (the foot is defined that way).
Step-by-step explanation:
Here, we want to find the length of the wire
from the diagram, we can identify two similar triangles
The smaller one and a bigger one
Mathematically, when two triangles are similar, the ratio of their corresponding sides are equal
We can start by getting the height of the tower
we have this as:
8/11 = 28/x
8 * x = 28 * 11
8x = 308
x = 308/8
x = 38.5 ft
So as we can see, the wire represents the hypotenuse of the larger triangle that measures 28 ft base and 38.5 ft height
So we can use the Pythagoras’ theorem to get the hypotenuse
Let us call this g mathematically;
According to the theorem, the square of the hypotenuse equals the sum of the squares of the two other sides
thus, we have it that;
g^2 = 38.5^2 + 28^2
g^2 = 2,266.25
g = √2,266.25
g = 47.61 which is approximately 48 ft
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Arianys wants to ride her bicycle 55.5 miles this week. She has already ridden 19 miles. If she rides for 5 more days, which equation could be used to determine, m, the average number of miles she would have to ride to meet her goal?
19m=55.5 − 5
5m=19−55.5
m=55.5−19/5
55.5=19m+5
The equation that could be used to determine, m, is 5m = 55.5 - 19. The correct option is C.
m = (55.5 - 19)/5
How do we come up with the Equation?We are to deduce from the question the equation that may be used to calculate, m, the average number of miles she would have to ride to reach her objective.
Given:
"Arianys wants to ride her bicycle 55.5 miles"
and
"She has already ridden 19 miles"
Hence,
The number of miles she needs to ride more is
55.5 miles - 19 miles
= 36.5 miles
'
If m is the average number of miles she must cycle in 5 days to reach her objective,
Then, we can write that
m = 36.5/5
OR
m = (55.5 - 19)/5
As a result, the equation that may be used to get m is 5m = 55.5 - 19.
C. m = (55.5 - 19)/5 is the right answer.
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Determine whether each equation represents a proportional relationship. If it does, identify the constant of proportionality
The proportional relationship of the given statement is a between x and y and the constant of proportionality is 0.5.
According to the statement
We have to find that the constant of proportionality.
So, For this purpose, we know that the
An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
From the given information:
each equation represents a proportional relationship.
Then
We know that the A proportional relationship is one in which two quantities vary directly with each other
That's why
An equation has a proportional relationship if it can be written in the form y=kx where k is the constant of proportionality.
A linear function of x is given as y = 0.5x - 2.
Yes, this function of x represents a proportional relationship between x and y.
The constant of proportionality is given to be 0.5 provided that the dependent variable y has an initial value of - 2 at x = 0.
The graph of this equation will be a straight line and that means that the equation consists of a proportional relationship.
So, The proportional relationship of the given statement is a between x and y and the constant of proportionality is 0.5.
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Disclaimer: This question was incomplete. Please find the full content below.
Question:
Determine whether each equation represents a proportional relationship. if it does identify the constant of proportionality. y = 0.5 x -2
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Bilquis wants to ride her bicycle 47.6 miles this week. She has already ridden 8 miles. If she rides for 6 more days, what is the average number of miles she would have to ride each day to meet her goal
The average number of miles she would have to ride each day to meet her goal is 4.7.
It is required to find the average number of miles she would have to ride each day to meet her goal.
What is algebra?A part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations.
Given:
Bilquis wants to ride her bicycle 47.6 miles this week. She has already ridden 8 miles. she rides for 6 more days.
According to given question we have,
8+4x=26
Rearrange unknown terms to the left side of the equation:
4x=26-8
4x=18
Divide 4 on both sides of the equation we get,
4x/4=18/4
x=18/4
x=4.7
Therefore, the average number of miles she would have to ride each day to meet her goal is 4.7.
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Suppose the Kalamazoo Brewing Company (KBC) currently sells its microbrews in a seven-state area: Illinois, Indiana, Michigan, Minnesota, Mississippi, Ohio, and Wisconsin. The company’s data analytics department has collected data from its distributors in each state. This data consists of the quantity and price (per case) of microbrews sold in each state, as well as the average income (in thousands of dollars) of consumers living in various regions of each state. The data for each state are available via the link below--please note there are multiple tabs at the bottom of the spreadsheet, each refers to one of the seven states selling the Kalamazoo Brewing Company’s microbrews. Excel Data File Assuming that the underlying demand relation is a linear function of price and income, use your spreadsheet program to obtain least squares estimates of Ohio’s demand for KBC microbrews.
Let's say that the Kalamazoo Brewing Company (KBC) now distributes its microbrews across a region that includes seven states: Illinois, Indiana, Michigan, Minnesota, Mississippi, Ohio, and Wisconsin. Data from the company's distributors in each state has been gathered by the data analytics division. This information includes the volume and cost of microbrews (per case) sold in each state as well as the average annual income (in thousands of dollars) of customers residing in various states' areas. Please notice that there are several tabs at the bottom of the spreadsheet, each of which corresponds to one of the seven states where the microbrews of the Kalamazoo Brewing Company are sold. The statistics for each state are available via the link below. Data File in Excel Using the supposition that the fundamental demand relationship is a linear function of price and income, utilize
With your spreadsheet application, you may predict Ohio's demand for KBC microbrews using least squares.
Ohio's demand for KBC microbrews is estimated using least squares as follows: Price + Income = 1.57 times the quantity.
Note: Sheet1 of the attached Excel document has a replication of the data from the question, and Sheet2 contains the results of the regression analysis.
Rounding to two decimal places, we find the following coefficients in the second column of the third table in Sheet2:
Price: $1.57 Profit: $15.00
Because a zero intercept was used for the analysis, the intercept is zero.
Based on the information above, the following is how the least squares estimations of Ohio's demand for KBC microbrews may be expressed:
Quantity = Price + Income + 1.57
To do the regression starting with Sheet1 to acquire
the results in Sheet, then replicate his steps:
Select the "Data Analysis" tab from the "Data" menu. Scroll down in the new window to "Regression," select it, then click "OK."
Select the column holding the Quantity data as the dependent variable by clicking in the "Input Y Range" box in the new window. Select the column holding the Price and Income data as the independent variables by clicking on the "Input X Range" box in the new window. Add "Labels," "Confidence level (95%)," and "Constant is Zero" to your selections. After that, click "OK" to get the results in Sheet2.
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Part 2 of 2
You invested $15,000 in two accounts paying 4% and 7% annual interest, respectively. If the total interest earned for
the year was $960, how much was invested at each rate?
$3000 and $12000
If you know the principal amount, the rate of interest, and the time periods, you can use the simple interest formula to calculate the interest.
The following is a simple interest formula:
SI = P *R * T /100
Where SI = simple interest
Principal (P)
interest rate, or R (in percentage)
T is the time period (in years)
The following formula is used to determine the overall sum:
Sum (A) = Principal (P) + Interest (I)
Where,
The entire amount repaid at the conclusion of the borrowing period is indicated by the letter (A).
You can also write the total amount formula for simple interest as follows:
A = P(1 + RT)
X+Y = $15000
0.04x + 0.07y = 960
0.04x + 0.07 ( 15000 - x) = 960
0.04x + 1050 - 0.07x = 960
0.03x = 90
x = $3000
y= $12000
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−4x+2y+z=1;x−y+3x=−5;3x+y−4z=10 Solve each system, HELP!
Answer: x = [tex]\frac{-31}{23}[/tex], y = [tex]\frac{-9}{23}[/tex], z = [tex]\frac{-83}{23}[/tex]
Step-by-step explanation :Solving system of equations
Given the following system of equation below
- 4x + 2y + z =1 .......(1)
x − y + 3x = −5 ........(2)
3x + y − 4z = 10.........(3)
first we solve eq 1 and eq 3:
we get, -13x + 9y = 14 ..........(4)
Now we solve eq 2 and eq 4 and get value of x and y:
we get, x = [tex]\frac{-31}{23}[/tex] and y = [tex]\frac{-9}{23}[/tex]
finally we put value of x and y in eq 1 and get the value of z:
-4(-31/23) + 2(-9/23) + z = 1
106/23 + z = 1
z = 1 - 106/23
z =[tex]\frac{-83}{23}[/tex]
Hence the solution of equation is ([tex]\frac{-31}{23}[/tex], [tex]\frac{-9}{23}[/tex], [tex]\frac{-83}{23}[/tex])
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what is the answer to 2 1/5 - 1 3/10=
Answer: 1/10
Step-by-step explanation:2 * 1/5 is 2/5 and 1*3/10 is 3/10 and when you subtract them you get 1/10
Answer:9/10
Step-by-step explanation:get 2 1/5 into a improper fraction thats 11/5 then multiply by to for the common demontor thats 22/10
get 1 3/10 into a improper fraction 13/10
then minus 22-13 thats 9
then bring in the bottom 9/10 hope that helps
If the terminal side of angle θ passes through a point on the unit circle in the first quadrant where y=0, what is the exact measure of θ?
Step-by-step explanation:
y of o but you dont have to use y
Square LMNO has vertices, L(−4, 2), M(1, 2), N(1, −3), and O(−4, −3). Square LMNO is translated 3 units left and 4 units up to produce square L′M′N′O′.
Which coordinates describe the vertices of the image?
L′(−7, 6), M′(−2, 6), N′(−2, 1), and O′(−7, 1)
L′(−1, 6), M′(4, 6), N′(4, 1), and O′(−1, 1)
L′(0, −1), M′(5, −1), N′(5, −6), and O′(0, −6)
L′(0, 5), M′(5, 5), N′(5, 0), and O′(0, 0)
PLS HELP ME IS ANYONE SPAMS I WILL REPORT YOU AND BAN YOU IMMEDIATLEY BUT IF CORRECT I WILL GIVE BRAINLIESTPLS HURRRY THIS IS DUE TODAY
The coordinate of the square L'M'N'O' will be L′(−7, 6), M′(−2, 6), N′(−2, 1), and O′(−7, 1). Then the correct option is A.
What is a transformation of a point?Each higher dimensional space mapping to itself constitutes a spatial transformation, which preserves some spatial link between the figures.
Square LMNO has vertices, L(−4, 2), M(1, 2), N(1, −3), and O(−4, −3). Square LMNO is translated 3 units left and 4 units up to produce square L′M′N′O′.
The transformation rule will be given as,
(x', y') = (x - 3, y + 4)
Then the coordinate of the square L'M'N'O' will be
L′(−7, 6), M′(−2, 6), N′(−2, 1), and O′(−7, 1)
Then the correct option is A.
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Transportation costs include car payments and costs for gasoline, repairs, and so on. What is their average monthly expenditures for transportation costs?
Group of answer choices
$879.92
$433.52
$278.50
$275.00
Answer: 433.52
Step-by-step explanation: it depends if you have the pass or not
writing linear equations: two points or graph worksheet, please help
Answer:
Q3
Slope = 5
Standard= y=5X+15
Q5
Slop =-4
Standard= y=-4x+4
Step-by-step explanation:
m=-5-5/-4+2=-10/-2= 5
y-5=5(x+2)
y-5=5x+10
y=5x+15
C=-
Convert the following temperatures from Fahrenheit to Celsius or vice versa.
F - 32
1.8
a. 50°F
F = 1.8C +32
b. 75°C
C. -40°C
Answer:
A)50°F = 10°C
B)75°C = 167°F
C)-40°C = -40°F
Step-by-step explanation:
A) 50°F = 10°C
Formula
(50°F − 32) × 5/9 = 10°C
B) 75°C = 167°F
Formula
(75°C × 9/5) + 32 = 167°F
C) -40°C = -40°F
Formula
(-40°C × 9/5) + 32 = -40°F
The distance from the tip of a slice of pizza to the crust is 7 inches.
Does this represent diameter, circumference, or radius?
ANSWER ASAP THANK YOU
Answer:
radius
Step-by-step explanation:
The fraction model below shows the steps that a student performed to find a quotient. Which statement best interprets the quotient?
The statement that best interprets the quotient is "there are 6( 1/4 ) two - third in 4( 1/6 )".
In mathematics, the sum obtained by diluting two numbers is known as a quotient. The term "quotient" is used frequently in mathematics and is sometimes known as the integer portion of a division, a fraction, or a ratio.
Any integer may be used as the larger number in a fraction, which is a number that represents a percentage of the larger number. It is shaped like a denominator and a numerator.
For step 1:
The shaded part is:
4 + 1/6 = 4(1/6) = 25/6
Now, dividing the shaded part by the unshaded part,
( 25/6 ) / ( 2/3 ) = ( 25/6 ) × ( 3/2 )
( 25/6 ) / ( 2/3 ) = 25/4
( 25/6 ) / ( 2/3 ) = 6( 1/4 )
Hence, there are 6( 1/4 ) two - third in 4( 1/6 ).
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For which equations below is x = -3 a possible solution? Select three options
x = -3 a possible solution for |x| = 3, |–x| = 3 and –|x| = –3.
Here, we are given 6 equations and we need to check for which of the following, x = -3 is a solution.
Let us look at each equation one by one,
1. |x| = 3
if x = -3
|-3| = 3
3 = 3
Hence, -3 is a solution to this equation.
2. |x| = –3
if x = -3
|-3| = -3
3 ≠ -3
Hence, -3 is not a solution to this equation.
3. |–x| = 3
if x = -3
|3| = 3
3 = 3
Hence, -3 is a solution to this equation.
4. |–x| = –3
if x = -3
|3| = -3
3 ≠ -3
Hence, -3 is not a solution to this equation.
5. –|x| = –3
if x = -3
-|3| = -3
-3 = -3
Hence, -3 is a solution to this equation.
4. –|x| = 3
if x = -3
-|3| = 3
-3 ≠ 3
Hence, -3 is not a solution to this equation.
Thus, equations |x| = 3, |–x| = 3 and –|x| = –3 are correct.
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Your solution was incomplete. Check for full content below-
For which equations below is x = –3 a possible solution? Check all that apply.
|x| = 3
|x| = –3
|–x| = 3
|–x| = –3
–|x| = –3
–|x| = 3
NO LINKS!! Please help me with these problems. Part 13a1
Answer's:
27. vertex: (-4, -5) and x-intercept: (√5 - 4, 0) and (-√5 - 4, 0)
28. vertex: (-5, -11) and x-intercept: (√11 - 5, 0) and (-√11 - 5, 0)
29. vertex: (1, 16) and x-intercept: (-3, 0) and (5, 0)
To find vertex, the quadratic function should be in f(x) = a(x - h)² + k form.
where:
(h, k) is the vertex27.
g(x) = x² + 8x + 11
completing square:
g(x) = (x² + 8x) + 11
g(x) = (x + 4)² + 11 - (4)²
g(x) = (x + 4)² - 5
g(x) = (x - (-4))² - 5
To find x-intercept: set y = 0
[tex]\sf (x + 4)^2 - 5 = 0[/tex]
[tex]\sf (x + 4)^2 = 5[/tex]
[tex]\sf x + 4 = \pm \sqrt{5}[/tex]
[tex]\sf x= \sqrt{5}-4, \ - \sqrt{5}-4[/tex]
Hence, the vertex is (-4, -5) and x intercepts (√5 - 4, 0) and (-√5 - 4, 0).
28.
f(x) = x² + 10x + 14
completing square:
f(x) = (x² + 10x) + 14
f(x) = (x + 5)² + 14 - (5)²
f(x) = (x + 5)² - 11
Find x intercept, so y = 0:
[tex]\sf (x + 5)^2 - 11 = 0[/tex]
[tex]\sf (x + 5)^2 = 11[/tex]
[tex]\sf x + 5 = \pm\sqrt{11}[/tex]
[tex]\sf x = \sqrt{11} -5, \ -\sqrt{11} -5[/tex]
Hence, the vertex is (-5, -11) and x intercepts (√11 - 5, 0) and (-√11 - 5, 0).
29.
f(x) = -(x² - 2x - 15)
f(x) = -((x² - 2x)) + 15
f(x) = -(x - 1)² + 15 + (-1)²
f(x) = -(x - 1)² + 16
Find the x-intercept's:
[tex]\sf -(x - 1)^2 + 16 = 0[/tex]
[tex]\sf -(x - 1)^2 = -16[/tex]
[tex]\sf (x - 1)^2 = 16[/tex]
[tex]\sf x - 1 = \pm\sqrt{16}[/tex]
[tex]\sf x - 1 = \pm4[/tex]
[tex]\sf x = -3, \ 5[/tex]
Hence, the vertex is (1, 16) and x intercepts (-3, 0) and (5, 0).
Answer:
[tex]\textsf{27.} \quad \textsf{Vertex}:(-4,-5) \quad \textsf{$x$-intercepts}: x=-4+\sqrt{5}, \:\:x=-4-\sqrt{5}[/tex]
[tex]\textsf{28.} \quad \textsf{Vertex}:(-5,-11) \quad \textsf{$x$-intercepts}: x = -5+\sqrt{11}, \:\:x=-5-\sqrt{11}[/tex]
[tex]\textsf{29.} \quad \textsf{Vertex}:(1,16) \quad \textsf{$x$-intercepts}: x = 5, \:\:x=-3[/tex]
Step-by-step explanation:
Vertex form of a quadratic function:
[tex]\boxed{y=a(x-h)^2+k}[/tex]
where:
(h, k) is the vertex.[tex]a[/tex] is some constant.Question 27Given function:
[tex]g(x)=x^2+8x+11[/tex]
Change to vertex form by completing the square.
Add and subtract the square of half the coefficient of x:
[tex]\implies g(x)=x^2+8x+11 +\left(\dfrac{8}{2}\right)^2-\left(\dfrac{8}{2}\right)^2[/tex]
[tex]\implies g(x)=x^2+8x+11 +16-16[/tex]
[tex]\implies g(x)=x^2+8x+16+11-16[/tex]
[tex]\implies g(x)=x^2+8x+16-5[/tex]
Factor the perfect trinomial:
[tex]\implies g(x)=(x+4)^2-5[/tex]
Therefore, the vertex is (-4, -5).
To find the x-intercepts, set the function to zero and solve for x:
[tex]\begin{aligned}g(x) & = 0\\\implies (x+4)^2 -5 & = 0\\(x+4)^2 & = 5\\\sqrt{(x+4)^2} & = \sqrt{5}\\x+4&=\pm\sqrt{5}\\x+4-4&=-4\pm\sqrt{5}\\x&=-4\pm\sqrt{5}\end{aligned}[/tex]
Therefore, the x-intercepts are:
[tex]x = -4+\sqrt{5}, \quad x = -4-\sqrt{5}[/tex]
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Question 28Given function:
[tex]f(x)=x^2+10x+14[/tex]
Change to vertex form by completing the square.
Add and subtract the square of half the coefficient of x:
[tex]\implies f(x)=x^2+10x+14+\left(\dfrac{10}{2}\right)^2-\left(\dfrac{10}{2}\right)^2[/tex]
[tex]\implies f(x)=x^2+10x+14+25-25[/tex]
[tex]\implies f(x)=x^2+10x+25+14-25[/tex]
[tex]\implies f(x)=x^2+10x+25-11[/tex]
Factor the perfect trinomial:
[tex]\implies f(x)=(x+5)^2-11[/tex]
Therefore, the vertex is (-5, -11).
To find the x-intercepts, set the function to zero and solve for x:
[tex]\begin{aligned}f(x) & = 0\\\implies (x+5)^2 -11 & = 0\\(x+5)^2 & = 11\\\sqrt{(x+5)^2} & = \sqrt{11}\\x+5&=\pm\sqrt{11}\\x+5-5&=-5\pm\sqrt{11}\\x&=-5\pm\sqrt{11}\end{aligned}[/tex]
Therefore, the x-intercepts are:
[tex]x = -5+\sqrt{11}, \quad x = -5-\sqrt{11}[/tex]
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Question 29Given function:
[tex]f(x)=-(x^2-2x-15)[/tex]
Change to vertex form by completing the square.
Add and subtract the square of half the coefficient of x:
[tex]f(x)=-\left(x^2-2x-15+\left(\dfrac{-2}{2}\right)^2-\left(\dfrac{-2}{2}\right)^2\right)[/tex]
[tex]f(x)=-\left(x^2-2x-15+1-1\right)[/tex]
[tex]f(x)=-\left(x^2-2x+1-15-1\right)[/tex]
[tex]f(x)=-\left(x^2-2x+1-16\right)[/tex]
Factor the perfect trinomial:
[tex]f(x)=-\left((x-1)^2-16\right)[/tex]
Simplify:
[tex]f(x)=-(x-1)^2+16[/tex]
Therefore, the vertex is (1, 16).
To find the x-intercepts, set the function to zero and solve for x:
[tex]\begin{aligned}f(x) & = 0\\\implies -(x-1)^2 +16 & = 0\\(x-1)^2 -16 & = 0\\(x-1)^2 & = 16\\\sqrt{(x-1)^2} & = \sqrt{16}\\x-1&=\pm4\\x-1+1&=1\pm4\\x&=5,-3\end{aligned}[/tex]
Therefore, the x-intercepts are:
[tex]x = 5, \quad x=-3[/tex]
