Answer:
1. Dividing the expressions [tex]ax^2+bx+c[/tex] by a is a different step.
2. Yes, both of these factorization are correct. They are not complete because they can be factored further.
3. The roots of the related quadratic equation are the x-intercepts of of the related function and factors of the expression are difference of x and roots of the related quadratic equation.
Step-by-step explanation:
1.
To factorize the quadratic expressions [tex]ax^2+bx+c[/tex] first we divide it by a. Then we factorize it same as [tex]x^2+bx+c[/tex].
It means all the steps of factoring a quadratic expressions [tex]ax^2+bx+c[/tex] and [tex]x^2+bx+c[/tex] are same except the first step, i.e., divide the expressions [tex]ax^2+bx+c[/tex] by a.
2.
The given quadratic expression is
[tex]P(x)=2x^2+6x-20[/tex]
[tex](2x-4)(x+5)=2x(x+5)-4(x+5)\Rightarrow 2x^2+10x-4x-20=2x^2+6x-20=P(x)[/tex]
[tex](x-2)(2x+10)=x(2x+10)-2(2x+10)\Rightarrow 2x^2+10x-4x-20=2x^2+6x-20=P(x)[/tex]
The product of factors is equal to the given expression. It means both of these factorization are correct.
They are not complete because they can be factored further.
[tex](2x-4)(x+5)=2(x-2)(x+5)[/tex]
[tex](x-2)(2x+10)=(x-2)2(x+5)[/tex]
3.
If the factored form of a quadratic expression is defined as
[tex](x-a)(x-b)[/tex]
Then the related quadratic equation is
[tex](x-a)(x-b)=0[/tex]
[tex]x=a,b[/tex]
The roots of the quadratic equation are a and b.
The related function is
[tex]f(x)=(x-a)(x-b)[/tex]
The x-intercepts of the function are a and b because at x=a and x=b the value of function is 0.
The roots of the related quadratic equation are the x-intercepts of of the related function and factors of the expression are difference of x and roots of the related quadratic equation.
It should be noted that in order to factorize the quadratic expression, one will have to divide it by a.
FactorizationThe factorization of the quadratic expression ax² + bx + c is different from factoring x² + bx + c as one has to first divide it by a.
Secondly, the factorization by the students isn't complete because they can be factored further.
Lastly, the relationship between the factors of a quadratic expression, the roots of the related quadratic equation is that the roots are the x-intercept of the related function.
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If I had a board that was 11 1/2 feet long and wanted to give it to 7 boys in equal pieces how long would each piece be?
If the radius of a sphere is doubled, then its volume is multiplied by _____. 2 4 8
You've decided you want a plant for your room. At the gardening store, there are 4 different kinds of plants (tulip, fern, cactus, and ficus) and 4 different kinds of pots to hold the plants (clay pot, plastic pot, metal pot, and wood pot).
If you randomly pick the plant and the pot, what is the probability that you'll end up with a tulip in a plastic pot?
Blue shaded 20 squares on his hundreds grid. Becca shaded 30 squares on her hundreds grid. Write two decimals greater than Luke decimal in less than Bekkas decimal
What is the relationship between the 6s in the number 7,664?
Answer:
10s
100s
Step-by-step explanation:
One of the 6s are in the 10th digit position
the other one is in the 100th digit position
Janelle was trying to find the distance between (3,7) and (9,6) in the coordinate plane. She knew the formula was D=√(9 - 3)^2 + (6 - 7)^2. So she took the square root and got (9-3)+(6-7)=5. Did she get the correct answer? Explain.
A gully can fly at a speed of 22 miles per hour about how many feet per hour can the gull fly?
A new restaurant is to contain two-seat tables and four-seat tables. Fire codes limit the restaurant's maximum occupancy to 72 customers. If the owners have hired enough servers to handle 22 tables of customers, how many of each kind of table should they purchase?
To adhere to the fire code and server capacity, the restaurant should ideally buy 18 four-seat tables and 4 two-seat tables. This results in a total of 22 tables and maximizes seating capacity at 72.
Explanation:This problem can be solved using a system of linear equations. Let's denote the number of two-seat tables as T and the number of four-seat tables as F.
From the information given, we can establish two equations:
The total number of tables must not exceed 22, so T + F ≤ 22The total number of seats cannot exceed 72, so 2T + 4F ≤ 72To figure out how many of each type of table they should purchase, we need to solve this system of equations.
The goal is to maximize the number of customers (seats) while not exceeding the limits on tables and seats. So, a possible solution to maximize seating would be to have 18 four-seat tables (F = 18) and 4 two-seat tables (T = 4). This gives a total of 22 tables and 72 seats.
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twice a number and 5 more is 100
Crystal reads 25 pages in 1/2 hours write an equation to represnets the relationship between the number of pages crystal reads and how much time she spends reading.
Crystal would read 37.5 pages in [tex]\( \frac{3}{4} \)[/tex] hour.
To represent the relationship between the number of pages Crystal reads and the time she spends reading, we can use the formula:
[tex]\[ \text{Pages Read} = \text{Reading Rate} \times \text{Time Spent Reading} \][/tex]
In this case, Crystal reads 25 pages in 1/2 hour, so her reading rate can be calculated as follows:
[tex]\[ \text{Reading Rate} = \frac{\text{Pages Read}}{\text{Time Spent Reading}} \][/tex]
[tex]\[ \text{Reading Rate} = \frac{25 \text{ pages}}{\frac{1}{2} \text{ hour}} \][/tex]
[tex]\[ \text{Reading Rate} = 25 \times 2 \][/tex]
[tex]\[ \text{Reading Rate} = 50 \text{ pages per hour} \][/tex]
Now, we can substitute this reading rate into the equation to represent the relationship:
[tex]\[ \text{Pages Read} = 50 \times \text{Time Spent Reading} \][/tex]
This equation describes the relationship between the number of pages Crystal reads and the time she spends reading.
To illustrate how to use this equation, let's say Crystal reads for [tex]\( \frac{3}{4} \)[/tex] hour. We can plug this value into the equation to find out how many pages she reads:
[tex]\[ \text{Pages Read} = 50 \times \frac{3}{4} \][/tex]
[tex]\[ \text{Pages Read} = 37.5 \][/tex]
So, Crystal would read 37.5 pages in [tex]\( \frac{3}{4} \)[/tex] hour.
Suppose a and b give the population of two states where a>b . Compare the expressions and tell which of the given pair is greater or if the expression are equal.
b/a+b and 0.5
which line would best fit the data shown in a scatterplot
** NEED THIS ANSWERED ASAP**
Find the indicated probability. Round to the nearest thousandth.
In a blood testing procedure, blood samples from 6 people are combined into one mixture. The mixture will only test negative if all the individual samples are negative. If the probability that an individual sample tests positive is 0.11, what is the probability that the mixture will test positive?
a. 0.503
b. 0.00000177
c. 1.00
d. 0.497
A bank withdraw of 50 dollars
Find the surface area of a cylinder with a diameter of 2 and an altitude of 16
If a number A is a 2 digit number and its digits are transposed to form number B, then the difference between the larger of the two numbers and the smaller of the two numbers must be divisible by:
The difference is a multiple of 9, so it is always divisible by 9.
What is an expression?An expression contains one or more terms with addition, subtraction, multiplication, and division.
We always combine the like terms in an expression when we simplify.
We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.
Example: so
1 + 3x + 4y = 7 is an expression.com
3 + 4 is an expression.
2 x 4 + 6 x 7 – 9 is an expression.
33 + 77 – 88 is an expression.
We have,
The difference between the larger of the two numbers and the smaller of the two numbers.
A - B or B - A (whichever is greater)
If we transpose the digits of a two-digit number A to form B, then:
A = 10a + b, where a is the tens digit and b is the one's digit
B = 10b + a, where b is the tens digit and a is the one's digit
The difference between the two numbers.
= A - B
= (10a + b) - (10b + a)
= 9a - 9b
= 9(a - b)
or
B - A
= (10b + a) - (10a + b)
= 9b - 9a
= 9(b - a)
Either way, the difference is a multiple of 9, so it is always divisible by 9.
Therefore,
The difference is a multiple of 9, so it is always divisible by 9.
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A baseball is hit with an initial upward velocity of 70 feet per second from a height of 4 feet above the ground. The equation h= −16t^2 +70t + 4 models the height in feet t seconds after it is hit. After the ball gets to its maximum height, it comes down and is caught by another player at a height of 6 feet above the ground. About how long after it was hit does it get caught?
By setting the given height (6 feet) in the height equation and using the quadratic formula to solve for time 't', we get two solutions. Since the ball reaches 6 feet twice in its ascension and decension, the latter value of t = 3.79 seconds would be the time it is caught.
Explanation:The question is regarding the time at which a baseball, hit with an initial upward velocity and caught at 6 feet above the ground, is caught. Firstly, input the given height of 6 feet into the height equation h= -16t^2 + 70t + 4 and solve for
t
. Based on the quadratic formula, we receive two solutions: t = 3.79 s and t = 0.54 s. Since the ball has two points at which it reaches the height of 6 feet during its trajectory - once while going up and once while coming down - the time when it is caught would be the larger value,
t = 3.79 s
. Therefore, approximately 3.79 seconds after being hit, the ball is caught.
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A mother gives birth to a 10 pound baby. Every 4 months, the baby gains 2 pounds. If x is the age of the baby in months, then y is the weight of the baby in pounds. Find an equation of a line in the form y = mx + b that describes the baby's weight.
The equation of the line that describes the baby's weight is y = (1/2)x + 10.
Explanation:To find the equation of the line that describes the baby's weight, we need to determine the slope and y-intercept of the line. The slope represents the rate at which the baby's weight increases, and the y-intercept represents the initial weight of the baby.
Since the baby gains 2 pounds every 4 months, the slope of the line is 2/4 = 1/2. This means that for every month that passes, the baby's weight increases by 1/2 pound.
To find the y-intercept, we can use the initial weight of the baby, which is 10 pounds. So the equation of the line is y = (1/2)x + 10.
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the hypotenuse of a right triangle is 24ft long. The length of one leg is 20ft more than the other. Find the length of the legs.
We can find the lengths of the legs of the right triangle using the Pythagorean theorem. One leg is x and the other leg is x+20. A quadratic equation can be solved to find x.
Explanation:The problem involves a right triangle, and we are given the length of the hypotenuse and a relationship between the lengths of the legs. We can solve it using the Pythagorean theorem, which for a right triangle with legs of lengths 'a' and 'b' and hypotenuse 'c' is stated as a² + b² = c².
Let's assign 'x' to the shorter leg. Given that the other leg is 20ft longer, it would be 'x + 20'. The hypotenuse is given as 24, hence the equation becomes: x² + (x + 20)² = 24².
By solving this equation, we find two potential values for 'x', but since a length can't be negative, we exclude the negative value. Hence, the length of the shorter leg is 'x' and of the longer leg is 'x + 20'.
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The process of using sample statistics to draw conclusions about population parameters is called
M(6, 6) is the midpoint of mc139-1.jpg. The coordinates of S are (8, 9). What are the coordinates of R?
A sealed rectangle or box measuring 8 x 6 x 18 contains 864 Sugar cubes each measuring one by one by one how many sugar cubes are touching the box
The arrangement of the 864 cubes would be 18 6 by 8 layers.
All 48 would be touching the bottom of the box on the bottom layer and all 48 would be touching the top of the box on the top layer.
The cubes along both lengths would be touching the sides of the box for the remaining 16 layers. That would be16 cubes per layer or 256 cubes after counting both sides.
The cubes along the width would be touching the ends of the box for those 16 layers. Those need to be eliminated from the count since the corner cubes were already counted as part of the ones touching the sides. 4 cubes have not been previously counted for each width of 6 cubes. Both ends of the 16 layers has 8 cubes per layer or 128 cubes.
Therefore, that is 256 (sides) + 128 (ends) + 48 (top) + 48 (bottom) and that totals to 480 cubes touching the box.
A store stocked 150 cans of popcorn for a weekend sale.
That weekend, 72 of the cans sold. What percent of the
cans of popcorn stocked were sold that weekend?
Answer:
48%
Step-by-step explanation:
In order to find the percentage we need to divide the sold cans by total cans and multiply the result by 100.
Total cans = 150
Sold cans = 72
→ 72/150 = 0.48
→ 0.48 * 100 = 48
The percentage of the cans of popcorn stocked were sold that weekend is 48%
The given parameters are:
Total can of popcorn = 150
Sold can of popcorn = 72
The percentage of can sold is then calculated as:
[tex]\%Sold = \frac{72}{150} *100\%[/tex]
Multiply 72 and 100
[tex]\%Sold = \frac{7200}{150}\%[/tex]
Divide 7200 by 150
[tex]\%Sold = \%48[/tex]
Hence, the percentage of the cans of popcorn stocked were sold that weekend is 48%
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On average, the merchandise shop sells 80 CDs for every 1 vinyl record. Estimate how many vinyl records they are likely to sell if the merchandise shop sells 760 CDs.
Answer:
10 vinyl records are expected to be sold.
Step-by-step explanation:
On average, the merchandise shop sells 80 CDs per 1 vinyl record. This is our conversion factor. To estimate the number of vinyl records likely to be sold when 760 CDs have been sold we will use proportions.
760 CD × (1 vinyl record/ 80 CD) = 9.5 ≈ 10 (we round it off because you cannot sell half a vinyl record).
10 vinyl records are expected to be sold.
Evaluate the surface integral. (give your answer correct to at least three decimal places.) s is the boundary of the region enclosed by the cylinder x2 + z2 = 1 and the planes y = 0 and x + y = 2
Simple interest formula: P=Irt
Solve for t
The value of t in the simple interest formula P = Irt is t = P / (Ir).
To solve the simple interest formula P = Irt for t, we need to isolate the variable t on one side of the equation.
The formula can be rearranged as follows:
P = Irt
First, divide both sides of the equation by I:
P/I = rt
Next, divide both sides of the equation by r:
(P/I) / r = t
Simplifying further:
t = P / (Ir)
Therefore, the value of t in the simple interest formula P = Irt is t = P / (Ir).
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The ages of Edna,Ellie,and Elsa are consecutive integers. The sum of their ages is 120. What are their ages?
x+x-1+x+1 =120
3x=120
x=40
40-1=39
40+1 =41
39 +40 +41 = 120
ages are 39 40 & 41
The area, a, of an ellipse can be determined using the formula a=TTxy where x and y are half the lengths of the largest and smallest diameters of the ellipse. Which is an equivalent equation solved for y
Answer:
Area of an ellipse(a), ,having x and y being the lengths of the largest and smallest diameters of the ellipse = π xy
The lengths of the largest and smallest diameters of the ellipse is called Major Axis and Minor axis of the ellipse.
[tex]\rightarrow a=\pi x y\\\\\rightarrow y=\frac{a}{\pi \times x}[/tex]
What effect does adding a constant have on a exponential function?
Adding a constant to an exponential function results in shifting the graph vertically or horizontally, depending on where the constant is added. The shift will be positive for a positive constant and negative for a negative constant.
Explanation:In Mathematics, when dealing with an exponential function, adding a constant can affect the function in two different ways, depending on where the constant is being added. If the constant is added to the exponent, this results in shifting the graph horizontally. However, if the constant is added outside the exponent (as in f(x) = 2x + k), this will result in the entire graph being shifted upward or downward vertically, based on whether the constant is positive or negative.
For example, consider the simple exponential function f(x) = 2x. If a constant 'c' is added - resulting in f(x) = 2x + c, the resulting graph will be the same as the original, but shifted 'c' units upward if 'c' is positive and downward if 'c' is negative. This is a fundamental principle of exponential functions.
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20 PTS!!!Each month, Matthew gets a $25 allowance and earns $100 mowing lawns. He uses the expression 25x + 100y to keep track of his earnings.
Part A: Identify the variables and coefficients in the expression. (3 points)
Part B: How many terms are in the expression, what are they, and how do you know? (4 points)
Part C: Which term in the expression shows the total earned from mowing lawns? (3 points)