Two students, Tony and Mike, factored the trinomial 8x2 − 12x − 8. Tony factored it as 4(x − 2)(2x + 1) and Mike factored it as (x − 2)(8x + 4). Indicate which student factored the trinomial completely and which student did not, and explain why. (10 points)
Tony and Mike, factored the trinomial [tex]8x^2 - 12x - 8[/tex]
Tony factored it as 4(x - 2)(2x + 1) and
Mike factored it as (x - 2)(8x + 4)
[tex]8x^2 - 12x - 8[/tex]
GCF is 4. We factor out 4
[tex]4(2x^2 - 3x - 2)[/tex]
2*-2=-4. We find out two factors whose product is -4 and sum is -3
two factors are -4 and 1. Split middle term -3x using two factors
[tex]4(2x^2 - 4x + 1x - 2)[/tex]
Group first two terms and last two terms
[tex]4[(2x^2 - 4x) + (1x - 2)][/tex]
Factor out GCF from each group
[tex]4[2x(x - 2) + 1(x - 2)[/tex]
4(2x+1)(x-2)
Tony factored it correctly
Mike factored it as (x − 2)(8x + 4)
Mike factor 8x+4 further. GCF of 8 and 4 is 4
So it becomes 4(2x+1)
Mike not factored it completely
Graph the function with the given description. A linear function h models a relationship in which the dependent variable increases 1 unit for every 5 units the independent variable decreases. The value of the function at 0 is 3.
Let's assume
independent variable is x
dependent variable is h
A linear function h models a relationship in which the dependent variable increases 1 unit for every 5 units the independent variable decreases.
so, we can use formula of linear function
[tex]h=mx+b[/tex]
where
m is slope
[tex]m=-\frac{1}{5}[/tex]
now, we can plug values
[tex]h=-\frac{1}{5}x+b[/tex]
The value of the function at 0 is 3
so, at x=0 , h=3
we can use it and find b
[tex]3=-\frac{1}{5}*0+b[/tex]
[tex]b=3[/tex]
now, we can plug back
[tex]h=-\frac{1}{5}x+3[/tex]
now, we can draw graph
Graph:
Answer:The y-intercept is 3,The slope is [tex]-\frac{1}{5}[/tex],and the x-intecept is 15.
Step-by-step explanation: Credit:The answer is above me.
Write " 240 miles i'm 6 hours " as a unit rate
if you traveled at 40 mph you would go 240 miles in 6 hours. your answer is 40 mph
Alex is building a rectangular fence around his yard. The total perimeter of the fence is 68 feet and the area of the yard is 240 square feet. Based on this information, what are the dimensions of the fence?
Perimeter (P) = 2L + 2w
68 = 2L + 2w
68 - 2L = 2w
2(34 - L) = 2(w)
34 - L = w
********************
Area (A) = L * w
240 = L (34 - L)
240 = 34L - L²
L² - 34L + 240 = 0
(L - 10)(L - 24) = 0
L = 10 or L = 24
w = 34 - L = 34 - 10 = 24 or w = 34 - 24 = 10
Answer: width = 10, length = 24 assuming length is bigger than the width
Perimeter (P) = 2L + 2w
68 = 2L + 2w
68 - 2L = 2w
2(34 - L) = 2(w)
34 - L = w
********************
Area (A) = L * w
240 = L (34 - L)
240 = 34L - L²
L² - 34L + 240 = 0
(L - 10)(L - 24) = 0
L = 10 or L = 24
w = 34 - L = 34 - 10 = 24 or w = 34 - 24 = 10
Answer: width = 10, length = 24 assuming length is bigger than the width
Sandra has a photo that is 9 inches by 12 inches. She wants to resize the photo by the scale factor of 3/4. What will be the dimensions of the new photo
The new dimensions are 6.75 inches by 9 inches
The dimensions of the new photo if, Sandra has a photo that is 9 inches by 12 inches, and The scale factor is 3 / 4, is 6.75 inches by 9 inches.
What is the scale factor?To adjust the size of a figure without altering its shape, a scale factor is a number or conversion factor that is utilized. It is employed to change the size of an object.
Given:
Sandra has a photo that is 9 inches by 12 inches,
The scale factor is, s = 3 / 4
Calculate the dimensions of the new photo as shown below,
The length = 9 × 3 / 4
The length = 27 / 4
The length = 6.75 inches,
The width of photo = 12 × 3 / 4
The width of the photo = 36 / 4
The width of the photo = 9 inches
Thus, the dimensions of the new photo will be 6.75 inches by 9 inches.
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Math graph please help 35 points
In order to better find the points on this graph, we need to convert it from standard form to slope-intercept.
We can do that by solving for y.
-7y + 8 = 21x - 6
Subtract 8 from both sides.
-7y = 21x - 14
Divide both sides by -7
y = -3x + 2
Now that the equation is in slope-intercept, we can find two points on the line.
We know that (0, 2) will be the first point, because 2 is the y-intercept.
We can plug 1 into the x value of the equation to find the corresponding y value.
y = -3(1) + 2
y = -3 + 2
y = -1
The second point is (1, -1)
It's the linear function.
The slope-intercept form: y = mx + b.
-7y + 8 = 21x - 6 subtract 8 from both sides
-7y = 21x - 14 divide both sides by (-7)
y = -3x + 2
We need only two points:
for x = 0 → y = -3(0) + 2 = 0 + 2 = 2 → (0, 2)
for x = 2 → y = -3(2) + 2 = -6 + 2 = -4 → (2, -4)
Which best describes the range of the function f(x)=2/3(6)^x after it's been reflected over the x-axis
When a function is reflected over the x-axis, the positive y-values become negative and the negative y-values become positive.
Explanation:When a function is reflected over the x-axis, the positive y-values become negative and the negative y-values become positive. In the case of f(x)=2/3(6)^x, when it is reflected over the x-axis, the positive y-values will become negative and the negative y-values will become positive. This means that the range of the reflected function will be the opposite of the original range. Since the original range is all positive y-values, the reflected range will be all negative y-values.
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Which equation represents y = x^2 − 8x + 5 in vertex form?
A) y = (x − 4)^2 − 9
B) y = (x − 4)^2 + 11
C) y = (x − 4)^2 + 21
D) y = (x − 4)^2 − 11
Answer: D
Step-by-step explanation:
y = x² - 8x + 5
-5 -5
y - 5 = x² - 8x complete the square by adding [tex](\frac{-8}{2})^{2}[/tex] to both sides
y - 5 + 16 = x² - 8x + 16 the right side is a perfect square: (x - 4)²
y + 11 = (x - 4)²
-11 -11
y = (x - 4)² - 11
which statement about -2h^2-15h-7 is true?
The answer is C) 2h + 1
First i factored out the negative sign, then i split the second term into two terms, then i factored out the common terms and i got -(h + 7) (2h + 1).
Answer:
One of the factor is (2h+1)
Step-by-step explanation:
[tex]-2h^2-15h-7[/tex]
Take out negative sign in common
[tex]-(2h^2+15h+7)[/tex]
Now factor the parenthesis
To factor this , we find out two factors whose product is 14 and sum is 15
[tex]-(2h^2+h+14h+7)[/tex]
Break first two terms and last two terms
[tex]-(2h^2+h)+(14h+7)[/tex]
[tex]-h(2h+1)+7(2h+1)[/tex]
[tex](-h+7)(2h+1)[/tex]
[tex](7-h)(2h+1)[/tex]
One of the factor is (2h+1)
Evaluate the expression 3xy-2x^2 x=4 and y=3
3 × 4 × 3 - 2 × 4²
36 - 2 × 4²
36 - 2 × 16
36 - 32
The answer is 4
Is that right? I hoped it helped...
The table shows the height of a soccer ball that has been kicked from the ground over time. (For reference: h(t) = −16t2 + 40t) Time (seconds) Height (feet) 0 0 0.5 16 1 24 1.25 25 1.5 24 2 16 2.5 0 Which statement describes the rate of change of the height of the ball over time? The rate of change is not constant and decreases over the entire time. Between 0 and 0.5 second the ball rises 16 feet, but between 0.5 and 1 second it rises only 8 more feet. The rate of change is not constant and increases over the entire time. Between 1.5 and 2 seconds the ball falls 8 feet, but between 2 and 12.5 seconds it falls 16 more feet. The rate of change is not constant and decreases then increases over time. The ball rises by 16 in the first half second, but only 8 feet over the next one. After it reaches 25 feet in the air, the ball drops. The rate of change is not constant and increases then decreases over time. The height of the ball above ground gets larger until 1.25 seconds and then gets smaller after that time.
Answer:
- Between 0 and 0.5 second the ball rises 16 feet, but between 0.5 and 1 second it rises only 8 more feet
- The ball rises by 16 in the first half second, but only 8 feet over the next one
- After it reaches 25 feet in the air, the ball drops
- The height of the ball above ground gets larger until 1.25 seconds and then gets smaller after that time.
Step-by-step explanation:
Let's rewrite the table:
Time (seconds) Height (feet)
0 0
0.5 16
1 24
1.25 25
1.5 24
2 16
2.5 0
By simply looking at the table, we can see that the following statements are all correct:
- Between 0 and 0.5 second the ball rises 16 feet, but between 0.5 and 1 second it rises only 8 more feet
- The ball rises by 16 in the first half second, but only 8 feet over the next one
- After it reaches 25 feet in the air, the ball drops
- The height of the ball above ground gets larger until 1.25 seconds and then gets smaller after that time.
use the function rule to complete the table
-10x+y=4
the chart is below i coppied it
X= -2, -1, 0, 1, 2
y=is blank so you have to complet the bottom part witch is y.
Describe the difference between vertical angles and linear pairs of angles
Vertical angles are opposite each other when two lines intersect and are equal. Linear pairs of angles are adjacent angles that form a straight line and their sum is always 180 degrees. The difference lies both in their position and their angle sum.
Explanation:The main difference between vertical angles and linear pairs of angles derives from their positioning and sums. Vertical angles are angles opposite each other when two lines intersect. Vertical angles are always congruent, meaning they have the same measure.On the other hand, linear pairs of angles are adjacent angles whose non-common sides are opposite rays or in other words, they form a straight line. The sum of a linear pair of angles is always 180 degrees.
So, in sum, the difference lies in their positioning (vertical angles are opposite, linear pair angles are adjacent) and in their sum (vertical angles are equal, linear pair angles sum to 180 degrees).
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If 1 yard = 3 feet and 1 mile = 5,280 feet, how many yards are there in 2 miles? 2,640 yards 3,520 yards 10,560 yards 31,680 yards
To solve this, you first have to find the number of yards in one mile.
If three feet = 1 yard, and 5280 ft = 1 mile, all you have to do to find the number of yards in a mile is divide 5280 feet by 3 feet.
You end up with 1760 yards in 1 mile.
To find the number of yards in 2 miles, all you would have to do is multiply 1760 by 2
1760 x 2 = 3520 yardsTo convert 2 miles into yards, first change miles to feet (2 miles * 5,280 feet/mile = 10,560 feet). Then, convert feet to yards (10,560 feet ÷ 3 feet/yard = 3,520 yards). So, 2 miles is equal to 3,520 yards. therefore, option b is correct
Explanation:
To calculate the number of yards in 2 miles, we must first know the relationship between miles, feet, and yards. As the question mentioned, 1 mile = 5,280 feet and 1 yard = 3 feet.
So the first step is to convert 2 miles into feet: 2 miles * 5,280 feet/mile = 10,560 feet.
The second step is to convert feet into yards: 10,560 feet ÷ 3 feet/yard = 3,520 yards.
Therefore, there are 3,520 yards in 2 miles.
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In a math class, the teacher asked the students to find the approximate value of one of the x-coordinates of a point of intersection of two functions: f(x) = 2x2 − 3x + 4 g(x) = 5x − 1 Her students gave her different answers. Which answer is the most accurate? A. 0.8 B. 0.9 C. 1.9 D. 1.1
Set the polynomial equal to each other.
x2 - 2x - 5 = x3 - 2x2 - 5x - 9
Move all the terms to the right side of the equation to make the left side equal to zero.
0 = x3 - 3x2 - 3x - 4
Now we use synthetic division to factor.
4 | 1 -3 -3 -4
4 4 4
___________________________
1 1 1 0
0 = (x - 4)(x2 + x + 1)
Now you can set the factors equal to zero and solve for x. Since the quadratic factor has no real solutions, we ignore and focus on the linear factor. x = 4
Evaluate any of the function when x=4 to get the y-coordinate of the intersection point.
Answer:
The answer is 0.8 so A
Step-by-step explanation:
A vessel sails 32 miles S 45 E. How far south has it sailed?
23 miles
hope this helps :)
If there is 45 angle from the south to the east α= 45° then the distance of d=32miles is
the diagonal of the square and your question how far south has it sailed is the side of the square let's call it x.
The relation is
d = x √2 => x= d/√2 = 32/1.41 ≈ 22.69 ≈ 22.7 miles
The correct answer is 22.7 miles
Good luck!!
Elephants drink 225 liters of water a day how many liters for 2 days
All we have to do here is multiply how many liters of water they drink in a day (225) by 2
225 x 2 = 450
Therefore, elephants drink 450 liters of water in 2 days
Hope this helps you
-AaronWiseIsBae
What is 0.275 0.20 0.572 and 0.725 greatest to least
Riley was scuba diving 12 feet below sea level.She dove 8 feet further down.What is her elevation now?
When Riley dove 8 feet further down from her initial position of 12 feet below sea level, her new location is 20 feet below sea level.
Explanation: Riley was initially scuba diving 12 feet below sea level. This can be represented as a negative number, so we start with -12. Then, she dove 8 feet further down. In mathematical terms, this means we subtract another 8 from -12.
Thus, her elevation now is -12 - 8 = -20. Therefore, Riley is now 20 feet below sea level.
Hence, it is proved here completely.
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please help on this one
A) False because it is not a line (it is a parabola)
B) False because it fails the horizontal line test
C) False because it passes the vertical line test.
D) True because it passes the vertical line test but fails the horizontal line test.
Answer: D
Lucy, Sam, and Bob served a total of 72 orders Monday at the school cafeteria. Bob served 3 times as many orders as Lucy. Lucy served 8 more orders than Sam. How many orders did they each serve?
let
x-------> the number of Lucy's orders
y-------> the number of Sam's orders
z-------> the number of Bob's orders
we know that
[tex]x+y+z=72[/tex]----> equation A
[tex]z=3x[/tex]-----> equation B
[tex]x=8+y-----> y=x-8[/tex] -----> equation C
substitute equation B and C in equation A
[tex]x+(x-8)+3x=72\\ 5x=72+8\\ 5x=80\\x=16\\[/tex]
find z
[tex]z=3*16=48[/tex]
find y
[tex]y=x-8------> y=16-8=8[/tex]
therefore
the answer is
the number of Lucy's orders is [tex]16[/tex]
the number of Sam's orders is [tex]8[/tex]
the number of Bob's orders is [tex]48[/tex]
What is the LCM of xy(x + 1) and x(x + 2)?
A.x2(x + 3)
B.x2(x + 1)(x + 2)
C.xy(x + 1)(x + 2)
D.xy(x + 3)
The answer is (C)
Explanation[tex]xy (x + 1), x(x+2)[/tex]
By multiplying x in equation
[tex]y(x^2 + x), x^2 + 2x[/tex]
[tex]y(x^2 + x), x(x+2)[/tex]
All the 4 factors are (x^2 + x), y, x, (x + 2)
So, the LCM is
[tex]xy(x + 1)(x + 2)[/tex]
Final answer:
The LCM of xy(x + 1) and x(x + 2) is found by including the highest powers of all variables and unique factors from both expressions, yielding x²y(x + 1)(x + 2).
Explanation:
The Least Common Multiple (LCM) of two algebraic expressions is the smallest expression that both original expressions can divide into evenly without leaving a remainder. To find the LCM of xy(x + 1) and x(x + 2), we look for the highest power of all the variables and factors included in both expressions.
The factor x appears in both expressions, but with different powers. The highest power of x in the given expressions is x² as seen in the second expression.
The factor y appears only in the first expression, so it must also be included in the LCM.
Next, the factors (x + 1) and (x + 2) are both unique, so they are also included in the LCM.
Therefore, the LCM of xy(x + 1) and x(x + 2) is x²y(x + 1)(x + 2), which corresponds to choice C. Hence, the correct answer is C.xy(x + 1)(x + 2).
The sun is about 93 * 10^6 miles from earth.What is this distance written as a whole number?
Multiplication time!
10^6 = 1,000,000
93 × 1,000,000 = 93,000,000
Hope this helps!!!
Find the interval(s) over which the function is constant
Answer: (-∞, -4)
To find the interval(s) over which the function is constant,
We look at the graph and pick the interval where the graph is neither increasing nor decreasing
When the graph is neither increasing nor decreasing means it should be a straight line
From the given graph , we have a straight line starts at -4 and goes to infinity
So constant interval is (-∞, -4)
The function is constant over the interval [0, 20].
Explanation:The function is constant when its graph is a horizontal line. In this case, the graph is a horizontal line between x = 0 and x = 20, inclusive. Therefore, the function is constant over the interval [0, 20].
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Which expressions are polynomials?
The denarius was a unit of currency in ancient Rome. Suppose it costs the Roman government 10 denarius per day to support 3 legionaries and 3archers. It only costs 3 denarius per day to support one legionary and one archer. Use a system of linear equations in two variables. Can we solve for a unique cost for each soldier?
Answer:
We can not solve for a unique cost for each soldier.
Step-by-step explanation:
Let x be the daily cost for legionaries and y be the daily cost for archers.
Upon using our given information we will get a system of linear equations as:
[tex]3x+3y=10...(1)[/tex]
[tex]x+y=3...(2)[/tex]
Now we will solve for x from our 2nd equation,
[tex]x = 3-y[/tex]
Now we will substitute this value in our 1st equation.
[tex]3(3-y)+3y=10[/tex]
[tex]9-3y+3y=10[/tex]
We can see that -3y cancels out with 3y and 9 is not equal to 10. So this is an unsolvable system. Therefore, we can not find a unique cost for each soldier.
Answer: No solutions
Step-by-step explanation:
1. Rodrigo has a ladder that is 13 ft long. The ladder is leaned against a vertical wall. The top of the ladder is 10.8 ft above the ground. The angle the ladder makes with the ground needs to be 60o or less for safety purposes. a. Is this ladder in a safe position? (1 point) b. Show your work (3 points) and draw a diagram (1 point) to support your answer. Answer:
Answer:
As per the given condition: Rodrigo has a ladder that is 13 ft long. The ladder is leaned against a vertical wall. The top of the ladder is 10.8 ft above the ground.
The Orientation of the ladder with the wall forms a right triangle.
The ladder length is the hypotenuse of the triangle,
the distance between the ladder at ground level and the base of the wall is the horizontal leg of the triangle,
and the height of the ladder is the vertical leg of the triangle.
⇒ Height of the ladder = 10.8 ft and hypotenuse = 13 ft
Using sine ratio formula;
[tex]\sin \theta = \frac{\text{Opposite side}}{\text{Hypotenuse side}}[/tex]
Opposite side = height of the ladder = 10.8 ft and
Hypotenuse side = 13 ft.
then;
[tex]\sin \theta = \frac{10.8}{13} =0.830769230769[/tex]
or
[tex]\theta = \sin^{-1}(0.830769230769)[/tex]
Simplify:
[tex]\theta = 56.2^{\circ}[/tex] (nearest to tenth place)
Since, it is given that the angle the ladder makes with the ground needs to be 60 degree or less for safety purposes.
(a)
Yes, this ladder in a safe position.
as [tex]\theta = 56.2^{\circ} < 60^{\circ}[/tex]
(b)
You can see the diagram as shown below in the attachment.
A grocery store gives away a $10 gift card to every 25th customer and a $20 gift card to every 60th customer. Which customer will be the first to receive both gift cards? Customer # b. After this customer receives both gift cards, what is the total amount in gift cards that the store has given away? $
A rectangular field is 65 meters wide and 115 meters long. Give the length and width of another rectangular field that has the same perimeter but a smaller area.
Answer: width = 30 m, length = 150 is one possible answer
Step-by-step explanation:
P = 2L + 2w A = L x w
= 2(65) + 2(115) = 65 x 115
= 130 + 230 = 7,475
= 360
360 ÷ 2 = 180
Find 2 numbers whose sum equals 180 and product is less than 7475
Sum: L + w = 180 ⇒ w = 180 - L
Product: Lw < 7475
Graph these 2 equations to see which coordinates of the sum fall into the shaded region of the product.
Vector ahs 4 orange picks for every 3 green picks.If 8 of the picks are orange, how many picks are green?
If 8 of the picks are orange, then 6 of the picks will be green