The solution set for the logarithmic equation log3(x + 21)-log3(x-5)=3 is x=9.
To solve the logarithmic equation log3(x + 21)-log3(x-5)=3, we can use the quotient rule of logarithms to rewrite the equation as log3[(x + 21)/(x-5)]=3. We know that the domain of a logarithmic function is only valid for positive values inside the parenthesis. Therefore, we must reject any value of x that makes the denominator (x-5) equal to 0. So, x cannot be equal to 5.
Next, we can rewrite the equation without logarithms as 3=3 log3[(x + 21)/(x-5)]. Using the property that a log a(x)=x, we can simplify the equation as 3=(x + 21)/(x-5)³. Multiplying both sides by (x-5)³, we get 3(x-5)³ = x+21.
Expanding the left side of the equation and simplifying, we get 3x³ - 72x² + 498x - 1089 = 0. We can then solve for x using synthetic division or long division, which gives us the solution x=9.
However, we must check if x=9 is a valid solution by plugging it back into the original equation. Since log3(9+21) = log3(30) and log3(9-5) = log3(4), we can simplify the original equation as log3(30/4) = log3(15/2) = 3. Therefore, x=9 is a valid solution.
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lcm(x-3)^2(x+4);(x-3)(x+4)^2
Answer:
[tex](x - {3})^{2} x(x + {4})^{2} [/tex]
Which function has the lowest minimum value, and what are its coordinates?
A. Function 1 has the lowest minimum value, and its coordinates are (−1, −3).
B. Function 1 has the lowest minimum value, and its coordinates are (0, 1).
C. Function 2 has the lowest minimum value, and its coordinates are (−1, 0).
D. Function 2 has the lowest minimum value, and its coordinates are (0, 2).
Answer:
A. Function 1 has the lowest minimum value, and its coordinates are (−1, −3).
Step-by-step explanation:
To find the minimum value for f(x), set its first derivative to 0 and solve for x.
First derivative of f(x) is
f'(x) = 8x + 8 + 0 = 8x + 8
Setting this equal to 0 gives
8x + 8 = 0
8x = -8
x = -1
Plugging in this value for x in 4x² + 8x + 1 gives
f(x) = 4(-1)² + 8(-1) = 4 · 1 - 8 + 1
= 4 - 8 + 1
-4 + 1
= -3
So minimum for f(x) occurs at (-1, -3)
For g(x) the minimum is 0 and is at x = -1
So f(x) has minimum of -3 at x = -1
g(x) has minimum of 0 and occurs at x = -1
Answer:
A. Function 1 has the lowest minimum value, and its coordinates are (−1, −3).
How do you find the center of dilation of two lines?
Finding the center of dilation of two lines involves determining the point that is equidistant from the two lines.
A dilation is a transformation that changes the size of an object, but not its shape, and the center of dilation is the point from which the object is scaled.
The center of dilation of two lines can be found by constructing a line perpendicular to both lines that bisects the angle between them. The point of intersection of this line with the segment connecting the two closest points on the two lines is the center of dilation.
To find the center of dilation, we can use the following steps:
1. Draw the two lines and find the angle between them.
2. Draw a line perpendicular to both lines and bisecting the angle between them.
3. Draw the segment that connects the two closest points on the two lines.
4. The intersection of the line from step 2 with the segment from step 3 is the center of dilation.
It's also possible to use vector and linear algebra to find the center of dilation for two lines. By finding the slope of each line and their intersection point, the center of dilation can be found as the midpoint of the intersection point of the two lines.
In conclusion, finding the center of dilation of two lines requires determining the point that is equidistant from the two lines.
It can be done by drawing a line perpendicular to both lines that bisects the angle between them, or by using vector and linear algebra to find the intersection point of the two lines and finding the midpoint of that point.
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What is the formula for square units?
The formula for the area of a square is A = s², where A is the area of the square and s is the length of one of its sides.
To calculate the area of a square, multiply the length of one side by itself. For example, if the side of a square is 5 meters, the area of the square would be 25 square meters (5² = 25). If the side of a square is 8 feet, the area of the square would be 64 square feet (8² = 64). The area of a square is always equal to the length of one side multiplied by itself.
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Algebra i need help asap
Answer:
Below
Step-by-step explanation:
Solutions are where the graphs intersect....they are equal at these points
(2,-3) and (4,-1)
There is likely one more point of intersection down and to the left which is not shown on the graph
if each quadrilateral below is a trapezoid, find the missing measures
If each of the quadrilateral is a trapezoid, then the measure of ∠Q = 89 and ∠S = 153.
What is a trapezoid?A polygon with only one set of parallel sides is called a trapezoid. The parallel bases of a trapezoid are another name for these parallel sides. Trapezoids have two additional sides that are not parallel and are referred to as their legs.
Given that in a trapezoid, the angle T is 91 and the angle R is 27.
For a trapezoid the measure of angle are as follows:
∠T + ∠Q = 180
∠R + ∠S = 180
Substituting the measures of ∠T and ∠R in the above equations we have:
91 + ∠Q = 180
∠Q = 89
27 + ∠S = 180
∠S = 153
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The nearest whole number.
12) Huong enlarged the size of a painting to a
width of 12 cm. What is the new height if
it was originally 4 cm wide and 3 cm tall?
The nearest whole number to which the new height of the image is enlarged is 9cm.
What is scaling?By scaling, we can create a drawing of an object that corresponds to the object's true size. In geometry, scaling refers to either growing or reducing figures in order to preserve their fundamental shape. Similar figures are what we call scaled figures.
Given that the original size was in the ratio 4: 3.
When the image is enlarged the enlarged image will be in the similar ratio to the original size.
Hence, the ratio in which the enlarged image will be increased is 4x : 3x.
The new width is given as 12,
The value of 4x = 12
x = 3
Then the new height of the image is:
3 (3) = 9cm.
Hence, the new height of the image is 9cm.
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Please help me find the value and the kind of angel this is?
Answer: the only line crossed is called a transversal line. x +3 would be vertically opposite to the other angle,and 2x-61 would also be vertically opposite. if you look at alternate and corresponding angles you would find the exact answer. this angle does not have a kind i guess.
hope it helps you!!!
Step-by-step explanation:
What is the formula for finding the area of a sector of a triangle?
The formula for a sector's area is: 1. A = (sector angle / 360 ) * (pi *r2)
2. A = (sector angle / (2*pi)) * (pi * r2) 3. A = (fraction of the circle) * (pi * r2)
The ratio used to find the area of the sector is that of how to know what fraction of the circle is the sector.
Ratio = (θ/360°)
θ is the angle subtended at the center of the circle by the arc, measured in degrees.
For this question,
θ = 60°
Ratio = (θ/360°) = (60°/360°) = (1/6)
Area of a circle is given as
Area of a circle = πr²
where
π = pi
r = radius of the circle = 8
Area of the circle = πr²
Area of the circle = π (8²) = 64π square units
Area of the sector is given as
Area of the sector = (Ratio) × (Area of the circle)
Ratio = (1/6)
Area of the circle = 64π square units
Area of the sector = (Ratio) × (Area of the circle)
Area of the sector = (1/6) × (64π) = (64π/6) = (32π/3) square units
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Is the triangle with sides 25cm 5 cm and 24 cm a right triangle give reasons for your answers?
No, a triangle with sides 25 cm, 5 cm and 24 cm does not make a right triangle as these lengths does not adhere to the Pythagorean theorem.
A right triangle has one angle that measures exactly 90 degrees. In a triangle, the longest side (the hypotenuse) is opposite the right angle.
The Pythagorean theorem states that in a right triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the hypotenuse.
In this case, the side lengths do not satisfy the Pythagorean theorem, as (24^2 + 5^2 = 601) ≠ (25^2 = 625)
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Which method of solving for the variable could be used instead of cross multiplication?.
Answer: multiplying by denominators
Step-by-step explanation: not sure if this is the right answer in this context but i hope it helps
technically cross multiplying basically is multiplying by denominators, but depending on the math level they are sometimes considered different methods
again, this may not be correct so if you want a more clear answer could you provide a bit of context? sorry, hope it helps anyway though.
You can work a total of no more than 40 hours each week at your two jobs. Housecleaning pays $4 hour and your sales job pays $8 per hour. You need to earn at least $120 each week to pay your bills. Write and graph a system of inequalities that shows the various numbers of hours you can work at each job.
Mrs. Peterson had lunch at Lunchee's. She paid $12.99 for her meal. She left a tip of 2.60. What percent did she leave?
answer = _____%
The percentage of the tip that she leaves will be 20.02%.
What is the percentage?The amount of any product is given as though it was a proportion of a hundred. The ratio can be expressed as a quarter of 100. The phrase % translates for one hundred percent. It is symbolized by the character '%'.
The percentage is given as,
Percentage (P) = [Final value - Initial value] / Initial value x 100
Mrs. Peterson had lunch at Lunchee's. She paid $12.99 for her meal. She left a tip of 2.60.
Then the percentage is given as,
P = [2.6 / 12.99] x 100
P = 0.2002 x 100
P = 20.02%
The percentage of the tip that she leaves will be 20.02%.
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Find the polynomial function in standard form that has the zeros listed. 2 (multiplicity 2), 3+i (multiplicity 1)
A. f(x) = x4 - 10x³ + 36x² - 56x +32
B. f(x) = x4 – 10x³ + 38x² - 64x + 40
C. f(x) = x³8x² +22x - 20
D. f(x) = x4 – 10x³ + 37x² − 60x + 36
The polynomial function in standard form that has the zeros - 2 (multiplicity 2), and 3+i (multiplicity 1) is option b: f(x) = x^4-10x^3+38x^2-64x+40.
What is a polynomial function?
In an equation such as the quadratic equation, cubic equation, etc., a polynomial function is a function that only uses non-negative integer powers or only positive integer exponents of a variable.
The zeros of the polynomial function are -
2 (multiplicity 2), and 3+i (multiplicity 1)
If the zero is 2, the factor is (x-2).
A multiplicity of 2 implies there are 2 factors -
(x-2) or (x-2)(x-2) = (x-2)^2
= (x^2-4x+4)
For the complex zero (3+i), there will also be a zero at the complex conjugate (3-i).
The factors are -
= [x-(3+i)] and [x-(3-i)] or
= (x-3-i)(x-3+i)
= x^2-3x+ix-3x+9-3i-ix-+3i-i^2
= x^2-6x+9-i
= x^2-6x+9-(-1)
= x^2-6x+10
Multiply all factors to find the polynomial in standard form -
f(x) = (x^2-4x+4)(x^2-6x+10)
= x^4-6x^3+10x^2-4x^3+24x^2-40x+4x^2-24x+40
= x^4-10x^3+38x^2-64x+40
Therefore, the polynomial function in standard form is x^4-10x^3+38x^2-64x+40.
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A ratio of 4 to 7 maybe written as the following, except 4:7 8:14 7/4 8/14 12:21
Answer:
except 7/4
Step-by-step explanation:
The ratio of 4 to 7 is the same as the fraction 4/7
7/4 is not equal to 4/7 and therefore they cannot be written in the same way.
Aira got 76, 85, 81 and 77 in her four quizzes. What average score must she get in her next two quizzes to obtain an average of 83?.
Aira should must score a total of 179 combined in her next two quizzes in order to make the average of all the marks of the quizzes as 83.
Aira has got 76, 85, 81 and 77 marks in her 4 quizzes.
I have to give two more quizzes and the marks obtained in those two quizzes should be such that the average of all the quizzes should be 83.
We know that we can find the average of the data by taking the sum of all the given values and dividing that particular value by the number of data given to us.
Here in this case let assume that she obtained X marks in her next to test combined. Now this way total number of observation are 6.
Now, we can simply apply the formula to find the average of the data.
Average = (76 + 85 + 81 + 77 + X)/6
83 = (76 + 85 + 81 + 77 + X)/6
X = 179
So, Aira should obtain 179 marks in her next to quizzes in order to make the average of all the score equal to 83.
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What is the factor theorem simple?
In mathematics, the factor theorem mostly used when factoring the polynomials totally. factorial theorem is that links factors and zeros of the polynomial.
By to factor theorem, if f(x) is a polynomial of degree n ≥ 1 and ‘a’ is any real number, then, (x-a) is a factor of f(x), if f(a)=0.
including, we can say, if (x-a) is a factor of polynomial f(x), then f(a) = 0. This also proves that the converse of the theorem. Let us see the proof and example.
Factor theorem is mostly used for factoring a polynomial and searching the roots of the polynomial. It is an important case of a polynomial remainder theorem.
As examined in the introduction, a polynomial f(x) has a factor (x-a), if and only if, f(a) = 0. It is one of the processes to do the factorization of a polynomial.
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Using the diagram below, find the measure of angle A.
A
101°
92°
B
68°
C
Answer: the missing angle is going to be 99 because the shape that is being formed it going to be a quaderdiral because that is the next number up.
Step-by-step explanation: add the degress together and then see what number is the closest
100 points
solve each system by graphing
x=-3
x-y=-2
Answer:
(-3, - 1)--------------------------
Graph each line:
x = - 3, this is a vertical line passing through x = - 3,x - y = - 2 is y = x + 2, which is the line y = x, translated 2 units up.The intersection of the lines is the solution:
(-3, - 1)What is the 3D distance formula?
The 3D distance formula is represented as √((x2 - x1)² + (y2 - y1)² + (z2 - z1)²)
The 3D distance formula, also known as the Euclidean distance formula, is used to calculate the distance between two distinct points in three-dimensional space. The formula is the square root of the sum of the squares of the differences in the x, y, and z coordinates of the two points. Mathematically it can be represented as:
Distance = √((x2 - x1)² + (y2 - y1)² + (z2 - z1)²)
Where (x1, y1, z1) and (x2, y2, z2) are the coordinates of the two points.
It's used to measure the distance between two points in 3-D space. It's the straight-line distance between two points. This formula can be applied in various fields such as physics, computer graphics, and robotics.
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What is final value theorem Mcq?
The steady state value of the system is determined using the final value theorem.
The main characteristics of the Laplace transform are the initial and final value theorems. Pierre Simon Marquis De Laplace, a French mathematician and scientist, presented these theorems. Limiting theorems refer to both the initial and final value theorems.
The final value theorem asserts that one can compute a system's ultimate value using. The function's Laplace transform is
[tex]\lim_{t\rightarrow\infty}f(t) = \lim_{s\rightarrow0}sF(s)[/tex]
(1) Final value theorem prerequisites are the functions f(t) and f(t)' should be Laplace transformable.
(2) sF(s), where F(s) is the Laplace transform of f(t), has no pole on the j-w axis and right half plane.
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Can someone help me with this please?
By using the triangle inequality theorem, the possible side lengths for the third side is equal to: D. 4 < c < 42.
What is the triangle inequality theorem?In Euclidean geometry, the Triangle Inequality Theorem states that the sum of any two sides of a triangle must be greater than or equal (≥) to the third side of the triangle.
Mathematically, the Triangle Inequality Theorem is represented by this mathematical expression:
b - c < a < b + c
Where:
a, b, and c represent the side lengths of this triangle.
By applying the Triangle Inequality Theorem, the possible side lengths of the triangle's third side is given by:
b - c < a < b + c
23 - 19 < a < 23 + 19
4 < a < 42.
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Triangle PQR was transformed to create the congruent triangle STU. Which describes how triangle PQR could have been transformed.
The term "congruent" refers to having exactly the same shape and size. Even if we flip, turn, or rotate the shapes, the shape and size should remain the same.
What exactly is SSS SAS RHS AAS?SSS (Side-Side-Side) (Side-Side-Side) SAS (Side-Angle-Side) (Side-Angle-Side) ASA (Angle-Side-Angle) (Angle-Side-Angle) AAS (Angle-Angle-Side) (Angle-Angle-Side) RHS (Right angle-Hypotenuse-Side) (Right angle-Hypotenuse-Side)
What is the distinction between SSS SAS and ASA postulates?The first two postulates, Side-Angle-Side (SAS) and Side-Side-Side (SSS), emphasise the side aspects, whereas the following lesson discusses two additional postulates that emphasise the angles. The Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS) postulates are what they are.
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the length of a rectangle is 6 feet longer than the width. the perimeter is 28 feet. find the dimensions of the regtangle
The length of rectangle :10 feet
The width of rectangle : 4 feet
Given
perimeter of the rectangle :28 feet
let width of the rectangle be :x
and length of the rectangle be : 6+x (as per given condition)
therefore
perimeter = 2(length +width)
28 = 2(6+x+x)
28 = 2(6+2x)
28 = 12+4x
28-12 = 4x
16 = 4x
x = 4
as width of rectangle is : 4
similarly length : 6+x
6+4
:10
The dimensions of the rectangle are : length =10
width = 4
verification :
perimeter =2(l+w)
28 = 2(10+4)
28 = 28
Write the coordinates of the vertices after a dilation with a scale factor of 1/3, centered at the origin.
The coordinates of the vertices after a dilation with a scale factor of 1/3 centered at the origin is E'(0, -2), F'(3, -2), G'(3, 1) and H'(0, 1).
What is Dilation?Dilation is a type of transformation where the figure is enlarged or made smaller such that it preserves the shape but not size.
Every dilated image are similar figures to the original figure.
Given that center of the dilation is origin.
Any coordinate (x, y) when dilated to a scale factor k is (kx, ky).
The coordinates of the quadrilateral are E(0, -6), F(9, -6), G(9, 3) and H(0, 3).
Dilation is with a scale factor of 1/3.
So all the coordinates become (1/3x, 1/3y).
E(0, -6) becomes E'(0, -2)
F(9, -6) becomes F'(3, -2)
G(9, 3) becomes G'(3, 1)
H(0, 3) becomes H'(0, 1)
Hence the coordinates are E'(0, -2), F'(3, -2), G'(3, 1) and H'(0, 1).
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i need help, i feel like i know how to do it but when i try it makes no sense. can someone explain this to me with steps?
Completing a square is a method for transforming a quadratic equation’s form so that its LHS becomes a perfect square, commonly known as vertex form. This is accomplished by rearranging the phrase.
What exactly is an X2?X2 is x multiplied by itself, which is symbolized by and may be expressed as xx or x(x) as an algebraic term. 2 is an exponent in. It denotes that x has been multiplied by itself twice. x 2 = x .
heres the answer is, X squared is a helpful notation that we use in math. It can be used in different kinds of algebraic expressions. It simplifies our calculations and allows us to develop formulas to solve problems quicker!
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What are the types of substitute?
Answer:
"Substitution comes in three flavors: nominal, verbal or clausal, depending on the item being substituted.
Step-by-step explanation:
Hope it helps!
9,365,000 rounded to the nearest million place
What can you conclude about the following figure?
What we can conclude by the given triangle about similarity is;
ΔPQR ∼ ΔPST
How to Identify Triangle similarity theorem?Similar triangles are defined as triangles that have the same shape, but their sizes may vary. Thus, if two triangles are similar, then it means that their corresponding angles are congruent and corresponding sides are in equal proportion.
Now, there are three types of similarity theorems for triangles namely;
Angle Angle(AA) Similarity
Side Angle Side(SAS) Similarity
Side Side Side(SSS) Similarity
We see that both triangles PQR and PST have equal angle R and angle T which are both 36 degrees. Thus, they are both similar by the AA Similarity theorem
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Jarrod helped his teachers pack textbooks into boxes for the summer. He packed 6 boxes of science textbooks with 14 books per box. He also packed math textbooks with 12 books per box. If x represents the total number of boxes he packed, which of the following equations can be used to find the total number of books that Jarrod packed into boxes?
The total number of books that Jarrod packed into boxes can be represented by 84 + x*12.
The total number of books that Jarrod packed into boxes can be found by multiplying the number of boxes he packed with the number of books per box.
In this case, Jarrod packed 6 boxes of science textbooks with 14 books per box, so the number of books in the science boxes is 614 = 84. He also packed math textbooks with 12 books per box, so the number of books in the math boxes can be represented by x*12, where x represents the total number of boxes he packed.
So the total number of books that Jarrod packed into boxes can be represented by 84 + x*12.
Therefore, the equation that can be used to find the total number of books that Jarrod packed into boxes is: 84 + x*12 = Total number of books.
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