Using algebraic identities, we found that for B=2 and B= -2 , the trinomial x² + Bx +1 makes a perfect square.
What are algebraic identities?Algebraic identities are the identities which aids in solving products or powers of algebraic expressions. Most common identities are :
(a+ b)² = a² + 2ab + b²
(a- b)² = a² - 2ab + b²
a² -b² = (a+ b) (a- b), where a and b can be variable or constants or algebraic expressions.
Let the perfect square of the trinomial be (x +a)²
x² + Bx +1 = (x+ a)² = x² + 2xa + a² (using identity (a+ b)² = a² + 2ab + b²)
⇒ x² + Bx +1 = x² + 2xa + a²
⇒ Bx = 2xa , a² =1
⇒B = 2a , a =± 1
if a =1 ⇒ B = 2
If a = -1 , B = -2
For B = 2 , x² + Bx +1 = x² + 2(1)x +1² = (x+1)² ( ∵(a+ b)² = a² + 2ab + b²)
For B = -2 , x² + Bx +1 = x² -2x + (-1)² = x² -2x +1 = ( x-1)² (∵(a- b)² = a² -2ab
+ b²)
Thus, the given trinomial x² + Bx +1 is perfect square when B =2 or -2
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Which expressions are equivalent to ✓-27?
3i√3
91√√/3
O √27i
O i√27
3√/91²
The expression for [tex]\sqrt{-27}[/tex] is 3i[tex]\sqrt{3}[/tex]
What is an expression?
Expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.
given [tex]\sqrt{-27}[/tex] we know that
3 multiplied by 9 gives 27 and the square root of 9 is 3
we can write
[tex]\sqrt{-3(3)(3)}[/tex]
or [tex]3\sqrt{-3}[/tex]
square root of minus gives iota
we get,
[tex]3i\sqrt{3}[/tex]
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HELPPPPP SAT QUESTION
The value of x in the angles is equal t0 15
Angles on a Straight line
To solve this problem, we can simply apply the theorem that states the sum of angles in a straight line is equal to 180 degrees.
This implies that;
[tex]AOF + FOG + GOB = 180^0[/tex]
substituting the values into the equation;
[tex]AOF + FOG + GOB = 180^0\\(5x -15) + 90 + 2x = 180[/tex]
Let's solve for x
[tex]5x - 15 + 90 + 2x = 180\\7x + 75 = 180\\7x = 105\\x = 15[/tex]
The value of x that makes the angles equal to 180 degrees is 15.
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Which point represents -(-7)−(−7)minus, left parenthesis, minus, 7, right parenthesis on the number line?
The final point where -(-7)−(−7) number after simplification lie is 14
In the above question it is given,
-(-7)−(−7)
We need to find the position of this number on the number line.
A number line is a visual depiction of numbers on a straight line in mathematics. A number line's numerals are arranged in a sequential manner at equal intervals along its length. It is often displayed horizontally and can be infinitely expanded in any direction.
To do that, we will first simplify the given numbers by using the sign rules of the real numbers
According to the sign rule of real numbers,
(i) (-) x (-) = +
(ii) (-) x (+) = -
(iii) (-) +(-) = -
(iv) (+) + (+) = +
Now using the above sign rules, we will simplify the given number
-(-7)−(−7)
-(-7)+ [-(−7)]
7 + 7
Now, on the number line, we'll move 7 steps in the positive direction of 0, we'll be on 7. Again from 7 we'll take next 7 steps in the same direction and we'll reach the point 14.
Hence, 14 is the final point where -(-7)−(−7) number after simplification lie.
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16. Given that Angle 1 and Angle 2 form a linear pair, find the measure of both angles if m<1 = (5x + 9)° and
m<2= (3x + 11).
x =
m<1 =
m<2=
Answer: [tex]x=20, m\angle 1=109^{\circ}, m\angle 2=71^{\circ}[/tex]
Step-by-step explanation:
Angles that form a linear pair add to 180 degrees, so:
[tex]5x+9+3x+11=180\\\\8x+20=180\\\\8x=160\\\\x=20\\\\\implies m\angle 1=5(20)+9=109^{\circ}\\\\\implies m\angle 2=180^{\circ}-109^{\circ}=71^{\circ}[/tex]
What is the slope of the line on the graph
State the power function that the graph of f resembles for large values of x. Find the end-behavior for the function. Write your results using limit notation
Notice that:
[tex]\begin{gathered} \lim _{x\rightarrow\infty}\frac{2x^2(x-5)^2}{x^4}=\lim _{x\rightarrow\infty}\frac{2(x-5)^2}{x^2} \\ =\lim _{x\rightarrow\infty}\frac{2(x^2-10x+25)}{x^2} \\ =\lim _{x\rightarrow\infty}(\frac{2x^2}{x^2}-\frac{10x}{x^2}+\frac{25}{x^2}) \\ =\lim _{x\rightarrow\infty}(2-\frac{10}{x}+\frac{25}{x^2}) \\ =\lim _{x\rightarrow\infty}2-\lim _{x\rightarrow\infty}\frac{10}{x}+\lim _{x\rightarrow\infty}\frac{25}{x^2} \\ =2-0+0 \\ =2 \end{gathered}[/tex]Which means that, for large values of x:
[tex]2x^2(x-5)^2\approx2x^4[/tex]Since the function:
[tex]f(x)=2x^4[/tex]is a 4th degree monomial, with positive coefficient, then it keeps growing as x grows. Then, we know that:
[tex]\lim _{x\rightarrow\infty}2x^2(x-5)^2=\infty[/tex]Which means that the end behavior is such that f(x) approaches infinity as x approaches infinity.
On the other hand, for large negative values of x, the function is also positive. Then:
[tex]\lim _{x\rightarrow-\infty}2x^2(x-5)^2=\infty[/tex]8. Find the area of the regular polygon.
14 in
5.2 in
The area of a regular polygon is 84.87 in²
What is the regular polynomial?A polygon with congruent sides and equal angles is referred to as a regular polygon. Equilateral triangles, squares, regular pentagons, and more are examples of regular polygons.
In this case Formula of the area of the equilateral triangle is ( [tex]\sqrt{3\\}[/tex]×a²)/4
where a is the length of the side of triangle.
Here the length of the side of the polygon(triangle) is 14in.
So the area of the equilateral triangle is (√3xa²)/4.
And a=14 in the above equation.
So the area of the regular polygon is: (√3x14x14)/4 =84.87 in²
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(2t + 5x 3) + (-9t - 4x + 7)
[tex]=2t+5x+3-9t-4x+7\\=2t-9t+5x-4x+3+7\\=-7t+x+10[/tex]
Be aware I used the plus sign for 3 since you DID NOT insert the sign. For Future purposes Be careful.
Answer: -7t+x+10 (if the 3 in your equation is positive)
Step-by-step explanation: Add all like variables and integers.
If cx-d+f, then x is equal to
(f + d)/c.
First get rid of -d on the left side by adding d to both sides
cx - d = f --> cx = f + d
Then remove c from the left side by dividing both sides by c.
cx = f + d --> x = (f + d)/c
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The compound interest formula states that if P dollars are invested at an annual interest rate of r, compounded n times per year, then A, the amount of money presentIf $10.500 is invested at 8 % compounded monthly, how much will this investment be worth in 22 years? Round youranswer to two decimal placesafter tyears, is given by A = P(1+2)
Step 1
State the compound interest formula
[tex]A=P(1+\frac{r}{n})^{t\times n}[/tex]where;
[tex]\begin{gathered} P=10500 \\ r=\frac{8}{100}=0.08 \\ t=22 \\ n=12 \end{gathered}[/tex]Step 2
Find the amount after 22 years
[tex]A=10500(1+\frac{0.08}{12})^{12\times22}[/tex][tex]\begin{gathered} A=10500\times5.778587511 \\ A=\text{ \$60675.16887} \\ A\approx\text{ \$60675.17} \end{gathered}[/tex]Answer;
[tex]\text{ \$60675.17}[/tex]The listed price of the pair of pants was $38, but the price with tax came to $39.14. Find the percent sales tax.
well, the difference is 39.14 - 38 = 1.14, so that's the tax.
if we take 38 to be the 100%, just the listed price alone, what is 1.14 off of it in percentage?
[tex]\begin{array}{ccll} amount&\%\\ \cline{1-2} 38 & 100\\ 1.14& x \end{array} \implies \cfrac{38}{1.14}~~=~~\cfrac{100}{x} \implies 38x=114\implies x=\cfrac{114}{38}\implies x=3[/tex]
Evaluate. Write your answer as a whole number or as a simplified fraction.
Answer:
121
Step-by-step explanation:
11^6/11^4
= 11^(6-4)
= 11^2
= 121
trucks in a delivery fleet travel a mean of 80 miles per day with a standard deviation of 26 miles per day. the mileage per day is distributed normally. find the probability that a truck drives at least 113 miles in a day. round your answer to four decimal places.
Answer:
0.1037
Step-by-step explanation:
HELP PLEASE!!A(5,1), B(7,1), C(7,6), D(5,6) What is the perimeter of rectangle ABCD?
Let f be the function given by f(x) = x + [tex]\frac{\sqrt{x+1} }{x-1}[/tex]
Find and simplify f(3).
Responses
A f(3) = 5
B f(3) = 3 + 2[tex]\sqrt{2}[/tex]
C f(3) = 3 + [tex]\sqrt{2}[/tex]
D f(3) = 4
By evaluating f(x) in x = 3, we will get that:
f(3) = 4
Thus the correct option is D:
How to evaluate the function in x = 3?Here we have the following function:
[tex]f(x) = x + \frac{\sqrt{x + 1} }{x - 1}[/tex]
We want to get f(3), and to get that, we just need to replace all the "x" in the equation by the number 3, we will get:
f(3) = 3 + (3 + 1)/(3 - 1)
f(3) = 3 + √4/2 = 4
With this, we can conclude that the correct option is D:
f(3) = 4.
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Create three probability application problems of your own. Solve your problems. Explain your answers.
The required problems on probability application are detailed below.
Probability is the study of chance represented as a proper fraction between 0 and 1 .
The manufacturing process of a particular rugby ball consists of two parts, a leather covering and a rubber bladder. The probabilities of defect in the leather covering is 5% and the probability of the rubber bladder is 15% respectively. What is the probability that the assembled Rugby ball will not have any defect.
Let A and B denote the event that a rugby ball is defective due to defects in the leather covering and the rubber bladder respectively.
Then we can say that P(A) = 5 / 100 = 1 / 20 and
P(B) = 15 / 100 = 3 / 20
Therefore P(A') = 1 - 1/20 = 19/20 and P(B') = 1 - 3 / 20 = 17 / 20.
P(A') and P(B') gives us the probabilities that the rugby ball will not have defective parts respectively.
Therefore the probability that the assembled product will not have any defect is:
P(A') × P(B')
[tex]=\frac{19}{20} \times \frac{17}{20}\\\\=\frac{323}{400} \\\\=0.8075[/tex]
Therefore the probability that the rugby ball will not have any defect is 80.75% .
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Find sin 8, cot0, and csc0, where 8 is the angle shown in the figure.
Give exact values, not decimal approximations.
Continue
25
0
sin 0
11
▬
cot 8 = 0
CSC8 = 0
믐
Answer:
sinTheta = 24/25
cotTheta = 7/24
cscTheta = 25/24
Step-by-step explanation:
Use pythagorean thm or pythagorean triples to find the missing side of the triangle.
7^2 + x^2 = 25^2
49 + x^2 = 625
x^2 = 576
x = 24
The third side of the triangle is 24. We'll need this in a sec.
The marked angle is called theta. That 0 with a belt is greek letter theta. They could just call it angleA or angleB or whatever. Its name is just theta no big deal.
25 is the hypotenuse (longest side). The side that's 7 units long is one of the pieces that makes up AngleTheta. So the 7 is "adjacent" to theta. The other leg, 24, is "opposite" from theta.
sinTheta is a ratio. It is OPPOSITE/HYPOTENUSE
sinTheta = 24/25
cotTheta is tanTheta flipped over.
tanTheta is OPP/ADJACENT
so...
cotTheta =
ADJ/OPP
=7/24
Last, cscTheta is sinTheta flipped over.
We already found sinTheta, so flip it over.
cscTheta = 25/24
SOH-CAH-TOA is supposed to help you memorize the ratios for sine (opp/hyp), cos (adj/hyp), and tan (opp/adj).
The math word for "flipped over" is reciprocal.
csc is the reciprocal of sin
sec is the reciprocal of cos.
cot is the reciprocal of tan.
Good luck! Hope this helps!
Answer the question in the photo for EASY points ( 20 points )
Answer: 7921 123569 5410
Step-by-step explanation:
1. 7912
2. 123569
3. 5210
How many blocks would be in the 10th figure? (figure 1 is 1 block) figure 11 figure 15 figure 19 figure 22
The 10th figure will have 19 blocks.
What is Arithmetic Progression?An arithmetic progression (AP) is a sequence in which the differences between each successive term are the same. The formula Tₙ = a+(n-1)d finds the general term (or) nth term of an AP whose first term is 'a' and the common difference is 'd'.
In the given figures,
Figure 1 has 1 block.
Figure 2 has 3 blocks.
Figure 3 has 5 blocks.
We have to find the number of blocks in the 10th figure.
The number of blocks in the figures are forming an arithmetic progression of 1,3,5.....with a common difference of 2.
So the 10th term of AP can be calculated as,
[tex]T_{10} = 1 + (10 -1)2[/tex]
[tex]T_{10} = 1 + 9*2[/tex]
[tex]T_{10} = 19[/tex]
Therefore, the 10th figure will have 19 blocks.
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i need the right answers please help
Answer:
To what please
Step-by-step explanation:
I am here to help you so please send your questions
DOES ANYONE KNOW THE ANSWERSSSS
65 pt
Answer:
3.75 cups of flour
2.5 cups of sugar
1.25 cups of butter
Step-by-step explanation:
Hello!
Let the amount of sugar used be x:
Flour = 1.5xButter = 0.5xThe total sum of ingredients should be 7.5 cups, so we can add x, 1.5x, and 0.5x and set it to 7.5.
Solve for xx + 0.5x + 1.5x = 7.53x = 7.5x = 2.5Clifford needs 2.5 cups of sugar.
The amount of flour can be found by multiplying 2.5 by 1.5: 3.75 cups
The amount of butter can be found by multiplying 2.5 by 0.5: 1.25
Clifford needs 3.75 cups of flour, 2.5 cups of sugar, and 1.25 cups of butter.
if three buckets of capacities 15 litre,18 litre and 24 litre can fill a drum in exact number of times, find the least capacity of the drum.
There are two questions to be solved, therefore, I'll solve the buckets one first and then I'll solve the other one. To solve both we need to apply LCM (least common multiple).
There are three buckets with differente capacities, one with 15 litres, one with 18 litres and one with 24 litres. Since they can fill a drum in an exact number of times, this means that the capacity of the drum is a multiple of the capacity of each bucket, therefore we should break each number into their multiples and find the common ones as shown below.
[tex]\begin{gathered} 15\text{ = 3}\cdot5 \\ 18\text{ = 2}\cdot3\cdot3 \\ 24\text{ = 2}\cdot2\cdot2\cdot3 \end{gathered}[/tex]We now need to multiply all the common multiples:
[tex]\text{least capacity = 2}\cdot2\cdot2\cdot3\cdot3\cdot5\text{ = }360\text{ litres}[/tex]For the second problem we have three items a book, a pen and a box. They cost respectively 36, 45 and 54. We need to then find the smallest sum of money that could buy a exact number o each item. We need to apply LCM once again as shown below:
[tex]\begin{gathered} 36\text{ = 2}\cdot2\cdot3\cdot3 \\ 45\text{ = 3}\cdot3\cdot5 \\ 54\text{ = 2}\cdot3\cdot3\cdot3 \end{gathered}[/tex]We now need to multiply all the common multiples:
[tex]\text{least sum of money = 2}\cdot2\cdot3\cdot3\cdot3\cdot5\text{ = }540[/tex]zimak bought 3 xylobars and 5 5 yackities from the intergalactic candy store for 1.75 jeejunu paid1.88 and got 4 xylobars and 1 yackity how much would dado pay for 1 xylobar and 2 yackities
For 1 xylobar and 2 yackities Dado would pay $0.61 .
In the question ,
it is given that
Zimak paid $1.75 for 3 xylobars and 5 yackities .
let the price of 1 xylobar = x
and let the price of 1 yackities = y
According to the question
3x+5y=1.75 ...(i)
Jeejunu paid $1.88 for 4 xylobars and 1 yackity .
So , equation becomes
4x+y=1.88 ...(ii)
y = 1.88-4x
substituting it in equation (i), we get
3x + 5(1.88 - 4x) = 1.75
3x + 9.4 - 20x = 1.75
20x-3x = 9.4 - 1.75
17x = 7.65
x = 7.65/17
x = 0.45
So, the price of 1 xylobar = $0.45
put x = 0.45 in y= 1.88-4x , we get
y = 1.88 - 4(0.45)
y = 1.88 - 1.8
y = 0.08
So , the price of 1 yackity is $0.08 .
So, For 1 xylobar and 2 yackities , Dado would pay
= x+2y
= 0.45 + 2(0.08)
= 0.45+0.16
= 0.61
Therefore , for 1 xylobar and 2 yackities Dado would pay $0.61 .
The given question is incomplete , the complete question is
Zimak bought 3 xylobars and 5 yackities from the intergalactic candy store for $1.75 . Jeejunu paid $1.88 and got 4 xylobars and 1 yackity . How much would dado pay for 1 xylobar and 2 yackities ?
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y = 2 |x|
Graph the following linear inequalities
Mathematical expressions known as linear inequalities compare two expressions using the inequality symbol. The expression may be either a numerical or an algebraic expression, or it may be both.
What is meant by linear inequalities?Addition, subtraction, multiplication, and division are the four types of operations that can be performed on linear inequalities. Equivalent inequality refers to linear inequalities with the same solution. Both equality and inequality are subject to laws. For inequality involving smaller than or equal to () and greater than or equal to (), all the principles listed below apply. Let's look at some of the key inequality rules for all these operations before learning how to solve linear inequalities.
As with multi-step linear equations, start by separating the variable from the constants when resolving multi-step linear inequalities with one variable. According to the laws of inequality, it is crucial that we remember to flip the inequality sign when multiplying or dividing with negative values while resolving multi-step linear inequalities.
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63= 4w + 15 solve for w and simplify your anwser as much as possible.
Answer:
w is 59
Step-by-step explanation:
63-4 =59 and this should be the answer your looking for
Answer:
w=12
Step-by-step explanation:
63+4w+15
collect like terms
63-15=4w
48=4w (divide all through by 4)
w=12
f(x)=x^2+x and g(x)=1/x+1 find the rules for (f o g)(x) and (g o f)(x) and give the domain of each composite function
The composite functions of (f o g)(x), (g o f)(x) and their domains in the function f( x ) = x² + x and g( x ) = 1/(x+1) are (x+2) / (x+1)², Domain: {x|x ≠ -1 } and 1/(x² + x + 1 ), Domain: {x|x ∈ R } respectively.
What are the composite result function and domain of (f o g)(x) and (g o f)(x) in the given function?A function is simply a relationship that maps one input to one output.
Given the data in the question;
f( x ) = x² + xg( x ) = 1/(x+1)(f o g)(x) = ?(g o f)(x) = ?First, set up the composite result function (f o g)(x).
(f o g)(x) = f( g(x) )
f( x ) = x² + x
f( g(x) ) = f( 1/(x+1) ) = ( 1/(x+1) )² + 1/(x+1)
Apply product rule to simply
f( g(x) ) = 1² /(x+1)² + 1/(x+1)
f( g(x) ) = (1 + x + 1 )/ (x+1)²
f( g(x) ) = (x+2) / (x+1)²
Next, find the domain by equating the denominator to 0 and solve for x.
x + 1 = 0
x = -1
Domain: {x|x ≠ -1 }
For (g o f)(x), set up the composite result function (g o f)(x).
(g o f)(x) = g( f(x) )
g( x ) = 1/(x+1)
g( f(x) ) = g( x² + x ) = 1/( (x² + x) + 1 )
g( f(x) ) = 1/( (x² + x) + 1 )
g( f(x) ) = 1/(x² + x + 1 )
Next, find the domain by equating the denominator to 0 and solve for x.
x² + x + 1 = 0
x = ( -1±√(1² - 4 ×(1×1) ) / (2 × 1 )
x = (-1±i√3)/2
x = (-1+i√3)/2, (-1-i√3)/2
Hence,
Domain is set of all real numbers.
Domain: {x|x ∈ R }
Therefore the composite functions and their domain are (x+2) / (x+1)², Domain: {x|x ≠ -1 } and 1/(x² + x + 1 ), Domain: {x|x ∈ R }.
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you are constructing a 12 cubic feet box that is open at the top. the material used to construct the bottom costs $3 per square foot and the material used to construct the sides is $1 per square foot. what dimensions should you use to minimize the cost of the materials? set up and solve using the lagrange method.
The length and breadth should be 2 feet each and height should be 3 feet to minimize the cost of the materials.
Let consider the length of the box =x
considering a square bottom ,
length = breadth = x
and let the height =h
The surface area of the box is
SA = x² + 4xh ( box is open from top )
cost of 1sq feet material for bottom =$3
cost of 1sq feet material for sides =$1
The cost functions can be written as
C = 3x² + 1(4xh)
C = 3x² + 4xh
Also the volume equation is
x²h = 12
h = 12/x²
substituting the value of h into the cost function.
C = 3x² + 4x (12/x²)
C = 3x² + 48/x
C = (3x³ + 48)/ x
Take the derivative of C and set it equal to zero. So we get
6x³ - 48 = 0
x³ = 48/6 = 8
x = 2
∴ h = 12/x²
h = 12/4
h = 3
This is the value of x and h that minimizes the cost.
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It took Elsa 18 minutes to make 3 bookmarks.
Which of the following pairs of equivalent fractions could be used to determine how many bookmarks she can make in 90 minutes?
y + 7 > 22 solve the inequality for y and simplify your answer as much as possible
Answer:
y > 15
Step-by-step explanation:
y + 7 > 22 ( subtract 7 from both sides )
y > 15
Bricklayers use the formula n = 7th to estimate the number n
of bricks needed to build a wall of length and height h, where and hare in
feet. Solve the formula for h. Estimate the height of a wall 28 ft long that requires
1568 bricks.
(Round to the nearest tenth if necessary)
Answer: the answer is ..
Step-by-step explanation: fec r/sm mkdnjervc