The greatest common factor (GCF) of 7a³, 14a, and -21a, is 7a.
In algebra, the greatest common factor (GCF) is the largest positive integer that divides two or more integers without leaving a remainder. Finding the GCF of algebraic terms involves factoring each term into its prime factors. The GCF of the terms is then the product of the common factors with the smallest exponents. In this problem, we had to find the GCF of 7a³, 14a, and -21a. By factoring each term, we found that the GCF is 7a.
It's important to simplify each term before finding the GCF to ensure that all the common factors are identified.
7a³ = 7 * a * a * a
14a = 2 * 7 * a-21
a = -3 * 7 * a
The GCF of these terms is the product of the common factors with the smallest exponents.
Therefore, the GCF is:
7 * a = 7a
So, the GCF of 7a³, 14a, and -21a is 7a.
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At the DVD rental store, Jamie found 6 DVDs that she wanted, but can only rent 4. How many possible choices can she make
At the DVD rental store, Jamie found 6 DVDs that she wanted, but can only rent 4. Jamie can make 15 possible choices, as per combinations of DVDs.
Number of Possible Combinations:
Given Information is as follows,
Total number of DVDs that Jamie wanted, n = 6
Number of DVDs Jamie can rent at a time, x =4
The Combinations formula is given as,
ⁿCₓ = n! / (n-x)! x!
Here, n = 6 and x = 4
Substituting these values of n and x in the Combinations formula, we get,
⁶C₄ = 6! / (6-4)! 4!
⁶C₄ = 6! / 2! 4!
⁶C₄ = 6×5×4! / 2! 4!
⁶C₄ = 6×5 / 2
⁶C₄ = 3×5
⁶C₄ = 15
Thus, Jamie can make 15 possible combinations of DVDs.
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Determine the (x, y) coordinates of the vertex of the parabola that represents each of the following functions:
The vertices of the parabolae are:
(h, k) = (- 3, - 1)(h, k) = (1, - 9)(h, k) = (4, 1)(h, k) = (3/4, 9/4)(h, k) = (- 3, 2)(h, k) = (0, 36)(h, k) = (7/2, 9/4)(h, k) = (5, - 1)(h, k) = (1, - 3)(h, k) = (- 1/2, 1)How to find the coordinates of the vertex of a parabola
Parabolae are represented by quadratic equations. In this problem we have parabolae in standard form and we need to determine its vertex form to find the needed information. Now we summarize the forms of quadratic equations:
Standard form
y = a · x² + b · x + c (1)
Vertex form
y - k = C · (x - h)² (2)
Please notice that (x, y) = (h, k) represents the vertex of the parabola.
To change quadratic equations from standard form into vertex form we need to apply algebraic handling:
y = x² + 6 · x + 8
y + 1 = x² + 6 · x + 9
y + 1 = (x + 3)²
(h, k) = (- 3, - 1)
y = x² - 2 · x - 8
y + 9 = x² - 2 · x + 1
y + 9 = (x - 1)²
(h, k) = (1, - 9)
y = - x² + 8 · x - 15
y = - 1 · (x² - 8 · x + 15)
y - 1 = - 1 · (x² - 8 · x + 16)
y - 1 = - 1 · (x - 4)²
(h, k) = (4, 1)
y = - 4 · x² + 6 · x
y = - 4 · [x² - (3/2) · x]
y + (- 4) · (9/16) = - 4 · [x² - (3/2) · x + 9/16]
y - 9/4 = - 4 · (x - 3/4)²
(h, k) = (3/4, 9/4)
y = x² + 6 · x + 11
y - 2 = x² + 6 · x + 9
y - 2 = (x + 3)²
(h, k) = (- 3, 2)
y = - x² + 36
y - 36 = - x²
(h, k) = (0, 36)
y = - x² + 7 · x - 10
y = - (x² - 7 · x + 10)
y + (- 1) · (9/4) = - (x² - 7 · x + 49/4)
y - 9/4 = - (x - 7/2)²
(h, k) = (7/2, 9/4)
y = x² - 10 · x + 24
y + 1 = x² - 10 · x + 25
y + 1 = (x - 5)²
(h, k) = (5, - 1)
y = 2 · x² - 4 · x - 1
y = 2 · (x² - 2 · x - 1/2)
y + 2 · (3/2) = 2 · (x² - 2 · x + 1)
y + 3 = 2 · (x - 1)²
(h, k) = (1, - 3)
y = - 4 · x² - 2 · x
y = - 4 · [x² + (1/2) · x]
y + (- 4) · (1/4) = - 4 · [x² + (1/2) · x + 1/4]
y - 1 = - 4 · (x + 1/2)²
(h, k) = (- 1/2, 1)
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Determine the number of terms in the sequence: –45, –32, –19, –6, ..., 124.
Step-by-step explanation:
[tex] - 32 + 45 = 13 \\ t_{n} = ( a_{1} + (n - 1)d) \\ \\ d = 13 \: \: a_{1} = - 45[/tex]
[tex] t_{n} = - 45 + (n - 1)13 = = = > \\ - 45 + 13n - 13 = = = > \\ t_{n} = 13n - 58[/tex]
and now
[tex]124 = 13n - 58 = = = > \\ 182 = 13n = = = > n = 14[/tex]
The number of terms in the sequence: –45, –32, –19, –6, ..., 124 = 9.
The common difference is -45 - (-32)= 13
d = 13.
What is arithmetic progress?AP is a sequence of numbers in order, in which the difference among any two consecutive numbers is a constant cost. it's also referred to as mathematics series.
using arithmetic progress:-
last term = (n-1)d
first term(a) = –45
term = a + (n-1)d
there is a difference of 13, so the sequence will be
–45, –32, –19, –6,7, 20, 33, 46, 59, 72, 85, 98, 111, 124.
∴ number of terms = 9
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A sculpture is formed from a square-based pyramid resting on a cuboid. The base of the cuboid and the base of the pyramid are both squares of side 3 cm. The height of the cuboid is 8 cm and the total height of the sculpture is 15 cm. The total mass of the sculpture is 738 g. The cuboid-part of the sculpture is made of iron with density 7.8 g/cm³. The pyramid is made from copper. Calculate the density, in g/cm³, of the copper.
Answer:
7.8g/cm
Step-by-step explanation:
Ellen has a bag with 3 red marbles and 2 blue marbles in it. she is going to randomly draw a marble from the bag 300 times, putting the marble back in the bag after each draw. how many times do you predict that the marble picked will be blue using the theoretical probability?
The blue marble is predicted to be picked 120 times, in the experiment of picking a marble from a bag containing 3 red and 2 blue marbles and performing this experiment 300 times, using the theoretical probability.
The theoretical probability of any event is the ratio of the number of favorable outcomes to the event, to the total number of possible outcomes in the experiment.
If we have an event A, the number of favorable outcomes to event A as n, and the total number of possible outcomes in the experiment as S, then the theoretical probability of event A is given as:
P(A) = n/S.
In the question, we are given that Ellen has a bag with 3 red marbles and 2 blue marbles in it. She is going to randomly draw a marble from the bag 300 times, putting the marble back in the bag after each draw.
We are asked the predict the number of times that the marble picked will be blue using the theoretical probability.
Let the event of picking a blue marble be A.
The number of favorable outcomes to event A (n) = 2 {The total number of blue marbles in the bag}.
The total number of possible outcomes in the experiment of picking a ball (S) = 5 {The total number of marbles in the bag}.
Thus, the theoretical probability of event A is,
P(A) = n/S = 2/5 = 0.4.
To predict the number of times marble picked was blue, we multiply the time's the experiment was performed by the theoretical probability of picking a blue ball.
Thus, the predicted number of times = 300 * 0.4 = 120.
Thus, the blue marble is predicted to be picked 120 times, in the experiment of picking a marble from a bag containing 3 red and 2 blue marbles and performing this experiment 300 times, using the theoretical probability.
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Please solve for X: 11/2-x = 1/3 + 1/6
Answer: 5
Step-by-step explanation:
11/2-x = 1/3 + 1/6
-x+11/2 = 1/2
- 11/2 - 11/2
-x = -5
Divide -1 by both sides
And it gives you the answer 5.
If x =2 y=3 z=4 solve the following
x² + y²
Answer:
13
Step-by-step explanation:
Given x = 2, y = 3 and z = 4. We'll evaluate the value of x² + y² with given condition.
First, remind that we are only given the expression of x-term and y-term only and therefore, z-term is not included - it's not to be considered.
Substitute x = 2 and y = 3 in the expression:
[tex]\displaystyle{2^2+3^2 = 4+9}\\\\\displaystyle{4+9 = 13}[/tex]
Hence, the value of x² + y² when x = 2 and y = 3 is 13.
Please let me know if you have any questions!
Solve for x in the diagram below.
Answer:
x = 20
Step-by-step explanation:
x°
2x°
(x + 10)°
little box on the bottom left means this is a right angle which is 90°
add up all the angles and make it equal to 90°
( x° + 2x° + (x + 10)° ) = 90°
4x + 10 = 90
4x = 80
x = 20
Since we can see the "square" at the bottom left corner of the angle, the square indicates that the angle is a right angle (which measures 90°).
We can also see three smaller angles forming the right angle. Therefore, the sum of the measure of the smaller angles = 90°.
According to the diagram, the measures of the smaller angles are x°, 2x°, and (x + 10)° respectively. Then we get the following equation:
[tex]\implies x + 2x + (x + 10) = 90\°[/tex]
Step-2: Solving the equation obtained in step-1Here, we had the equation: [tex]\underline{x + 2x + (x + 10) = 90\°}[/tex]
[tex]\implies x + 2x + x + 10 = 90\°[/tex] [tex]\text{(Ope} \text{ning the parentheses)}[/tex]
[tex]\implies 4x + 10 = 90\°[/tex] [tex]\text{(Combining like terms)}[/tex]
[tex]\implies 4x + 10 - 10 = 90 - 10[/tex] [tex]\text{(Subtracting 10 on both sides)}[/tex]
[tex]\implies 4x = 80[/tex] [tex]\text{(Simplifying both sides)}[/tex]
[tex]\implies \dfrac{4x}{4} = \dfrac{80}{4} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{(Dividing 4 on both sides)}[/tex]
[tex]\implies \boxed{x = 20}[/tex] [tex]\text{(Simplifying both sides)}[/tex]
Therefore, the value of x, in the diagram provided, is 20.
The prime factors of 13195 are 5, 7, 13 and 29. What is the largest prime factor of the number 600851475143 ?
Answer:
The prime factors of 13195 are 5, 7, 13 and 29. The largest prime factor of the number 600851475143 is 6857
Area=
Help me please asap thanks
Answer:
10 unit^2.
Step-by-step explanation:
The base of the triangle (CE)
= 6 - 2 = 4 units
The height = 5
Area = 1/2 * 4 * 5
= 10 unit^2.
Answer:
10
Step-by-step explanation:
area = hight ×base÷2
hight = 5-0
= 5
base = 6-2
= 4
area = 4×5÷2
=10
Solve kx-2=7 for x. A. x=5/k B. x=9k C. x=9-k D. x=9/k
Answer:
the answer is D- 9/k
Answer:
D- 9/k
Step-by-step explanation:
Please help me i need to turn this in soon and i don't understand
circumference of the mirror = 175.84 cm
area of the mirror = 2461.76 cm²
The measure needed to find the amount of wire round the mirror is the circumference.
The measure needed to find the amount of glass needed is the area.
How to find area and circumference of a circle?The circumference and area of a circle can be found as follows:
circumference of the mirror = 2πr
circumference of the mirror = 2 × 3.14 × 28 = 175.84 cm
area of the mirror = πr²
area of the mirror = 3.14 × 28²
area of the mirror = 2461.76 cm²
The measure needed to find the amount of wire round the mirror is the circumference.
The measure needed to find the amount of glass needed is the area.
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I think of a number. i add 7 to it and then divide by 10. my answer is 3.8. what number did i start with
Answer:
31
Step by step:
You need to do it in reverse:
Devidng Becomes multiplying so its 3.8x10=38
Then same for this + = - do its 38-7=31
So answer is 31
find the equation of the straight line with ng gradient and points. b) - 1/2, (2,3)
Answer:
[tex]y = -\frac{1}{2} x+4[/tex]
Step-by-step explanation:
Standard form for equation of line: y = mx + c
where m = gradient or slope
c = y-intercept
From the question we know that,
x = 2
y = 3
m = - 0.5
We will substitute these values into the equation to find c.
3 = (-0.5)(2) + c
3 = - 1 + c
c = 3 + 1 = 4
Therefore the equation of this line is [tex]y = -\frac{1}{2} x+4[/tex]
Which quadratic function is in standard
form?
Answer:
h(x) = 2x² -8x -10
Step-by-step explanation:
In the US, a quadratic is in standard form when the terms are listed in order of decreasing degree.
In the UK, a quadratic is in standard form when it is written in vertex form.
__
USThe only function with terms in order of decreasing degree is ...
h(x) = 2x² -8x -10
UKAll of the functions except the last are written in vertex form:
f(x) = (x +1)² +0 . . . . . . . . would usually be written f(x) = (x +1)²y(x) = -(x -6)² +16g(x) = -3(x -3)² +4__
Additional comment
Since the question asks about one function, we assume it is from the perspective of the US understanding of standard form. The point here is that "standard form" may vary from one tradition to another. (The "standard form" for numerical values varies by tradition, as well.)
Show your steps in evaluating each of the following expressions. The steps count 4 points each, the answer is 1
The steps in evaluating each of the following expressions is shown below.
What is an expression?An expression is a mathematical equation which shows the relationship that exist between two or more numerical quantities or variables.
How to evaluate the given expressions?15 - 35/7 - 2 + 3 - 4
15 - (35/7) - 2 + 3 - 4 (bracket and division)
15 - 5 - 2 + 3 - 4 (regroup)
15 + 3 - 5 - 2 - 4 (subtract and add)
18 - 11 = 7.
Expression 2.10 + 2(9 - 5) - 16/18
10 + (2 × 4) - 8/9 (bracket and division)
10 + 8 - 8/9 (add)
18 - 8/9 (subtract)
162/9 - 8/9 = 17 1/9 or 154/9.
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Complete Question:
Show your steps in evaluating each of the following expressions. The steps count 4 points each, the answer is 1 point.
A. 15 - 35/7 -2 + 3 -4
B. 10 + 2(9 - 5) - 16/18
What is the logarithmic form of 13^2=169
The logarithmic form of 13^2=169 is [tex]\log_{13}(169) = 2[/tex]
How to determine the logarithmic form?The equation is given as
13^2=169
Take the logarithm of both sides
log(13^2) = log(169)
This gives
2log(13) = log(169)
Divide both sides by log(13)
[tex]\log_{13}(169) = 2[/tex]
Hence, the logarithmic form of 13^2=169 is [tex]\log_{13}(169) = 2[/tex]
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Rearrange the equation so q is the independent variable.
-7q+12r=3q-4r
Rearranging the equation so q is the independent variable is; q = -1.6r
How to change the subject of a subject?We are given the expression;
-7q + 12r = 3q - 4r
Since we want to make q the subject, let us rearrange the equation with variables having q on the left and others on the right side to get;
-7q - 3q = -4r - 12r
-10q = -16r
q = 16r/10
q = -1.6r
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Simplify: 2n (n^2 + 3n + 4)
Answer:
2n^3 + 6n^2 + 8n
Answer: It's 2n^3 + 6n^2 + 8n
Step-by-step explanation:
Find the missing side lengths. Show work.
x=______ y=_____
At a coffee shop, the manager recorded the number of customers who visited the store at the end of each hour. The graph shows the recordings for a 24-hour period. The function describing this graph is a transformation of the parent sine function, y=sin(x)
Which value is closest to the amplitude of the transformed function?
O 83 customers
O 27 customers
O 54 customers
O 30 customers
Based on the transformed function of y = sin(x) and the given parameters, the value that is closest to the amplitude of the transformed function is 54
Which value is closest to the amplitude of the transformed function?The amplitude of the function is calculated
Amplitude = Highest - Lowest
From the graph, we have the following points
Highest = 84
Lowest = 30
Substitute the known values in the above equation
Amplitude = 84 - 30
Evaluate the difference
Amplitude = 54
Based on the transformed function of y = sin(x) and the given parameters, the value that is closest to the amplitude of the transformed function is 54
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A quantity with an initial value of 310 grows continuously at a rate of 8.5% per second. what is the value of the quantity after 1.2 minutes, to the nearest hundredth?
The value of the quantity after 1.2 minutes, to the nearest hundredth, is 313.56
What is the calculated quantity?A quantity is a measurable attribute of a particular thing or group of objects. One quantity might be more than, less than, or equal to another quantity when comparing them. Quantity is a notion that is used frequently in both mathematics and the sciences.
A quantity cannot be a property that cannot be compared. The conventional format for presenting a quantity is as the product of a magnitude and a unit.
According to the question,
Initial value of the quantity=310
Rate of growth= 8.5% per second
Value of the quantity after 1.2 minutes,
=[tex]310(1+\frac{8.5}{100}) ^{1.2*60}[/tex]
=[tex]310(\frac{108.5}{100})^{72}[/tex]
=313.56
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Answer:
141008.06
Step-by-step explanation:
Is there a way to find the intersections only by the equations?
I just know how to find the intersections by graphing them.
Answer:
see explanation
Step-by-step explanation:
to find the intersection equate g(x) and h(x) , that is
(x + 7)(x - 5) = x - 5 ← expand left side using FOIL
x² + 2x - 35 = x - 5 ( subtract x - 5 from both sides )
x² + x - 30 = 0 ← in standard quadratic form
(x + 6)(x - 5) = 0 ← in factored form
equate each factor to zero and solve for x
x + 6 = 0 ⇒ x = - 6
x - 5 = 0 ⇒ x = 5
substitute these values into either g(x) or h(x) for corresponding values of y
substituting into h(x)
h(- 6) = - 6 - 5 = - 11 ⇒ (- 6, - 11 )
h(5) = 5 - 5 = 0 ⇒ (5, 0 )
Select the correct answer. Which function has a domain of all real numbers?
y = -x + 5 y = -2(3x) ³ O A. B. OC. y = CD. y = (x + 2)² (2x) ³ (2x) - 7
The function that has a domain of all real numbers is:
A. [tex]y = 2x^{\frac{1}{3}} - 7[/tex].
What is the domain of a function?The domain of a function is the set that contains all possible input values for the function.
If a function has an even root, equivalent to an exponent of [tex]\frac{1}{n}[/tex] with n even, the domain is only positive values, while if the exponent is odd, the domain is all real values.
Researching the problem on the internet, the function with odd exponent is:
A. [tex]y = 2x^{\frac{1}{3}} - 7[/tex].
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help i’ll give brainliest!
Which system models this situation? y = 26x and y = 8,400(x-500)2 15,900 y = 26x and y = -0.030(x-500)2 15,900 y = x/26 and y = -0.030(x-500)2 15,900 y = x/26 and y =8,400(x-500)2 15,900
The option B is a correct option which is y = 26x and y = -0.030(x-500)2 15,900 represent the quadratic and linear function.
According to the statement
we have given that Vertex
(h,k) = (500, 15900)
And One of the points on the graph
(x,y) = (0,8400)
FIRST PART: we have to find the quadratic function
We can find the quadratic function by parabola's equation formula
y = a (x - h)² + k -(1)
Input the numbers to the formula, to find the value of a
y = a (x - h)² + k
8400 = a(0 - 500)² + 15900
8400 = a (500)² + 15900
8400 = 250000a + 15900
-250000a = 15900 - 8400
-250000a = 7500
a = 7500/-250000
a = 0.03
Now,
Submit a to the formula (1)
y = a (x - h)² + k
y = 0.03 (x - 500)² + 15900
this is the quadratic equation.
SECOND PART: Find the linear function
the total cost = cost each helmet × the number of helmet
y = 26x
So,
The option B is a correct option which is y = 26x and y = -0.030(x-500)2 15,900 represent the quadratic and linear function.
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Disclaimer: This question was incomplete. Please find the full content below.
Question: A company plans to sell bicycle helmets for $26 each. The company's business manager estimates that the cost, y, of making x helmets is a quadratic function with a y-intercept of 8,400 and a vertex of (500, 15900)
x= number of helmets
y = amount in dollars
Which system models this situation?
a) y = 26x and y = 8,400(x-500)2+15,900
b) y = 26x and y = -0.030(x-500)2+15,900
c) y = x/26 and y = -0.030(x-500)2+15,900
d) y = x/26 and y =8,400(x-500)2+15,900
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Answer:
B. y = 26x and y = -0.030(x-500)2+15,900
Step-by-step explanation:
On the 405 freeway one can travel 65 miles per hour (mph), how many kilometers per hour (kph) is the car travelling
Answer:
104.607kph
Step-by-step explanation:
Conversion from mph to kph = 1.609mph
In Circle P, chord AB measures 4x - 6 centimeters and chord CD measures 6x - 12
centimeters. If Segment AB and Segment CD are each 4 centimeters from P, find AP.
Answer:
5 cm
Step-by-step explanation:
When two chords are equidistant from the center of the circle, the two chords have equal length. Therefore, the length of chord AB, 4x - 6, is equal to the length of chord CD, 6x - 12.
4x - 6 = 6x - 12
6 = 2x
x = 3
Now that we know x = 3, we can substitute the value back into the original expression, 4x - 6, to find the length of chord AB.
AB = 4x - 6 = 4*3 - 6 = 12 - 6 = 6 cm
When measuring the distance between a line and a point, we create a segment through the point perpendicular to the line. In Geometry, we also learn that the perpendicular bisector of a chord in a circle contains the center of the circle.
In this case, the gray line perpendicularly bisects segment AB. PX is 4 cm long and AX is 3 cm long (because AX is half the length of AB). Notice that triangle AXP is a right triangle, so we can use Pythagorean Theorem to find AP.
[tex]AP^{2} = AX^{2} +PX^{2}[/tex]
[tex]AP = \sqrt{3^{2} +4^{2} }[/tex]
[tex]AP =\sqrt{9+16}[/tex]
[tex]AP = \sqrt{25}[/tex]
[tex]AP = 5[/tex]
Please help and explain!!!
As x increases, y value decreases.
The rate of change for y as a function of x is decreasing, therefore the function is a decreasing function.
For all values of x, the function value y, decreases to 0.
The y intercept of the graph is the function value y=8
When x=1, the function value y=5.
From the given graph, it is clear that the curve is decreasing for all values of x. Hence, the rate of change of the give curve is decreasing.
Therefore, the giving function is called as decreasing function.
Since, the function is decreasing for all values of x, the value of y decreases to 0.
Form the graph, it is clear that the y-intercepts is at y=8.
Also, the value of y at x=1 is 5 from the graph.
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The equation i= ;δv / r is called __________.
Answer:
Step-by-step expl
2.73 A,16.3 V