What methods can be used to rewrite square trinomials and difference of squares binomials as separate factors?
Answers
Difference of squares is simply written in the form of (a² - b²) and to find its roots, you simply simplify the previous expression to be (a+b)(a-b) where a and b are the roots
i.e. (a² - b²) = (a+b)(a-b)
2- As per square trinomials:
Let's assume that the square trinomial is written in the form of (ax² + bx +c) where a,b and c are constants.
Now the basic formula to get the two roots of this expression is given as shown in the below picture.
This formula is valid for almost all expressions, nevertheless, there is a special case (perfect square) in which we do not have to use to use this formula.
Perfect Square:
Let's assume that we have an expression written in the form of (ax² + cx +b) where a,b and c are constants. If you find that the value of a is the square of a first binomial, b is the square of a second binomial and c is the twice the value of multiplying a and b (i.e. c=2ab), then this expression is said to be a perfect square and can simply be simplified as (a+b)²
Note: If the expression of the perfect square was (ax² - cx +b), then the simplification would be (a-b)²
Factoring Polynomials with Four Terms, and Factoring Trinomials by Grouping can be used to rewrite square trinomials and difference of squares binomials as separate factors
Related Questions
Find the solution of this system of equations.
Separate the x- and y-values with a comma.
x= 5 + y
28x – 9y= -12
Answers
28x – 9y = -12
substitute x= 5 + y into 28x – 9y = -12
28x – 9y= -12
28(5 + y) – 9y = -12
140 + 28y - 9y = -12
19y = -152
y = -8
x = 5 + y
x = 5 - 8
x = -3
solution (-3, -8)
Which is the slope of the line that connects points (3,17) and (7,25) ?
Answers
slope=(y2-y1)/(x2-x1) where (x1,y1) and (x2,y2) are any two points...
slope=(25-17)/(7-3)
slope=8/4
slope=2
8/4, reduced to 2/1.
2/1 is the correct answer.
What value should go in the empty box to complete the calculation for finding the product of 62.834 × 0.45?
Answers
Table that organizes two categorical variables is called?
It says it has 13 letters
Answers
Please Help! Given that line s is perpendicular to line t, which statements must be true of the two lines? Check all that apply.
a.Lines s and t have slopes that are opposite reciprocals.
b.Lines s and t have the same slope.
c.The product of the slopes of s and t is equal to -1
d.The lines have the same steepness.
e.The lines have different y intercepts.
f.The lines never intersect.
g.The intersection of s and t forms right angle.
h.If the slope of s is 6, the slope of t is -6
Remember, it is check all that apply, so there will be multiple answers.
Answers
the products of the slopes of s and t is equal to -1
they have different y int's
In geometry, when line s is perpendicular to line t, statements a, c, and g are true: Lines s and t have slopes that are opposite reciprocals, the product of the slopes of s and t equal -1, and the intersection of s and t forms a right angle.
Explanation:In geometry, if line s is perpendicular to line t, several facts about these two lines can be stated:
a. Lines s and t have slopes that are opposite reciprocals. This is true. If the slope of one line is m, the slope of the line perpendicular to it is -1/m.b. Lines s and t have the same slope. This is false as orthogonal lines have slopes that are negative reciprocals of each other.c. The product of the slopes of s and t is equal to -1. This is true. When two lines are perpendicular, the product of their slopes is -1.d. The lines have the same steepness. This is false because perpendicular lines have different slopes.e. The lines have different y intercepts. This assertion is not necessarily true. Perpendicular lines may or may not have different y-intercepts.f. The lines never intersect. This is false. Perpendicular lines intersect once, forming a 90 degrees angle.g. The intersection of s and t forms a right angle. This is true. The definition of perpendicular lines states that they intersect at a right angle.h. If the slope of s is 6, the slope of t is -6. This is false. If the slope of s is 6, the slope of t, being a negative reciprocal, would be -1/6, not -6.Learn more about Perpendicular Lines here:https://brainly.com/question/18271653
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in a book 3/8 of the pages have pictures on them.Given that 72 pages have a picture on, work out the number of pages in the book.
Answers
Multiply and that is your answer
8x24 = 192
Hence 192 pages in the book
Find the indicated terms of the sequence defined by each of the following recursive formulas:
a3 = −11 and an = 2an − 1 − 1
a2 =
a4 =
a4 = −36 and an = 2 an − 1 − 4
a3 =
a2 =
Answers
Answer:
1.
Given the recursive formula:
[tex]a_3 = -11[/tex] and
[tex]a_n = 2a_{n-1} -1[/tex]
For n = 3:
[tex]a_3=2a_2 -1[/tex]
Substitute [tex]a_3 = -11[/tex] we have;
[tex]-11=2a_2 -1[/tex]
Add 1 to both sides we have;
[tex]-10 = 2a_2[/tex]
Divide both sides by 2 we have;
[tex]-5 = a_2[/tex]
or
[tex]a_2 = -5[/tex]
For n = 4, we have;
[tex]a_4=2a_3 -1[/tex]
Substitute [tex]a_3 = -11[/tex] we have;
[tex]a_4 = 2 \cdot -11 -1 = -22-1 = -23[/tex]
⇒[tex]a_4 = -23[/tex]
2.
Given:
[tex]a_4 = -36[/tex] and [tex]a_n = 2a_{n-1} -4[/tex]
For n = 4, we have;
[tex]a_4=2a_3 -4[/tex]
Substitute [tex]a_4 = -36[/tex] we have;
[tex]-36 = 2a_3 -4[/tex]
Add 4 to both sides we have;
[tex]-32 = 2a_3[/tex]
Divide both sides by 2 we have;
⇒[tex]a_3 =-16[/tex]
For n = 3:
[tex]a_3=2a_2 -4[/tex]
Substitute [tex]a_3 = -16[/tex] we have;
[tex]-16=2a_2 -4[/tex]
Add 4 to both sides we have;
[tex]-12 = 2a_2[/tex]
Divide both sides by 2 we have;
[tex]-6 =a_2[/tex]
or
⇒[tex]a_2 = -6[/tex]
The indicated terms of the sequence defined by each of the following recursive formulas are as follows:
[tex]\mathbf{a_{2} = -5}[/tex][tex]\mathbf{a_4 = -23}[/tex][tex]\mathbf{{a_3}=-16}[/tex][tex]\mathbf{{a_2}=-6}[/tex]What are recursive formulas?A recursive formula is one that describes each term in a series in terms of the term before it. The general term for an arithmetic sequence by using a recursive formula is [tex]\mathbf{a_n = a_{n-1} + d}[/tex]
From the given information:
[tex]\mathbf{a_3 = -11}[/tex] [tex]\mathbf{a_n = -2a_{n-1} -1}[/tex]Now, when n = 3
[tex]\mathbf{a_3 = -2a_{3-1} -1}[/tex]
[tex]\mathbf{-11= -2a_{2} -1}[/tex]
[tex]\mathbf{2a_{2} = -10}[/tex]
[tex]\mathbf{a_{2} = -5}[/tex]
When n = 4
[tex]\mathbf{a_4= -2a_{4-1} -1}[/tex]
[tex]\mathbf{a_4 = 2(-11) -1}[/tex]
[tex]\mathbf{a_4 = -23}[/tex]
Second Part:
[tex]\mathbf{a_4 = -36}[/tex][tex]\mathbf{a_n = 2_{an-1}-4}[/tex]When n = 4
[tex]\mathbf{a_4 = 2_{a4-1}-4}[/tex]
[tex]\mathbf{a_4= 2_{a3}-4}[/tex]
[tex]\mathbf{-36+4= 2_{a_3}}[/tex]
[tex]\mathbf{2_{a_3}=-32}[/tex]
[tex]\mathbf{{a_3}=-16}[/tex]
When n = 3
[tex]\mathbf{a_3= 2_{a3-1}-4}[/tex]
[tex]\mathbf{a_3= 2_{a2}-4}[/tex]
[tex]\mathbf{-16= 2_{a_2}-4}[/tex]
[tex]\mathbf{2_{a_2}=-12}[/tex]
[tex]\mathbf{{a_2}=-6}[/tex]
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HELP If f(x)=1/7x-9,then f^-1(x)=?
Answers
Answer:
Plato/Edementum
Step-by-step explanation:
this is the correct answer if you are taking Inverse of a Function: Mastery Test just type in 7x+63 type it in like that and it will be correct
if lines are parallel or perpendicular 4x-8y=9 and 8x-7y=9
Answers
-8y=9-4x
8y=-9+4x
y=0.5x-9/8
8x-7y=9
-7y=-8x+9
7y=8x-9
y=8/7x-9/7
neither, they aren't parallel or perpindicular. I tested by graphing them as well.
The variable Z is directly proportional to X. When X is 3, Z has the value 54. Wha is the value of Z when X= 12
Answers
Z = kX (Where k is the constant of proportionality)
To find k
54 = k3
k = 54/3
k = 18
now find new Z
Z = 18X
Z = 18(12)
Z = 216
What time is 5 3/4 hours after 9:22 PM?
Answers
3/4 hours = 45 minutes
22 + 45 = 67 = 1 hrs 7 minutes
15 hrs 7 minuts = 3:07 AM
Patrick spins the spinner 9 times. What is the theoretical probability that it stops on the brown sector on the last spin?
1 over 45
1 over 25
1 over 9
1 over 5
Answers
A small school has 110 students who occupy three classrooms: a, b, and
c. after the first period of the school day, half the students in room a move to room b, one-fifth of the students in room b move to room c, and one-third of the students in room c move to room
a. nevertheless, the total number of students in each room is the same for both periods. how many students occupy each room?
Answers
There are 20, 50 and 30 in each room respectively.
What is equation?An equation is a mathematical statement that shows that two mathematical expressions are equal.
Given that, there are 110 students in total in all the class.
According to question,
a+b+c = 110
After the room changes, we have
a/2 + c/3 = a
4b/5 + a/2 = b
2c/3 + b/5 = c
or,
a/2 = c/3
a/2 = b/5
b/5 = c/3 = a/2
so, substituting in,
a + 5a/2 + 3a/2 = 110
2a + 5a + 3a = 220
a = 22
b = 55
c = 35
Hence, there are 20, 50 and 30 in each room respectively.
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Find the function y = f(t) passing through the point (0,12)
Answers
Using the parkland formula, calculate the hourly rate of fluid replacement with lactated ringerâs solution during the first 8 hours for a client weighing 75 kg with total body surface area (tbsa) burn of 40%. record your answer using a whole number.
Answers
V = 4mA
where
V = volume of replacement fluid (mL) required within the first 24 hours after a burn injury,
m = body mass, kg
A = percent of body surface area covered with burns.
The first half of the fluid should be delivered in the first 8 hours.
Because we are concerned with fluid replacement within the first 8 hours, the Parkland formula is valid to be used.
Given:
m = 75 kg
A = 40%
Therefore
V = 4*75*40 = 12,000 mL = 12 L
This is the fluid replacement required in the first 24 hours.
Because half of this amount should be delivered in the first 8 hours, the amount of fluid replacement is
6,000 mL in 8 hours, or
6000/8 = 750 mL per hour.
Answer: 750 mL every hour
A video game is on sale for 30% off the regular price of 50$. What is the sale price of the game?
Answers
You have to do: $15 times .30
Find the fifth roots of 243(cos 240° + i sin 240°)
Answers
zⁿ = rⁿ(cos.n.Ф + i.sinn.Ф).
The 5th root, that means that n = 1/5 ↔↔↔ rⁿ = r¹/₅ = ⁵√r
zⁿ = ⁵√(243) [cos (240+2kπ)/5 +i.sin(240+2kπ)/5]
with k = 1,2,3,..,n-1. Note that ⁵√(243) = 3
1st root : for k=0 → 3[cos(240/5)+isin(240/5) = 3(cos48°+isin48°)
2nd root : for k=1→3[cos(240+360)/5)+isin(240+360)/5]=3(cos120°+isin120°)
3rd root : for k=2 →→ 3(cos192 + isin192)
4th root : for k=3→→3(cos264+isin264)
5th root : for k =4↔↔3(cos336+isin336)
A student took a test which had 6 questions. He would score 8 points on the test if all his answers are correct. If y represents the student's score when he got x questions incorrect, which graph best represents this situation?
Answers
m=(8-0)/(6-0)
m=8/6
m=4/3, and since we know we have the point (0,0) we know the line is just:
y=mx, using m found above...
y=4x/3
So the graph that will match this line is one that has a point on the origin and has a positive slope of 4/3. So another point other than the origin that will be quite clear is the point (3, 4)
Dishwashers are on sale for 25% off the original price (d), which can be expressed with the function p(d) = 0.75d. Local taxes are an additional 14% of the discounted price, which can be expressed with the function c(p) = 1.14p. Using this information, which of the following represents the final price of a dishwasher with the discount and taxes applied?
c(p) ⋅ p(d) = 0.855pd
c(p) + p(d) = 1.89d
c[p(d)] = 0.855d
d[c(p)] = 1.89p
Answers
We are given the functions:
P (d) = 0.75 d ---> 1
C (P) = 1.14 P ---> 2
The problem asks us to find for the final price after discount and taxes applied; therefore we have to find the composite function of the two given functions 1 and 2. To solve for composite function of the final price of the dishwasher with the discount and taxes applied, all we have to do is to plug in the value of P (d) with variable d into the equation of C (P). That is:
C (P) = 1.14 (0.75 d)
C (P) = 0.855 d
or
C [P (d)] = 0.855 d
Answer:
0.855d
Step-by-step explanation:
I took the test. I also checked a bunch of other answers by completing the test. at least 2 other people had the same answer as me.
Suppose that 66% of people own dogs. if you pick two people at random, what is the probability that they both own a dog?
Answers
The probability of two people owning a dog when 66% of people own dogs is found by squaring the probability (0.66), giving us a result of 0.4356 or 43.56%.
Explanation:The question asks to calculate the probability that both people picked at random own a dog when it is known that 66% of people own dogs. To find this probability, we can simply multiply the probability of the first person owning a dog by the probability of the second person owning a dog, assuming that these events are independent.
The probability that one person owns a dog is given as 66%, which is 0.66 in decimal form. Thus, the probability that both people own a dog is 0.66 × 0.66 (or 0.66 squared). Calculation yields:
0.66 × 0.66 = 0.4356
Therefore, the probability that both people own a dog is 0.4356, or 43.56%.
Let g(x) = 2x and h(x) = x2 + 4. Evaluate (g ∘ h)(3).
Answers
(g○h)(x)=g(h(x))
(g○h)(x)=2(x^2+4)
(g○h)(x)=2x^2+8
(g○h)(3)=2(3^2)+8
(g○h)(3)=2(9)+8
(g○h)(3)=18+8
(g○h)(3)=26
Which of the following graphs represents the function f(x) = −2x3 − x2 + 3x + 1?
graph with 3 real zeros, down on left, up on right
graph with 3 real zeros, up on left, down on right
graph with 2 real zeros, down on left, down on right
graph with 2 real zeros, up on left, up on right
Answers
The function f(x) = −2x3 − x2 + 3x + 1 is represented by the graph with 3 real zeros, down on the left and up on the right, due to the negative leading coefficient and odd degree.
Explanation:The function you're looking at is f(x) = −2x3 − x2 + 3x + 1. To determine the correct graph, we consider the leading term, −2x3. Since the coefficient of the highest-degree term (which is -2) is negative, the graph will start down on the left and go up on the right. The degree of the function is the highest power of x, which is 3, a odd number. For polynomials, if the degree is odd and the leading coefficient is negative, the end behavior will be down on the left and up on the right. Therefore, the correct graph will have the description: Graph with 3 real zeros, down on the left, up on the right.
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Two times the least of three consecutive odd integers exceeds two times the greatest by 15. what are the integers
Answers
1 3 5
2 * 1 = 2
2 * 5 = 10
the smaller number cannot exceed the larger number.
The Pythagorean Theorem applies to ANY triangle in determining the length of an unknown side or leg given two of the other side or leg measures.
True or False?
Answers
The pythagorean theorem only applies to right triangles.
Mike and adam left a bus terminal at the same time and traveled in opposite directions. mike’s bus was in heavy traffic and had to travel 20 mi/h slower than adam’s bus. after 3 hours, their buses were 270 miles apart. how fast was each bus going?
Answers
mike's bus speed = m
total distace = t
m = a - 20
after 3 hour:
3 * (m + a) = t
3 * (m + a) = 270
3m + 3a = 270
[ 3 * (a - 20) ] + [ 3 * a ] = 270
[ 3a - 60 ] + 3a = 270
3a - 60 + 3a = 270
3a + 3a = 270 + 60
6a = 330
their speed:
a = 55
m = 55 - 20 = 35
Given a soda can with a volume of 15 and a diameter of 2, what is the volume of a cone that fits perfectly inside the soda can? (Hint: only enter numerals in the answer blank).
Answers
that is 5
Answer:
5 cubic units.
Step-by-step explanation:
We have been given that a can soda can has a volume of 15 cubic units and a diameter of 2.
First of all let us find the height of cylinder using volume of cylinder formula.
[tex]\text{Volume of cylinder}=\pi r^2 h[/tex], where,
r = radius of cylinder,
h = Height of cylinder.
Now let us divide our diameter by 2 to get the radius of cylinder.
[tex]\text{radius of cylinder}=\frac{2}{2}=1[/tex]
Let us substitute our given values in volume of cylinder formula to get the height of cylinder.
[tex]15=\pi*1^2*h[/tex]
[tex]15=\pi*h[/tex]
[tex]\frac{15}{\pi}=\frac{\pi*h}{\pi}[/tex]
[tex]\frac{15}{\pi}=h[/tex]
Now we will use volume of cone formula to find the volume of our given cone inscribed inside cylinder.
[tex]\text{Volume of cone}=\frac{1}{3}\pi*r^2h[/tex]
Since the height and radius of the largest cone that can fit inside the can will be equal to height and radius of can, so we will substitute [tex]\frac{15}{\pi}=h[/tex] and [tex]r=1[/tex] in the volume formula of cone.
[tex]\text{Volume of cone}=\frac{1}{3}\pi*1^2*\frac{15}{\pi}[/tex]
[tex]\text{Volume of cone}=\frac{1}{3}*1*15[/tex]
[tex]\text{Volume of cone}=5[/tex]
Therefore, volume of our given cone will be 5 cubic units.
Solve the inequality. 8x-5>_27. A.x>_4. B.x>_11/4. C.x<_4. D.x<_11/4
Answers
8x ≥ 32 <-- Add 5 to both sides
x ≥ 4 <-- Divide both sides by 8
So, to be in solution set, x has to be greater or equal to 4.
In interval notation: [4, ∞)
In a set builder notation: {x | x ∈ R, x ≥ 4}
The solution for the inequality 8x - 5 ≥ 27 can be written as x [4, ∞) or x ≥ 4, so option A is correct.
What is inequality?An inequality is a relation that compares two numbers or other mathematical expressions in an unequal way. It is most frequently used to compare the sizes of two numbers on the number line.
Given:
8x - 5 ≥ 27
Solve the above inequality as shown below,
Add 5 to both sides of an inequality,
8x - 5 + 5 ≥ 27 + 5
8x ≥ 32
Divide both sides by 8,
8x / 8 ≥ 32 / 8
x ≥ 4
x [4, ∞)
Thus, x can be any real number greater than 4 or equal to four.
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What is the y intercept of Y = 3x - 4?
Answers
y = mx + b
y = 3x + (-4)
so ur y intercept is -4
A triangle has coordinates A (1, 5), B (-2, 1) and C (0, -4). What are the new coordinates if the triangle is dilated with a scale factor of 1/5?
Answers
Find the value of each variable.
A. a = 15, b = 5, c = 8, d = 4
B. a = 15, b = 4, c = 8, d = 5
C. a = 14.5, b = 5, c = 6, d = 4
D. a = 14.5, b = 4, c = 6, d = 5
Answers
Answer:
(A)
Step-by-step explanation:
From the figure, since RT is parallel to QU, therefore ΔSQU is similar to ΔSRT, thus using the basic proportionality theorem, we get
[tex]\frac{SR}{SQ}=\frac{ST}{SU}[/tex]
[tex]\frac{c}{12+c}=\frac{10}{25}[/tex]
[tex]25c=120+10c[/tex]
[tex]15c=120[/tex]
[tex]c=8[/tex]
Also, QU is parallel to PV, therefore from ΔPVS and ΔSRT, we have
[tex]\frac{SR}{SP}=\frac{ST}{SV}[/tex]
[tex]\frac{c}{c+12+d}=\frac{10}{30}[/tex]
[tex]\frac{8}{20+d}=\frac{1}{3}[/tex]
[tex]24=20+d[/tex]
[tex]d=4[/tex]
Now, from ΔSRT and SQU, we have
[tex]\frac{RT}{QU}=\frac{ST}{SU}[/tex]
[tex]b=\frac{10{\times}12.5}{25}[/tex]
[tex]b=5[/tex]
Also, from ΔSQU and SPV,
[tex]\frac{12.5}{a}=\frac{25}{30}[/tex]
[tex]a=15[/tex]
Thus, value of a,b,c and d are 15,5,8 and 4 respectively.