The grouping (a · c) method is another method for factoring trinomials of the form ax² + bx + c.
The process of factoring a trinomial using the grouping method is,
First, find a GCF and then factor it out.The coefficient of the leading phrase an is then multiplied by the constant term c. List the elements of this product (a · c) to discover the pair of factors, f1, and f2, that add to b, the middle-term coefficient. When c is positive, the sign of the components of (a · c) is the same - Find the pair of factors that adds to b if the middle-term bx is positive. If the middle-term bx is negative, both factors are negative. When c is negative, the elements of (a · c) have the inverse sign - Find the pair of factors that subtracts to b. The largest of these factors has the same sign as the middle term.Rewrite the middle term bx with the components f1 and f2 from Step 2. The phrase currently has four terms: ax² + bx + c = ax² + f1x + f2x + cThen, as illustrated, group the expression's terms into binomial pairs: (ax² + f1x) + (f2x + c) Then, for each pair, factor out a GCF The terms will have a common binomial factor if the equation can be factored by grouping. To write the factorization, subtract the common binomial factor.Finally, double the result to confirm.Learn more about factoring trinomials at
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48% of people in a certain country like to cook and 54% of people in the country like to shop, while 26% enjoy both activities. What is the probability that a randomly selected person in the country enjoys cooking or shopping or both? The probability that a randomly selected person in the country enjoys cooking or shopping or both is ________
The probability that a randomly selected person in the country enjoys cooking or shopping or both will be;
⇒ P (C ∪ S) = 0.76
What is mean by Probability?
The term probability refers to the likelihood of an event occurring.
Given that;
48% of people in a certain country like to cook.
And, 54% of people in the country like to shop, while 26% enjoy both activities.
Now,
Probability to like the cook P (C) = 48%
Probability to like the shop P (S) = 54%
Probability to like the both activity P (C∩S) = 26%
So, Probability to like the both activity P (C ∪ S) is calculated as;
P (C ∪ S) = P (C) + P (S) - P (C ∩ S)
P (C ∪ S) = 48% + 54% - 26%
P (C ∪ S) = 102% - 26%
P (C ∪ S) = 76%
P (C ∪ S) = 0.76
Thus, The probability that a randomly selected person in the country enjoys cooking or shopping or both will be;
⇒ P (C ∪ S) = 0.76
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A restaurant serves sandwiches using slices of bread in the shape of a right triangle. If the legs of the
triangle are each 6 inches, how long is the hypotenuse in inches? Round your answer to the nearest inch.
6 inches
8 inches
10 inches
12 inches
Answer: 8 inches
Step-by-step explanation:
1) Set up the equation
If you know both sides of the triangle are 6 inches you can set a Pythagorean equation. C is the hypotenuse
[tex]6^2+6^2=c^2\\36+36=c^2\\72=c^2\\\sqrt{72} = \sqrt{c^2} \\8.485 = c[/tex]
2) Rounding
After you found 8.485, the tenth place is less the 5, so it will round down to 8 inches.
The answer is 8 inches
Assume that Lines L and M are parallel. Find a pair of acute, corresponding angles.
Select all that apply.
A. q and w
B. r and y
C. x and s
D. q and y
E. s and y
F. r and x
G. t and w
H. t and z
Answer:
Choice E and F
Step-by-step explanation:
acute = less than 90
Using the line AB, name two line segments.ACBA,BoAB, CAC, CBCB,B
Point C is between points A and B, this point divides the line AB into two segments:
One segment goes from point A to point C and can be named line segment AC
The second segment goes from point C to point B and can be named line segment CB
The correct option is the third one AC,CB
what is the equation for the line passing through the points: (3,-10.5) and (1,-7.5)
The equation of the line passing through the points: (3,-10.5) and (1,-7.5) is [tex]3x+2y+12=0.[/tex]
How to find equation of a line by two-point form?Every point on a line fulfills the equation of that line, which means that every point on the line represents every other point on the line.
The equation of a line passing through two-points [tex](x_{1} ,y_{1} )[/tex] and [tex](y_{1} ,y_{2} )[/tex] is given by the formula,
[tex]y-y_1= \frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]
So, the equation of the line passing through the points (3,-10.5) and (1,-7.5) will be-
⇒[tex]y-(-10.5)=\frac{-7.5-(-10.5)}{1-3} *(x-3)[/tex]
⇒[tex]y+10.5=\frac{-7.5+10.5}{-2} *(x-3)[/tex]
⇒[tex]y+10.5=\frac{3}{-2} *(x-3)[/tex]
⇒[tex]y+10.5=\frac{-3x}{2} +\frac{9}{2}[/tex]
⇒[tex](y+10.5)*2=-3x+9[/tex]
⇒[tex]2y+21=-3x+9[/tex]
⇒[tex]2y+3x+21-9=0[/tex]
⇒[tex]3x+2y+12=0[/tex]
Hence, the equation of the line passing through the points (3,-10.5) and (1,-7.5) is [tex]3x+2y+12 =0[/tex].
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A runner sprinted 365.21 ft to finish a race.Use the table of facts to find the distance she sprinted in yards.Round your answer to the nearest tenth.
we have the following:
[tex]365.21ft\cdot\frac{1y}{3ft}=121.74[/tex]Therefore, the answer is 121.7 yards
The diameter of the earth at its equator is about 8000 miles. The diameter of Saturn at its equator is about 74,000 miles. Saturn's diameter is about how many times as large as the Earth's diameter?
A. 9.25×10^3
B. 9.25 × 10^2
C. 9.25×10^1
D. 9.25 x10^0
Answer:
C
Step-by-step explanation:
[tex]\frac{74000}{8000}=9.25 \times 10^1[/tex]
The number of Americans over age 100 can be approximated by P(t) = 0.07e^0.54t, where is P(t) measured in millions and t is measured in decades. Also, t=0 corresponds to 2000.
a. How many Americans over age 100 were there in 2000?
b. How fast was the number of Americans over age 100 changing in 2000? Use the correct units.
There are 7 % of Americans over age 100 who were there in 2000 and the number of Americans growing by 54% over age 100 changed in 2000.
What is an exponential function?An exponential function is defined as a function whose value is a constant raised to the power of an argument is called an exponential function.
It is a relation of the form y = aˣ in mathematics, where x is the independent variable
The number of Americans over age 100 can be approximated by
[tex]P(t) = 0.07e^{0.54t}[/tex]
where is P(t) measured in millions and t is measured in decades.
Also, t = 0 corresponds to 2000.
⇒ [tex]P(t) = 0.07e^{0.54t}[/tex]
Substitute the t = 0 in the above function, and we get
⇒ [tex]P(t) = 0.07e^{0.54\times0}[/tex]
⇒ P(t) = 0.07 e°
⇒ P(t) = 0.07
Convert decimals into fractions, and we get
⇒ P(t) = 7 / 100
This represents 7 out of 100 which is 7 % of Americans over age 100 who were there in 2000.
Thus, the number of Americans growing by 54% over age 100 changed in 2000.
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Hallie has 10 times as many pages to read for her
homework assignment as Janet. Altogether, they have
to read 264 pages. How many pages
does each girl
have to read?
Answer:
24
Step-by-step explanation:
10x + x = 264
11x = 264
x = 264 ÷ 11
x = 24
therefore hallie has 240 pages and Janet 24
total 264
Is the coordinate (5,-3) a solution to the following system of equations?
y=2x-13
2x+6y=-8
Yes, the coordinate is a solution to both equations
No, the coordinate is not a solution for either equaiton
No, the coordinate is only a solution for y-2x-13
ordinate is only a solution for 2x+6y=-8
Option (a) is correct.
What is a equation?statement of equality between two expressions consisting of variables and/or numbers. In essence, equations are questions, and the development of mathematics has been driven by attempts to find answers to those questions in a systematic way.
Given that,
The given equation are
y=2x-13 ........(1)
2x+6y=-8 ........(2)
Solve the equation by elimination method
From equation (1)
y-2x = -13
From equation (2)
Now,
y-2x = -13
6y+2x = -8
adding both equation
7y = -21
y = -3
substitute the value of y in equation (1)
y = 2x-13
-3 = 2x-13
-3+13 = 2x
2x = 10
x = 5
Hence, The coordinate (5,-3) is a solution to both equations.
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What is the least common multiple of 4 and 8?
Enter your answer in the box.
Step-by-step explanation:
answer is 8
detail is in the picture
Create an image of rectangle ABCD that is
translated 6 units to the right and 3 units up
by plotting the corresponding vertices.
Plot the vertex that corresponds to A.
The image of the rectangle ABCD translated 6 units to the right and 3 units up is shown below .
In the question ,
a rectangle is given
the vertex of the rectangle ABCD is given as
A(-10,-6) , B(-10,-2) , C(-6,-2) , D(-6,-6) .
On applying the translation 6 units to right and 3 units up ,
the rule is (x,y)→(x+6,y+3)
by applying the rule for the translation
we get image as
A'(-4,-3) ,B'(-4,+1) , C'(0,1) , D'(0,-3)
The image is plotted below .
Therefore , The image of the rectangle ABCD translated 6 units to the right and 3 units up is shown below .
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The students in Hugh Logan's math class took the Scholastic Aptitude Test. Their math scores areshown below. Find the mean score. Round your answer to the nearest tenth.563 524 357 347 637358 351 528 470 482pls be fast
Answer:
[tex]461.7\Rightarrow\text{ Option \lparen D\rparen}[/tex]Explanation: We have to find the mean of the total scores shown, the formula for the mean is as follows:
[tex]A=\frac{a_1+a_2+a_3+a_4+...+a_N}{N}\rightarrow(1)[/tex]Using the formula (1) the mean of the score is calculated as follows:
[tex]\begin{gathered} A=\frac{563+524+357+347+637+358+351+528+470+482}{10} \\ \\ A=\frac{4617}{10}=441.7 \\ \\ A=461.7\rightarrow\text{ \lparen D\rparen} \end{gathered}[/tex]Therefore the answer is Option(D).
A linear function r models the growth of your right index fingernail. The length of the fingernail increases 0.7 millimeter every week. Graph r when r(0)=12.
A nail's length in x weeks of m=0.7 slope with a growth rate of mm per week b starting length follows.
The slope-intercept form of a straight-line equation is indicated by y = mx + b. X and y denote the line's distance from the x- and y-axes, respectively. When x = 0, the value of b equals y, and m indicates how steep the line is.
Given, r(0) = 12
According to the question, fingernail increases by 0.7 millimeters every week. So, m = 0.7 which is the slope of the equation of a straight line.
r(0) = 12 is the b of the y = mx + b.
Now, the final equation is:
r(x) = 0.7x + 12
As a result, the slope of r(x) length of a nail in x number of weeks of m=0.7, with growth rate in mm per week b starting length.
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What type of reasoning is used to find the next amount of train cars? Explain your answer.
The type of reasoning used to find the next amount of train cars is an arithmetic sequence, it can be assumed that the next amount of train cars that in the sequence is 6.
What is an arithmetic sequence?The difference between successive words in an arithmetic series is known as the common difference y. The existence of the nth term of an arithmetic sequence is given by the following rule:
x1 = x1 + (n-1)y;
Where x1 is the first term.
In the context of this problem, from the image, the number of train cars can be modeled by the following sequence exists (0, 2, 4, ...).
Which exists an arithmetic sequence with:
The first term of 0.A common ratio of 2.Therefore the next term of the sequence is given by:
4 + 2 = 6.
This is because 4 is the last term and 2 is the common ratio.
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8. For every 1,000 feet above sea level, the air temperature decreases by 4°F.The highest natural point in Texas is Guadalupe Peak at 8,751 feet. If thetemperature at sea level is 76°F, what is the temperature near the summit at8,000 feet? (Example 4)
Answer
The temperature at an altitude of 8,000 feet is 44° F
SOLUTION
Problem Statement
The question tells us that air temperature drops by 4 degrees Fahrenheit for every 1,000 feet above sea level. We are asked to calculate the temperature at 8,000 feet, close to the summit of Guadalupe Peak.
Solution
To solve this question, we simply compute the temperature changes for each 1,000 increase in height and then generalize. With the generalization, we can find the temperature at any height.
The temperature at 0 feet is 76°F (At sea level)
The temperature at 1,000 feet is 76°F - 4°F = 72° F
The temperature at 2,000 feet (2 x 1,000 feet) is 76°F - 4°F - 4°F (76°F - 2 x 4°F)
The temperature at 3,000 feet (3 x 1,000 feet) is 76°F - 4° F - 4° F - 4° F (76° F - 3 x 4° F)
And so on.
We can see a pattern developing and with this pattern, we can make a generalization about the temperature decrease for every increase in altitude.
[tex]\begin{gathered} The\text{ temperature at }n\times1,000\text{ feet is (}76^o-n\times4^o)F \\ \text{where,} \\ n=A\text{ whole number starting from zero} \end{gathered}[/tex]Thus, we can solve the question by substituting n = 8 for an altitude of 8,000 feet. This is done below:
8 x 1000 feet = 8,000 feet is:
[tex]\begin{gathered} 76^oF-8\times4^{o\text{ }}F \\ 76^oF-32^oF \\ =44^oF \end{gathered}[/tex]Final Answer
The temperature at an altitude of 8,000 feet is 44° F
Find the constant of proportions k as a fraction in simplest form. The enter an equation for the relation ships between x and y.The constant of proportions, k is equal to?The equation is y = ?
Notice that for every 6 units increment in the x-values in the table, there is a 2 units increment in the y-values.
Therefore, the constant of proportions is:
[tex]\begin{gathered} \frac{2}{6}\rightarrow\frac{1}{3} \\ k=\frac{1}{3} \end{gathered}[/tex]To find the equation, we'll use this constant as the slope. Using this, point
(6, 2) of the table and the slope-point form:
[tex]\begin{gathered} y-2=\frac{1}{3}(x-6) \\ \rightarrow y-2=\frac{1}{3}x-2 \\ \rightarrow y=\frac{1}{3}x \end{gathered}[/tex]This way the equation is:
[tex]y=\frac{1}{3}x[/tex]Find the value of x and y in which each quadrilateral mist be a parallelogram.
Recall the property of parallelograms that tell us that the diagonals intercept producing equal segments in each of them. Then, as the diagonals intercept in the midpoint of each diagonal, then we can create the following equations and solve them one at a time (making the measure of one segment equal the value on the other half of the diagonal):
5 y - 4 = 36 then we solve here for y
add 4 to both sides
5 y = 36 + 4
5 y = 40
divide both sides by 5
y = 40 / 5
y = 8
Now the second equation:
4 x - 10 = 18
add 10 on both sides
4 x = 28
divide both sides by 4 to isolate x
x = 28 / 4
x = 7
part 2..number 2...evaluate the expressions using numbers and evaluate the expression for the numbers provided...please read the direction to make sure I'm understanding what's asked of me..I'm a 7th grader can u please help me with my math summer package and explain it in a way that a 7th grader can understand it
Given:
Each story is 13ft tall.
Explanation:
a) To find: The height of the skyscraper if it has 55 stories.
Since the number of stories increases the height of the skyscraper also increases.
That is,
[tex]\begin{gathered} h=Number\text{ of stories}\times Height\text{ of each story} \\ h=55\times13 \\ =715ft \end{gathered}[/tex]b) To find: The height of the skyscraper if it has 65 stories.
The height of the skyscraper would be,
[tex]\begin{gathered} h=Number\text{ of stories}\times Height\text{ of each story} \\ h=65\times13 \\ =845ft \end{gathered}[/tex]c) To find: The height of the skyscraper if it has 75 stories.
The height of the skyscraper would be,
[tex]\begin{gathered} h=Number\text{ of stories}\times Height\text{ of each story} \\ h=75\times13 \\ =975ft \end{gathered}[/tex]d) To find: The equation of the height if it has f stories
The height of the skyscraper would be,
[tex]\begin{gathered} h=Number\text{ of stories}\times Height\text{ of each story} \\ h=f\times13 \\ h=13f \end{gathered}[/tex]Final answer:
a) The height of the skyscraper if it has 55 stories is 715ft.
b) The height of the skyscraper if it has 65 stories is 845ft.
c) The height of the skyscraper if it has 75 stories is 975ft.
d) The expression that represents the height of f stories is h = 13f.
Mowgli usually walk from home to the beach, and returns on an elephant It takes him 40 minutes altogether. One day he traveled on the elephant from homebto the beach and back, which took 32 minutes. How much time would he need to travel the same distance on foot?
The time it would take Mowgli from the home to the beach and from the beach to the home on foot is 48 minutes.
What is the time it would take to travel on foot?Based on the assumption that the average speed that Mowgli travels from the home to the beach and from the beach to the home is constant, the time it takes from the home to the beach and from the beach to the home should be equal.
Time from the home to the beach = 32 /2 = 16 minutes
The time it took Mowgli to walk from the home to the beach = 40 - 16 = 24 minutes
Time it would take Mowgli to travel from the home to the beach and from the beach to the home = 24 x 2 =48 minutes
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Write this in a equation. 1. three increased by 6 times a number2. ten more than twice a number3. the difference between 4 times a number and 14. 12 less than 3 times a number is negative 65. three times the difference of a number and 126. 2 more than 5 times a number is 17
Answer:
3. the difference between 4 times a number and 1
[tex]4x-1[/tex]5. three times the difference of a number and 12
[tex]3(x-12)[/tex]Explanation:
Writing the equation of the given expressions;
Let x represent the missing number;
3. the difference between 4 times a number and 1
[tex]4x-1[/tex]5. three times the difference of a number and 12
[tex]3(x-12)[/tex]What will it cost to buy fencing to go around a rectangular garden with a length of 25 ft. and a width of 30 ft.? The fencing costs $3.17 per linear foot.
It will cost $348.7 to buy fencing to go around the rectangular garden.
The length of the rectangular garden is 25 feet.The width of the rectangular garden is 30 feet.The cost of fencing per linear foot is $3.17.We need to buy fencing to go around the whole rectangular garden.This means we need the total length of the boundary.The boundary is the same as calculating the perimeter of the rectangular garden.The perimeter of the rectangular garden is two times the sum of its length and its breadth.The perimeter is 2*(25+30) = 110 feet.The total cost of fencing the whole rectangular garden is the product of the cost of fencing per foot and its perimeter.The total cost of fencing the rectangular garden is 3.17*110 = $348.7.To learn more about perimeter, visit :
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hey please help. this has to do with exponential functions. i have to do 225 problems before midnight. it’ll mean a lot!!!
Answer:
Explanation:
To complete the missing values, we have to evaluate the function at the x-values given.
For example, to find the number below x = -1, we will put in x = -1 into y = (1/6)^x.
Putting in x = - 1 in y = (1/6)^x gives
[tex]y=(\frac{1}{6})^{-1}[/tex]which simplifies to give
[tex]y=\frac{1}{1/6}[/tex][tex]\rightarrow y=6[/tex]Similarly, for the missing value below x = -2, we put in x = - 2 into y = (1/6)^x to get
[tex]y=(\frac{1}{6})^{-2}[/tex]which simplifies to give
[tex]y=\frac{1}{(1/6)^2}[/tex][tex]\begin{gathered} \rightarrow y=6^2 \\ \rightarrow y=36 \end{gathered}[/tex]Similarly, for the missing value below x = 2, we put in x = 2 in y = (1/6)^x to get
[tex]y=(\frac{1}{6})^2[/tex]which simplifies to give
[tex]y=\frac{1}{36}[/tex]Hence, the missing value to be put in is 36.
To summerise, the missing values are
x | -2
Plot the point then make a staircase using the given slope connect the points to form a line answer the questions below
Find the area of the figure
Answer:
126.5 cm²
Step-by-step explanation:
Hello!
This shape can be split into a rectangle and a triangle.
The rectangle's dimensions are 8 and 11, and the base and height of the triangle are 11 and 7 respectively.
Area of Rectangle:A = b * hA = 8 * 11A = 88Area of Triangle:A = b*h/2A = 11 * 7/2A = 77/2A = 38.5The combined area is 88 + 38.5 which is 126.5 cm².
Find the complement of the set given that U= {3,4,5,6,7,8,9,10,11} (use the roster method to write the set. Enter empty or 0 the empty set.) {5,7,9,10}
Let
[tex]\begin{gathered} U=\mleft\lbrace3,4,5,6,7,8,9,10,11\mright\rbrace \\ \text{and} \\ A=\mleft\lbrace5,7,9,10\mright\rbrace \end{gathered}[/tex]Then,
[tex]\begin{gathered} A^C=U-A=\lbrace3,4,5,6,7,8,9,10,11\rbrace-\lbrace5,7,9,10\rbrace=\mleft\lbrace3,4,6,8,11\mright\rbrace \\ \Rightarrow A^C=\lbrace3,4,6,8,11\rbrace \end{gathered}[/tex]Thus, the answer is {3,4,6,8,11}
b) Describe the function over each part of its domain. State whether it is constant,
increasing, or decreasing, and state the slope over each part.
Description of the function over each part of its domain is:
Slope = 0 and the domain is constant at (0,12000).Domain (12000, 38000) is declining with a negative slope.What is a function?A function is a type of rule that produces one output for a single input. This is illustrated by the equation y=x2. Any input for x results in a single output for y. Considering that x is the input value, we would say that y is a function of x.So, we examine the displayed graph.
The y values are constant from x = 0 to x = 12,000.
There is no increase or decrease in y's value; it is constant.Consequently, the graph's cost is constant from x=0 to x=12000.So slope = 0.
Then,
Slope = 0 and the domain is constant at (0,12000).
The y values are decreasing from x = 12,000 to x = 38,000.The slope is negative for y values that are decreasing.The slope is therefore negative.Domain (12000, 38000) is declining with a negative slope.
Therefore, description of the function over each part of its domain is:
Slope = 0 and the domain is constant at (0,12000).Domain (12000, 38000) is declining with a negative slope.Know more about functions here:
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A manufacturer throws out gaskets with weights that have an absolute deviation of more than 0.06 pound from the mean weight of the batch. The weights (in pounds) of the gaskets in a batch are 0.58, 0.63, 0.65, 0.53, and 0.61. Which gaskets should be thrown out? Use an absolute value inequality to justify your answer.
If the weights (in pounds) of the gaskets in a batch are 0.58, 0.63, 0.65, 0.53, and 0.61. The gaskets that should be thrown out is 0.53 and the inequality is: 0.54 ≤ w ≤ 0.66.
Means and inequalityUsing this formula to determine the mean
Mean = ∑x/n
Mean = 0.58 + 0.63 + 0.65 + 0.53 + 0.61 /5
Mean = 3/6
Mean = 0.6
Since the mean is 0.6 the gasket that are less than 0.53 and more than 0.66 (0.6±0.06) should be thrown out which in turn means that 0.53 should be thrown out.
Inequality :
0.54 ≤ w ≤ 0.66
Therefore 0.53 should be thrown out.
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Shaun estimated that the attendance at a college baseball game was 1,000. The actual Q3: attendance was 1,788. What is the percent error of Shaun's estimate? Round to the nearest whole percent.
The percentage error of Shaun's estimate is 44%
Here, we want to get the percentage error
We can get this by using the formula below;
[tex]\frac{error}{\text{actual value}}\text{ }\times\text{ 100\%}[/tex]We have the error as the difference between the estimated value and the actual value
The difference is;
[tex]1,788\text{ - 1,000 = 788}[/tex]The percentage error is;
[tex]\frac{788}{1,788}\text{ }\times\text{ 100 \% = 44.07 }[/tex]This is approximately 44%
Calculate the correlation coefficient for the given data below:
x y
2 21
3 22
48
5 11
6 15
7 14
Round your final result to two decimal places.
The correlation coefficient of the data provided is -0.52.
What is the correlation coefficient?Correlation is a measure that is used in statistics to determine the linear relationship that exists between two variables. Correlation can be positive, negative or zero.
Positive correlation is when two variables move in the same direction. Negative correlation is when two variables move in opposite directions. Zero correlation is when there is no relationship between the variables.
Correlation can be determined using a financial calculator. When the financial calculator is used, the correlation coefficient is -0.52
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