Answer:
[tex](x-2)^5 = x^5 - 10x^4 + 40x^3 - 80x^2 + 80x - 32[/tex]
Step-by-step explanation:
Binomial Theorem
[tex](a+b)^n=a^n+\dfrac{n!}{1!(n-1)!}a^{n-1}b+\dfrac{n!}{2!(n-2)!}a^{n-2}b^2+...+\dfrac{n!}{r!(n-r)!}a^{n-r}b^r+...+b^n[/tex]
Given expression:
[tex](x-2)^5[/tex]
Compare with the Binomial Theorem:
[tex]a=x, \quad b=-2, \quad n=5[/tex]
Substitute the values into the formula:
[tex]\begin{aligned}(x-2)^5 & = x^5+\dfrac{5!}{1!4!}x^{4}(-2)+\dfrac{5!}{2!3!}x^{3}(-2)^2+\dfrac{5!}{3!2!}x^{2}(-2)^3+\dfrac{5!}{4!1!}x^{1}(-2)^4+(-2)^5\\\\& =x^5+\dfrac{120}{24}x^{4}(-2)+\dfrac{120}{12}x^{3}(4)+\dfrac{120}{12}x^{2}(-8)+\dfrac{120}{24}x(16)-32\\\\& =x^5-\dfrac{240}{24}x^{4}+\dfrac{480}{12}x^{3}-\dfrac{960}{12}x^{2}+\dfrac{1920}{24}x-32\\\\& =x^5-10x^{4}+40x^{3}-80x^{2}+80x-32\end{aligned}[/tex]
Pascal's Triangle
Pascal's triangle is an arrangement of binomial coefficients in triangular form. Each row has 1 at both ends, and each remaining number in the row is found by adding together the two numbers directly above it.
[tex]\phantom{-b)))))))}1\\\phantom{-b)))))}1 \:\quad 1\\\phantom{-b)))}1 \:\quad 2\: \quad 1\\\phantom{-b)}1 \:\quad 3\: \quad 3\: \quad 1\\\phantom{))}1 \quad 4 \:\: \quad 6 \quad \;4 \: \quad 1\\1 \quad 5 \quad 10 \quad 10 \quad 5 \quad 1[/tex]
(Note: The initial row with the single number 1 is row zero)
Each term in the binomial expansion using Pascal's triangle is the product of a coefficient, a with an exponent, and b with an exponent.
As we move through each term from left to right, the exponent of a decreases from the exponent of the expression down to zero, and the exponent of b increases from zero up to the exponent of the expression.
The coefficients of each term are the numbers which appear in the number of the row that corresponds with the exponent of the expression.
From the given expression, the power that we are expanding the bracket to is 5, so we need to look at line 5 of Pascal's triangle, which is:
1 5 10 10 5 1
Therefore, the expansion using Pascal's triangle to the power 5 of is:
[tex](a+b)^5=1a^5b^0+5a^4b^1+10a^3b^2+10a^2b^3+5a^1b^4+1a^0b^5[/tex]
Substituting the values of a and b:
[tex]\begin{aligned}(x-2)^5 & = 1x^5(-2)^0 + 5x^4(-2)^1 + 10x^3(-2)^2 + 10x^2(-2)^3 + 5x^1(-2)^4 + 1x^0(-2)^5\\\\ & = x^5 - 10x^4 + 40x^3 - 80x^2 + 80x - 32\end{aligned}[/tex]
20 points!!
Show that for any real numbers a and b, and any integers x and y so that x≠0, y≠0, x≠y, and x≠-y,
(y/x-x/y)((ax+by)/(x+y)-(ax-by)/(x-y))=2(a-b).
See below for the proof of the equation [tex]\frac{ax + by}{x + y} - \frac{ax - by}{x - y}= 2(a - b)[/tex]
How to prove the equation?The equation is given as:
[tex]\frac{ax + by}{x + y} - \frac{ax - by}{x - y}= 2(a - b)[/tex]
Take the LCM
[tex]\frac{(ax + by)(x - y) -(ax - by)(x + y)}{(x + y)(x - y)}= 2(a - b)[/tex]
Expand
[tex]\frac{ax^2 - axy + bxy - by^2 -ax^2 - axy + bxy + by^2}{(x + y)(x - y)}= 2(a - b)[/tex]
Evaluate the like terms
[tex]\frac{-2axy + 2bxy }{(x + y)(x - y)}= 2(a - b)[/tex]
Rewrite as:
[tex]\frac{-2axy + 2bxy }{x^2 - y^2}= 2(a - b)[/tex]
Factorize the numerator
[tex]\frac{2(a - b)(x^2 - y^2)}{x^2 - y^2}= 2(a - b)[/tex]
Divide
2(a - b)= 2(a - b)
Both sides are equal
Hence, the equation [tex]\frac{ax + by}{x + y} - \frac{ax - by}{x - y}= 2(a - b)[/tex] has been proved
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What function does this graph represent? f(x) 6 -4 -2 = 6- 4 2- A -6 lo 2 OA. f(x) = -3(x - 2)² + 4 OB. f(x) 3(x + 2)² + 4 Oc. f(x) = 3(x - 2)² + 4 OD. f(x) = -3(x + 2)² + 4
This graph represents Option D = [tex]-3(x+2)^{2} + 4[/tex]
As the given figure is a parabola, it will have quadratic equation because the general equation of parabola corresponds to [tex]x^{2} = 4ay[/tex] which has highest power of the variable equals to 2, thus its a quadratic equation.
The maximum value of a quadratic equation [tex]ax^{2} + b x +c[/tex] is at points
[tex](-b/2a, c-\frac{b^{2} }{4a} )[/tex]
By looking at the graph we can observe the point at which the graph has achieved maxima is (-2,4).
∴ ([tex]-b/2a = -2[/tex] , [tex]c-\frac{b^{2} }{4a} = 4[/tex]
The coefficient of [tex]x^2[/tex] should be negative the graph opens downwards.
∴ Option B and C are eliminated.
On putting x = -2 only option D yields 4.
Thus this graph represents Option D = [tex]-3(x+2)^{2} + 4[/tex]
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(2, -7) and (4, -13)
Answer:
-14 or (6,20)
Find the missing angles. will give brainliest if done correctly.
Answer:
x = 69 v = 103
Step-by-step explanation:
Angles in a triangle add up to 180 degrees:
34 + 77 + x = 180
x = 69
Angles on a straight line add up to 180:
77 + v = 180
v = 103
Answer:
x = 69°V = 103°Step-by-step explanation:
the sum of the interior angles in a triangles is 180°, take away from 180 the know angles (34° and 77°) and you will have your answer
180 - 34 - 77 = 69°
Now we find angle V
The angle V and the angle of 77 ° are on a straight line, so the sum is 180°, from 180° you subtract 77 ° and you have the value of V
180 - 77 = 103 °
Can someone help me with these 4 geometry questions? Pls it’s urgent, So ASAP!!!!
Question 4
1) [tex]\overline{BD}[/tex] bisects [tex]\angle ABC[/tex], [tex]\overline{EF} \perp \overline{AB}[/tex], and [tex]\overline{EG} \perp \overline{BC}[/tex] (given)
2) [tex]\angle FBE \cong \angle GBE[/tex] (an angle bisector splits an angle into two congruent parts)
3) [tex]\angle BFE[/tex] and [tex]\angle BGE[/tex] are right angles (perpendicular lines form right angles)
4) [tex]\triangle BFE[/tex] and [tex]\triangle BGE[/tex] are right triangles (a triangle with a right angle is a right triangle)
5) [tex]\overline{BE} \cong \overline{BE}[/tex] (reflexive property)
6) [tex]\triangle BFE \cong \triangle BGE[/tex] (HA)
Question 5
1) [tex]\angle AXO[/tex] and [tex]\angle BYO[/tex] are right angles, [tex]\angle A \cong \angle B[/tex], [tex]O[/tex] is the midpoint of [tex]\overline{AB}[/tex] (given)
2) [tex]\triangle AXO[/tex] and [tex]\triangle BYO[/tex] are right triangles (a triangle with a right angle is a right triangle)
3) [tex]\overline{AO} \cong \overline{OB}[/tex] (a midpoint splits a segment into two congruent parts)
4) [tex]\triangle AXO \cong \triangle BYO[/tex] (HA)
5) [tex]\overline{OX} \cong \overline{OY}[/tex] (CPCTC)
Question 6
1) [tex]\angle B[/tex] and [tex]\angle D[/tex] are right angles, [tex]\overline{AC}[/tex] bisects [tex]\angle BAD[/tex] (given)
2) [tex]\overline{AC} \cong \overline{AC}[/tex] (reflexive property)
3) [tex]\angle BAC \cong \angle CAD[/tex] (an angle bisector splits an angle into two congruent parts)
4) [tex]\triangle BAC[/tex] and [tex]\triangle CAD[/tex] are right triangles (a triangle with a right angle is a right triangle)
5) [tex]\triangle BAC \cong \triangle DCA[/tex] (HA)
6) [tex]\angle BCA \cong \angle DCA[/tex] (CPCTC)
7) [tex]\overline{CA}[/tex] bisects [tex]\angle ACD[/tex] (if a segment splits an angle into two congruent parts, it is an angle bisector)
Question 7
1) [tex]\angle B[/tex] and [tex]\angle C[/tex] are right angles, [tex]\angle 4 \cong \angle 1[/tex] (given)
2) [tex]\triangle BAD[/tex] and [tex]\triangle CAD[/tex] are right triangles (definition of a right triangle)
3) [tex]\angle 1 \cong \angle 3[/tex] (vertical angles are congruent)
4) [tex]\angle 4 \cong \angle 3[/tex] (transitive property of congruence)
5) [tex]\overline{AD} \cong \overline{AD}[/tex] (reflexive property)
6) [tex]\therefore \triangle BAD \cong \triangle CAD[/tex] (HA theorem)
7) [tex]\angle BDA \cong \angle CDA[/tex] (CPCTC)
8) [tex]\therefore \vec{DA}[/tex] bisects [tex]\angle BDC[/tex] (definition of bisector of an angle)
In the following diagram, PQRS is a square of sides 21 cm. PTS and QTR are two semicircles touching each other at T.
Calculate the area, in cm², of the shaded region.
A 94.5 C 141.8 B 113.4 D 189.0 S
Answer:
[tex]94.6cm^{2}[/tex]
Step-by-step explanation:
First we will find the area of Square PQRS.
Area of Square PQRS = Length x Width = 21 x 21
= [tex]441cm^{2}[/tex]
Next we will found the Area of Semicircles PS and QR.
Note: Area of Semicircle PS = Area of Semicircle QR
Area of Semicircle = [tex]\frac{1}{2} \pi r^{2}[/tex]
Total Area of Semicircles PS and QR combined = [tex]2(\frac{1}{2} \pi r^{2} )\\=\pi r^{2}[/tex]
We know that the diameter of PS = QR = 21 cm (due to the length of the square)
Radius = Half of Diameter = 0.5 x 21cm = 10.5cm
Total Area of Semicircles PS and QR = [tex]\pi (10.5)^{2} \\=110.25\pi cm^{2}[/tex]
Finally,
Area of Shaded Region = Area of Rectangle PQRS - Total Area of Semicircles PS and QR
= [tex]441 - 110.25\pi\\= 94.6cm^{2} (1dp)[/tex]
In this case , you can choose the nearest answer as there might be some rounding differences.
9 times 2/3=
SIMPIFIED
Answer:
6
Step-by-step explanation:
9 x 2/3
9/1 x 2/3 = 18 / 3 = 6
Jamie has a restaurant bill of $70.70. About how much money should she leave for a 10% tip?
$7.00
$0.70
$1.00
$7.70
A test started at 10:55. the teacher collected the answer scripts 1 hour and 15 minutes later . At what time did she collect them?
Answer:
10+1=11 but the 55 so it is now 11:55+15min so it makes it 12:10
Answer:
Hello! The answer to your question is 12:10.
Step-by-step explanation:
1 hour after 10:55 is equivalent to 11:55. Now, we have 15 minutes remaining. We can add in increments of 5 to get to 15 in terms of time:
11:55 + 5 minutes
= 12:00 + 5 minutes
= 12:05 + 5 minutes
= 12:10
5 + 5 + 5 = 15, so we reached 15.
The teacher collected the answer scripts at 12:10.
For the given equation, find the values of a, b, and c, determine the direction in which the parabola opens, and determine the y-intercept. Decide which table best illustrates these values for the equation: y = negative 4 x squared
The values of a, b, and c are -4,0 and 0 and the parabola opens downward with y-intercept (0,0).
we have
y=-4x^2
What is the standard form of the quadratic equation?The quadratic equation in standard form is equal to
ax^2+bx+c=0
So,In this problem
a=-4, b=0 and c=0
The y-intercept is the value of y when the value of x is equal to zero
y=-4(0)^2=0
The y-intercept is the point (0,0)
The coefficient a is negative, therefore the parabola opens down.
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Please answer with everything needed i appreciate everyones help:))
Answer:
[tex]a = (h)(2h - 5)[/tex]
Answer:
Area of the rectangle = 2h² - 5
Step-by-step explanation:
• base length l = 5 less than twice the height h
= 2h - 5
• height = h
Area of rectangle = base × height
⇒ l × h
⇒ (2h - 5)(h)
⇒ 2h² - 5
The perimeter of a square field is 336 yards. How long is each side?
Question:
The perimeter of a square field is 336 yd. How long is each side?
Answer:
1,344 yd
Which quadratic equation does not have a real solution? (1 point)
3x2 − 6x + 3 = 0
−4x2 − 4x − 6 = 0
−x2 − 6x − 9 = 0
2x2 − 10x − 5 = 0
Since it has a negative discriminant, the quadratic equation that does not have a real solution is given by:
−4x2 − 4x − 6 = 0.
What is the discriminant of a quadratic equation and how does it influence the solutions?A quadratic equation is modeled by:
[tex]y = ax^2 + bx + c[/tex]
The discriminant is:
[tex]\Delta = b^2 - 4ac[/tex]
The solutions are as follows:
If [tex]\mathbf{\Delta > 0}[/tex], it has 2 rational solutions.If [tex]\mathbf{\Delta = 0}[/tex], it has 1 rational solutions.If [tex]\mathbf{\Delta < 0}[/tex], it has 0 real solutions.We want [tex]\Delta < 0[/tex], hence, for equation -4x² - 4x - 6, we have that a = -4, b = -4, c = -6, hence:
[tex]\Delta = (-4)^2 - 4(-4)(-6) = 16 - 96 = -80[/tex]
Negative discriminant, hence it does not have a real solution.
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Answer:
The answer above is correct. The answer was confirmed by a teacher, and was marked as correct on the test
Step-by-step explanation:
Classify the following triangle. Check all that apply.
To collect data from a number of people in a sample, use alan
The ways to collect data in a sample include:
Use of questionnaire.Through observations.Through surveys.What is a sample?It should be noted that a sample simply means a subset of a population that has the characteristics of the entire population.
In this case, the ways to collect data in a sample include the use of questionnaire, through observations, and through surveys.
l
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A searchlight uses a parabolic mirror to cast a beam of light in parallel rays where the light bulb is located at the focus of the parabola in order to give the best illumination. The mirror is modeled by y^2=36(x-10), where the measurements are in cm. What is the location of the light bulb?
Answer:
the third option
Step-by-step explanation:
Our equation is
[tex] {y}^{2} = 36(x - 10)[/tex]
Equation of a vertical parabola is
[tex]( {y - k)}^{2} = 4p(x - h)[/tex]
where (h,k) is the center
The focus is
(h+p, k)
Equation of directrix is
x= h-p,
Here the center is (10,0)
Next, we factor out 4 in the original equation
[tex]4(9)[/tex]
So we have
[tex] {y}^{2} = 4(9)(x - 10)[/tex]
So our p=9,
So our focus is
(10+9,0) or (19,0)
The third option is the answer
The location of the light bulb is at point A. (10, 0).
Option A is the correct answer.
We have,
To find the location of the light bulb, we need to identify the coordinates (x, y) at which the light bulb is located.
In the given parabolic mirror equation, y² = 36(x - 10), the vertex form of the equation for a parabola with a vertical axis is (h, k), where h is the
x-coordinate of the vertex and k is the y-coordinate of the vertex.
Comparing the given equation with the standard form
(y - k)² = 4a(x - h),
we can see that h = 10 and k = 0.
So, the location of the light bulb is at the point (10, 0).
Therefore,
The location of the light bulb is at point A. (10, 0).
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A searchlight uses a parabolic mirror to cast a beam of light in parallel rays where the light bulb is located at the focus of the parabola in order to give the best illumination.
The mirror is modeled by y² = 36 (x-10), where the measurements are in cm.
What is the location of the light bulb?
A. (10, 0)
B. (14, 0)
C. (19, 0)
D. (36, 0)
Select the correct graph of −x − 2y = 6.
Answer:
x = -6-2y
Step-by-step explanation:
Caraline is planting flowers in her garden she plants 7 flowers in 4 minutes after 12 minutes she has planted 21 different flowers
Answer:
A.28 minutes
B.40 minutes
C.56 minutes
D.70 minutes
Step-by-step explanation:
look at image.................
Answer:
-1/3
Step-by-step explanation:
When x is 1, y is 5
When x is 4, y is 4
5-4=1
1-4=-3
1/-3=-1/3
PLEASE HELP!!!!!!!!!!!!I BEG YOU
pls step by step on how to do that
wth is that wdym ???
The Surface Area of a square pyramid is 3600 ft2. The slant height is 80 feet and the base is 20 feet. At most how many 1 foot cubic blocks can be stuffed in the pyramid?
(*Some of the blocks might have to be cut up to fit into crevices*)
The number of 1 foot cubic blocks that can be stuffed in the pyramid are; 10,583 cubes
How to find the surface area of a Pyramid?The equation for the surface area of a square pyramid is;
Surface area = A + 1/2·p·s
where;
A = area of base
p = perimeter
s = slant height
The volume of the pyramid = (1/3) × A × h
Where:
A = Area of the base = 20 × 20 = 400 ft²
h = The height of the pyramid = √((slant height)² - ((base side length)/2)²)
h = √(80² - (20/2)²) = 30√7
The volume of the pyramid = 1/3*400*30·√7 = 4000·√7 ft³ = 10,583.005 ft³
If the cubes are flexible, then there are approximately 10,583 cubes
For rigid cubes we have;
Given that the height of the pyramid = 30·√7
The slope of the pyramid = (30/10)√7 = 3·√7
A increase in height of 1 foot gives a reduction in width of 2/(3√7)
The bottom can hold 400 cubes, Then next layer can hold;
19 × 19 = 361
If we continue to iterate, we will get a total of approximately 9915 rigid cubes in the pyramid.
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omg help me please!!!!
if you anwser the question i give you brainly
The graph is in a reciprocal relationship with Option C
What is a Function ?A function is a mathematical statement used to relate a dependent and n independent variable.
The function is represented by the graph
The nature of the graph is exponential
and the equation is determined by the points on the graph
( 1,1) ( 2,4) '
The equation is
y = ( 1/4) 4ˣ
The inverse of the function will be found by replacing the y variable with x and vice versa and then solving to bring the function into the form of x.
x = ( 1/4) [tex]\rm 4^y[/tex]
Plotting this gives the answer as Option C
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According to a report in USA Today, more and more parents are helping their young adult children get homes. Suppose eight persons in a random sample of 60 young adults who recently purchased a home in Kentucky received help from their parents. You have been asked to construct a 95% confidence interval for the population proportion of all young adults in Kentucky who received help from their parents. What is the margin of error for a 95% confidence interval for the population proportion
The margin of error for a 95% confidence interval for the population proportion is 0.0815.
Given sample size=60 and confidence interval of 95%.
We have to calculate margin of error for a confidence interval of 95%.
Margin of error is the difference between actual values and calculated values. Margin of error is a part of z test.
We have been given sample size=60.
8 people have received help from their parents from the sample.
p=8/60=0.13
which is sample proportion.
z=1-0.13
=0.87
To calculate standard error=[tex]\sqrt{p*z/n}[/tex]
=[tex]\sqrt{0.13*0.8/60}[/tex]
=0.0416
at 95% confidence interval
z(α/2)=1.96
therefore margin of error=1.96*0.0416
=0.081536
=0.0815
Hence the margin of error for 95% confidence interval is 0.0815
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help me i give you brainly
Answer:
False
Step-by-step explanation:
I'm not a 100 percent though
what is something you do that can be set up as a two- step equation? how would you set that up as an equation to explain to someone else how it can be solved?
What you can do that can be set up as a two- step equation is to follow the form which is [tex]ax + b = c[/tex] and should be noted that a, b, c are real numbers.
Two- step equation can be solved by following these
Perform the Simplification by getting rid of all brackets and parenthesis. perform Addition or subtraction to isolate the variable.Simplify to determine the value of the variable.Perform verification through substituting of answers in the given equation.What is two- step equation?Two step equations serves as algebraic equations which can be solved in exactly two steps to get the final value of the variable .
Example of this is [tex]2x + 3 = 7.[/tex]
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the tens digit of a two digit number is 5 greater than the units digit. if you subtract the reversed number from the original number you will get 1/4 of the original number
what is the original number
From the given information, the original number is 180. 02
How to determine the original numberLet the unit digit be x
The unit in tens is x+ 5
The original number = 10(x+5) + x = 11x + 50
The reversed number = 10x+ (x+5) = 11x+5
From the information given, we have that
(11x + 50) - (11x + 5) = 1/4 (11x + 50)
Expand the expression
11x + 50 - 11x - 5 = 1/ 4(11x + 50)
Collect like terms
11x - 11x + 50 - 5 = 1/ 4 (11x + 50)
45 = [tex]\frac{11x + 50}{4}[/tex]
Cross multiply
45 * 4 = 11x + 50
180 = 11x + 50
11x = 180 -50
11x = 130
x = 130/11
x = 11. 82
To find the original number, we have
Original number = 11x + 50
Substitute the value of x
Original number = 11(11.82) + 50
Original number = 130. 02 + 50
Original number = 180. 02
Therefore, the original number is 180. 02
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find f(-1) and find one value of x for which f(x)=0
[tex]f( - 1) = - 3[/tex]
[tex]f(x) = 0 \\ for \: x = 2 \: or \: x = - 2[/tex]
Point O is the center of dilation. Triangle A B C is dilated to create triangle A prime B prime C prime. The length of O A is 3. The length of A A prime is 9.
Triangle ABC is dilated to create triangle A'B'C' using point O as the center of dilation.
What is the scale factor of the dilation?
One-third
3
4
One-fourth
The scale factor of the dilation shown in the image is: C. 4.
How to Find the Scale Factor of Dilation?Scale factor = New dimension/corresponding old dimension.
From the image given, we have:
Scale factor = OA'/OA
OA' = 3 + 9 = 12 units
OA = 3 units
Scale factor of dilation = 12 units/3 units
Scale factor of dilation = 4
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C. 4
Next one is 6 if you have the option on edge