The total number of possible four-digit PINs that cannot begin or end with a 0 and must be an even number is 10,000 - 900 - 4,500 = 4,500. Thus, the correct answer is c. 4,500.
To determine this:
There are 10 options for the first digit (1-9) and 10 options for the last digit (2, 4, 6, 8). For the middle two digits, there are 10 options for each digit (1-9). Therefore, the total number of possible four-digit PINs is 10 x 10 x 10 x 10 = 10,000.
However, we must subtract the number of PINs that begin or end with a 0, which is 9 x 10 x 10 x 1 = 900. We must also subtract the number of PINs that are odd numbers, which is 9 x 10 x 10 x 5 = 4,500.
Therefore, the total number of possible four-digit PINs that cannot begin or end with a 0 and must be an even number is 10,000 - 900 - 4,500 = 4,500. Thus, the correct answer is c. 4,500.
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Andre says he can find the length of the third side of triangle ABC and it is 13 units. Mai disagrees and thinks that the side length is unknown. Who do you agree with? Show or explain your reasoning
On soloing the provided question we can say that - Area of the Circle - Area of the Triangle - Area of the Square is the area of the darkened area. A = 210-123 = 87 sq unit
what is triangle?Three sides and three vertices make up a triangle, which is a polygon. It is among the fundamental forms in geometry. Triangle ABC is a graphical representation of a triangle having vertices A, B, and C. When the three points are not collinear, a unique plane and triangle in Euclidean geometry are found. A triangle is a polygon that has three sides and three corners. The spots where the three sides join end to end make up the triangle's corners. Three triangle angles added together equal 180 degrees.
the sum of the areas of the circle, the triangle, and the square.
Area of the Circle - Area of the Triangle - Area of the Square is the area of the darkened area.
A = 210-123 = 87 sq unit
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Graph the image of this figure after a dilation with a scale factor of 2 centered at the origin.
Use the polygon tool to graph the dilated figure.
Polygon
+Move
10
9
8
7
6
5
4
3
2
1
-10-9-8-7-6-5-4-3-2-10
Undo
y
2
10
4
Redo
5
6
7
x Reset
8
00
9 10
The polygon tool to graph the dilated figure will be drawn below.
How to draw the graph of the function?The collection includes all locations on the surface of the shape (x, f(x)) that make up a function of f's graph. We may alternatively say that the graph of f is the curve of y = f. (x). As a result, the diagram of an equation is a particular instance of the graphs of functions.
It is a polygon with four sides. The total interior angle is 360 degrees. A rectangle's opposite sides are parallel and equal, and each angle is 90 degrees. Its diagonals are all the same length and intersect in the center.
Graph the image of this figure after a dilation with a scale factor of 2 centered at the origin.
The polygon tool to graph the dilated figure will be drawn below.
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You have $4.50. Each piece of candy at the candy store costs $0.15. Write and solve an inequality that represents the number of gum balls you can buy.
Answer:
So in the first case we buy the candy which costs 1 and take candies worth 3 and 4 for free, also you buy candy worth 2 as well. So min cost = 1 + 2 = 3. In the second case we buy the candy which costs 4 and take candies worth 1 and 2 for free, also We buy candy worth 3 as well. So max cost = 3 + 4 = 7.
Step-by-step explanation:
To get to his parents' house, Frank would have to drive due north 1.4 miles. To get to his grandparents' house, he would have to drive due east 5.7 miles. What is the straight-line distance between the parents' and grandparents' houses? If necessary, round to the nearest tenths?
The straight-line distance between the parents' and grandparents' houses is given as follows:
5.9 miles.
How to obtain the distance?One of the distances given is due north, while the other is due east, meaning that the situation can be represented by a right triangle, in which the straight-line distance is the hypotenuse.
Hence the Pythagorean Theorem can be used to obtain the distance, which states that the length of the hypotenuse squared is equals to the sum of each of the sides of the triangle squared.
Thus the distance, in miles, is then calculated as follows:
d² = 1.4² + 5.7²
d = square root(1.4² + 5.7²)
d = 5.9 miles.
(rounding to the nearest tenth).
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In which decade (10-year interval) did the population grow the fastest? Explain how you know.
The population grow fastest in the 5th decade (1950) because that is the highest point in the graph
In which decade (10-year interval) did the population grow the fastest?A graph is a visual representation of a set of data or a mathematical function. It is a way to represent data or relationships between different variables in a clear and concise way.
There are several types of graphs, including line graphs, bar graphs, scatter plots, and pie charts. Each type of graph is used to represent different types of data and convey different types of information.
Examining the given graph:
The population grow fastest in the 5th decade (1950)
You will notice that the peak point (highest point) on the graph with a population of 45 thousand is in 1950 i.e. 5th decade (Check the attached image)
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How many elements does a quadrilateral have?
Every quadrilateral has 4 vertices, 4 angles, and 4 sides.
Properties of Quadrilaterals It has four sides: AB, BC, CD, and DA It has four vertices: Points A, B, C, and D It has four angles: ∠ABC, ∠BCD, ∠CDA, and ∠DAB ∠A and ∠B are adjacent angles ∠A and ∠C are the opposite angles AB and CD are the opposite sides AB and BC are the adjacent sidesA quadrilateral is a 4-sided plane figure. Below are some important properties of quadrilaterals :
Every quadrilateral has 4 vertices, 4 angles, and 4 sides The total of its interior angles = 360 degrees Square Properties All the sides of the square are of equal measure The sides are parallel to each other All the interior angles of a square are at 90 degrees (i.e., right angle) The diagonals of a square perpendicular bisect each other Rectangle Properties The opposite sides of a rectangle are of equal length The opposite sides are parallel to each other All the interior angles of a rectangle are 90 degrees. The diagonals of a rectangle bisect each other. Rhombus Properties All the four sides of a rhombus are of the same measure The opposite sides of the rhombus are parallel to each other The opposite angles are of the same measure The sum of any two adjacent angles of a rhombus is equal to 180 degrees The diagonals perpendicularly bisect each other Parallelogram Properties The opposite side of the parallelogram are of the same length The opposite sides are parallel to each other The diagonals of a parallelogram bisect each other The opposite angles are of equal measure The sum of two adjacent angles of a parallelogram is equal to 180 degrees Properties of Trapezium Only one pair of the opposite side of a trapezium is parallel to each other The two adjacent sides of a trapezium are supplementary (180 degrees) The diagonals of a trapezium bisect each other in the same ratio Properties of Kite The pair of adjacent sides of a kite are of the same length The largest diagonal of a kite bisect the smallest diagonal Only one pair of opposite angles are of the same measure.
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WILL GIVE BRAINLIEST!!
A line has a slope of 1/3 and passes through the point (–9, –3). What is its equation in slope-intercept form?
Answer:
y = [tex]\frac{1}{3}[/tex] x
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
here slope m = [tex]\frac{1}{3}[/tex] , then
y = [tex]\frac{1}{3}[/tex] x + c ← is the partial equation
to find c substitute (- 9, - 3) , that is x = - 9, y = - 3 into the partial equation
- 3 = [tex]\frac{1}{3}[/tex] (- 9) + c = - 3 + c ( add 3 to both sides )
0 = c
y = [tex]\frac{1}{3}[/tex] x + 0 , that is
y = [tex]\frac{1}{3}[/tex] x
Represent and connect Andre wrote the expression 15(1.5x-3) to represent the relationship shown in the table. Write two other expressions that represent the relationship shown in the table .
X ..................value of expression
0 ..............................-45
2 ............. ...................0
5 ................ ................. 67.5
8 ......................................135
The two expressions which are similar to the given expressions are 45 (0.5x - 1) and 22.5 ( x - 2 ) .
What is linear Equation ?
linear equation can be defined as the equation in which the highest degree is one.
Given expression,
15(1.5x-3)
expression 15(1.5x-3) can also written as 15 (15/10 -3)
that is
15 ( 3/2 - 3 )
15 * 3 (1/2 x -1 )
45 (0.5x - 1)
And
45 ( 0.5x -1 )
45 (x-2/2)
22.5 ( x - 2 )
Hence , The two expressions which are similar to the given expressions are 45 (0.5x - 1) and 22.5 ( x - 2 ) .
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if a wave has a wavelength of 25.4 cm and a frequency of 1.63 khz, its speed is closest to
Its speed is closest to 414m/s
given that
A waveform signal that is carried in space or down a wire has a wavelength, which is the separation between two identical places (adjacent crests) in the consecutive cycles. This length is typically defined in wireless systems in metres (m), centimetres (cm), or millimetres (mm) (mm). The wavelength is more frequently described in nanometers (nm), which are units of 10-9 m, or angstroms (), which are units of 10-10 m, for infrared (IR), visible light (UV), and gamma radiation ().
Frequency, which is defined as the quantity of wave cycles per second, and wavelength have an inverse relationship. The wavelength of the signal decreases with increasing signal frequency.
a wave has a wavelength = 25.4cm
a frequency = 1.63khz
now, we need to find speed of the wave
speed = wavelength × frequency
speed = 25.4 × 16.3
= 414m/s
speed = 414m/s
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Katie is 1.58 m tall. George is 9 cm shorter than Katie. How tall is George?
Give your answer in cm
WHAT IS IT
Answer:
George is 149 centimeters
Step-by-step explanation:
We need to change meters to centimeters
1.58 meters * 100 cm/meters
158 cm
158 cm - 9 cm = 149 cm
George is 149 centimeters
Are the following two functions inverses of each other? Use function composition to explain your answer.
To determine whether two functions are inverses of each other, we can use function composition to check if the composition of the two functions is the identity function. The identity function is a function that always returns its input.
What does the term "inverse function" mean?
A function that reverses the effects of another function is called an inverse function. When y=f(x) and x=g, a function g is the inverse of a function f. (y). Applying f and then g is equivalent to doing nothing, in other words. This can be expressed as g(f(x))=x in terms of the relationship between f and g.
The composition of two functions f and g is denoted by f(g(x)) or f o g.
If f(g(x)) = x and g(f(x)) = x for all x in the domain of the functions, then f and g are inverse functions.
In this case,
f(g(x)) = f((x-2)/5) = 5((x-2)/5) + 2 = x
so, f and g are inverse functions.
and also
g(f(x)) = g(5x+2) = (5x+2-2)/5 = x
so, f and g are inverse functions.
Therefore the functions f(x)= 5x+2 and g(x) = x-2/5 are inverses of each other.
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a shipment contains 10 televisions, of which two are defective. a sample of three televisions is selected at random. what is the probability that the sample contains no defective televisions?
The combination of samples is 80% likely to contain no defective televisions.
A combination appears to be a statistical technique for estimating the number of potential configurations in a collection of objects where the sequence of anything like the option is immaterial. Organize the elements in every order to choose. Permutations & combinations are commonly misunderstood.
Evaluate just non-faulty TVs to identify the same amount of ways in which the picked TVs are not defective. Because there are two faulty televisions out of 16, the amount of non-faulty sets is two.
= 10 - 2
= 8
Two non-defective televisions must be picked from a pool of twelve. There are several ways to accomplish this:
= ((10 × 9 × 8 × 7 ....) ÷ ((2 × 1) × (8 × 7 × 6...)) = 8!/2!((10- 2)!) = 80.
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The endpoints of the diameter are (3,1) and (9,5) what’s the equation
The equation of the line that passes through the two points (3,1) and (9,5) is y= (5-1)/(9-3) * (x-3) + 1.
What’s the equation ?The equation of a line that passes through two points (x1, y1) and (x2, y2) is given by:y - y1 = (y2 - y1)/(x2 - x1) * (x - x1)
In this case, given the two points (3,1) and (9,5), we have x1 = 3, y1 = 1, x2 = 9, and y2 = 5.Plugging these values into the equation above gives us:y - 1 = (5 - 1)/(9 - 3) * (x - 3)
Simplifying this equation gives us:y - 1 = 4/6 * (x - 3)
Which can be written as:y = 4/6 x + 5/3
This is the equation of the line passing through the two points (3,1) and (9,5).Since this is the equation of a diameter, it is also the equation of the circle that contains both points.To learn more about The equation of the diameter refer to:
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13 Jitesh made a square using 8 toothpicks of the same length as shown below. Arun made a bigger square. He also used some toothpicks of the same length. Which of these could be the number of toothpicks Arun used to make his square? A. 14 B. 16- D. 22
The number of toothpicks Arun could have used to make his bigger square than Jitesh's 8-toothpick square was B. 16.
How is the number of toothpicks determined?We are told that Jitesh made a square using 8 toothpicks.
The toothpicks have the same length as Arun's. We also know that Arun made a bigger square than Jitesh.
The same number of toothpicks must be on the four sides to make a square. Otherwise, it cannot form a perfect square.
Thus, it is unlikely that 14 toothpicks can make a square because 14 is not evenly divisible by 4, just like 22. However, 16 is evenly divisible by 4 and forms a perfect square of 4, giving each side an equal size.
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Find the area of the following trapezoid:
22cm, 12cm, 10cm, and 8cm
Answer:
Area of a Trapezoid = [tex]136cm^{2}[/tex]
Step-by-step explanation:
Area of a Trapezoid = [tex]\frac{1}{2} X(a+b)h[/tex]
Area of a Trapezoid = [tex]\frac{1}{2} X(12+22)8[/tex]
Area of a Trapezoid = [tex]\frac{1}{2} X34X8[/tex]
Area of a Trapezoid = [tex]136cm^{2}[/tex]
How do you find the third side of an inequality of a triangle?
To find the third side of an inequality of a triangle, you must first use the Triangle Inequality Theorem.
This theorem states that for any triangle, the sum of any two sides of the triangle must be greater than the third side. This means that in order to find the length of the third side, you must subtract the sum of the two known sides from the smaller of the two sides, then the length of the third side will be equal to the difference between these two numbers. For example, if two sides of a triangle have lengths of 4 and 3, the third side must be greater than 1 (4 + 3 = 7 and 4 - 3 = 1). Therefore, the length of the third side must be greater than 1.
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Mia's family wants to go on a tour to Brazil. A tour company charges $150 bus rental fee plus $35 per person for private group tours. If the total charge can be modeled by the function, where represents the amount of people, identify each of the following.
Create a function that models Sidney's earnings
Function that represents the expression is f (y) = $150 + $35Y
what is function?
In mathematics, function is an expression that creates a relationship between an independent variable and a dependent variable.
The set of values of independent variable is called input and the set of values for dependent variable is called output.
What is the function that can represent the above statement?
given, bus rental fee = $150
fee for private group tour per person = $35
let, the amount of people included in this tour = Y
total fee for private group tour for Y people= $35Y
Therefore, the total charge for the Brazil tour = bus rental fee + fee for private group
total charge = $150 + $35Y
if total charge for Brazil tour can be modelled by a function
then function f (y) = $150 + $35Y
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what is the image of the point (-5,-4) after the rotation of 180 degrees counter clockwise?
Please help me ASAP!!!
Answer:
(5, 4 )
Step-by-step explanation:
under a counter- clockwise about the origin of 180°
a point (x, y ) → (- x, - y ) , then
(- 5, - 4 ) → (- (- 5), - (- 4) ) → (5, 4 )
I don’t understand my homework
The Seed, where the range is 100, is the initial value of an algorithm used to produce a set of numbers.
Describe Range.Difference between the greatest and lowest value is referred to as the range. To get the range, find the variable's greatest observed value and deduct its lowest observed value. The data points in the center of the distribution are not taken into account; the range only considers these two values. Between the lowest and greatest values, there is a range of values. Values at each end of the range make up the range. Use the data set 4, 6, 10, 15, 18 as an example. Its range is 18-4 = 14, and its minimum and maximum values are 4, 4, and 4.
The term "seed" refers to the initial value of an algorithm used to produce a set of integers.
[tex]20, 25, 30, 45, 55, 90, 110,115,120[/tex]
Range = 120 - 20 = 100
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a local lighthouse is 170ft tall. During a visit to the lighthouse, Tommy who is 5ft tall, cast a shadow that is 1.8ft long. How long in ft is the lighthouse shadow
The length of the lighthouse shadow is: 61.2 ft
______________________________________
To find the length of the lighthouse shadow given similar triangles, solve a proportion with corresponding sides.
Let's follow these steps:
Draw a diagram to understand the question (see picture).Write the pairs of corresponding sides from similar triangles.Write and solve a proportion.Similar TrianglesIn the diagram, LH is the lighthouse. TY is Tommy. HE is the lighthouse shadow. YE is Tommy's shadow. Notice that the green triangle, ΔTYE, fits onto the larger triangle, ΔLHE.
Using AA similarity, we know that the two triangles in the diagram are similar. To say the triangles are similar, write: ΔLHE ~ ΔTYE
ProportionalitySimilar triangles have proportionate sides. This means that the length of corresponding sides will be longer or shorter by the same multiplication factor, which we can write as K.
Our pairs of corresponding sides are:
LH ~ TYHE ~ YELE ~ TECorresponding sides are related to each other by the multiplication factor, K:
LH ÷ TY = KHE ÷ YE = KLE ÷ TE = KThis makes any pair of corresponding sides written as a fraction equal to each other. Thus, we can write a proportion using these pairs.
Write a proportionWe should choose two pairs of corresponding sides to write a proportion from:
a pair that we know both lengthsa pair that includes 'HE' (the lighthouse shadow)[tex]\displaystyle { \frac{LH}{TY} = \frac{HE}{YE} }[/tex]
Note that this works because: [tex]\displaystyle { K = \frac{LH}{TY} = \frac{HE}{YE} }[/tex]
Solve for the missing side[tex]\displaystyle { \frac{LH}{TY} = \frac{HE}{YE} }[/tex] Write the proportion.
[tex]\displaystyle { \frac{170}{5} = \frac{HE}{1.8} }[/tex] Substitute known lengths.
[tex]\displaystyle { 34= \frac{HE}{1.8} }[/tex] Simplify. Now, isolate HE.
[tex]\displaystyle { 34*1.8 = \frac{HE}{1.8}*1.8}[/tex] Multiply both sides by 1.8.
[tex]\displaystyle { 34*1.8= HE}[/tex] 1.8 cancels out on the right side.
[tex]\displaystyle { HE = 34*1.8}[/tex] Keep the missing variable on the left side.
[tex]HE = 61.2[/tex] Solved for HE.
Remember to write the final answer with units.
Therefore, the lighthouse shadow is 61.2 ft long.
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What are angles for Grade 6?
In Grade 6, students typically learn about angles and their properties. Some of the key concepts that students may study include:
Understanding that an angle is formed by two rays with a common endpoint called the vertex.Identifying and labeling angles, using the vertex and the two rays that form the angle.Understanding that angles can be measured in degrees (°).Measuring angles with a protractor.Classifying angles as acute (less than 90°), right (exactly 90°), obtuse (between 90° and 180°), or straight (exactly 180°).Identifying and drawing complementary and supplementary angles.Understanding the relationship between adjacent angles formed by parallel lines and transversals.Using angle properties to solve problems and prove geometric theorems.These are some of the key concepts that students may learn about angles in Grade 6, but the specific curriculum can vary depending on the school district and state.
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3-
You are given a baseball card as a gift, and over the next 9 years, it
follows the polynomial function of f(x) = x³-13x² + 40x + 15 to
determine its value. Sketch a graph of the function over the range of
the 9 years.
a. What is the card's starting value?
b. When does it reach its maximum value, and what value is that?
c. When does it reach its minimum value, and what value is that?
d. How much is it worth at the end of the 9 year period?
The graph of the given polynomial function representing a baseball card is shown.
What is a polynomial function?A polynomial function is a function that applies only integer dominions or only positive integer powers of a value in an equation such as the monomial, binomial and trinomial, etc. ax+b is a polynomial.
here,
Graph of the polynomial function of f(x) = x³-13x² + 40x + 15, is shown,
From the graph,
a.
The card's starting value is 15.
b.
It reaches it maximum value when x is 2 and 9, the value at that point is 51.
c. It reaches it minimum value when x is 6.67, the value at that point is 0.
d.
As shown in the graph at the end worth of the card is 51.
Thus, the graph of the given polynomial function representing a baseball card is shown.
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If your total was $25.78 and the charge an initial fee of $5 and it’s 1.75 per mile how many miles did you travel
Answer: The answer is about 12 miles.
Step-by-step explanation:
The total was $25.78 so 25.78=
The initial value is 5 so +5
1.75 per mile so 1.75x (since thats the missing value)
equation= 25.78=1.75x+5
25.78-5=1.75x+5-5
25.78=1.75x
25.78/1.75=1.75/1.75x
12=x
You would travel about 12 miles.
how many times does 5 go into 2.5
Answer:
.5
Step-by-step explanation:
2.5/5 = .5
What does the AAS triangle congruence theorem tell you about triangles?
The AAS Triangle Congruence Theorem states that if two angles and the side between them of one triangle are equal to two angles and the side between them of another triangle, then the two triangles are congruent.
The AAS Triangle Congruence Theorem states that if two angles and the side between them of one triangle are equal to the two angles and the side between them of another triangle, then the two triangles are congruent. This theorem is an important tool in geometry that can be used to determine if two triangles are congruent and to find missing side lengths or angles. This theorem is based on the idea that if two sides and the included angle of one triangle are equal to the two sides and the included angle of another triangle, then the two triangles must be congruent. This is because the third side of the triangle, and the remaining angles must also be equal in order for the triangles to be congruent. Therefore, the AAS Triangle Congruence Theorem can be used to determine if two triangles are congruent, and to find missing side lengths or angles.
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The points (2, u) and (3, 10) fall on a line with a slope of 8. What is the value of u?
Please I have a unit test tomorrow I need help.
Answer:
u = 2
Step-by-step explanation:
The slope of a line between points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex] is equal to [tex]\frac{y_2-y_1}{x_2-x_1}[/tex], so, given our slope of 8, we need to find the first y-coordinate:
[tex]m=\frac{y_2-y_1}{x_2-x_1}\\\\8=\frac{10-u}{3-2}\\\\8=\frac{10-u}{1}\\\\8=10-u\\\\-2=-u\\\\2=u[/tex]
Hence, the value of "u" is 2.
Answer:
To find the value of u, you can use the slope formula:
slope = (y2 - y1) / (x2 - x1)
In this case, (x1, y1) = (2, u) and (x2, y2) = (3, 10). Plugging these values into the formula gives:
slope = (10 - u) / (3 - 2)
Simplifying this expression gives:
slope = (10 - u) / 1
Since the slope is 8, we can set this expression equal to 8 and solve for u:
8 = (10 - u) / 1
8 = 10 - u
u = 10 - 8
u = 2
Therefore, the value of u is 2.
Certainly! The slope of a line is a measure of how steep the line is. It is calculated by finding the ratio of the difference in the y-coordinates of two points on the line to the difference in the x-coordinates of those same two points. For example, in the line you gave, the two points are (2, u) and (3, 10). The difference in the x-coordinates of these two points is 3 - 2 = 1, and the difference in the y-coordinates is 10 - u. The slope of the line is then (10 - u) / 1. We know that the slope of this line is 8, so we can set the expression for the slope equal to 8 and solve for u. This gives us the value of u, which is 2. I hope this helps! Let me know if you have any questions.
29. After a reflection in the y-axis, the quadratic function listed below would have which of the
following formulas?
f (x) = 2x² - 8x +3
A. y=-2a² + 8x - 3
B. y=-2x²8x+3
C. y = 2x² + 8x +3
D. y=2x² +8x-3
Answer:
After a reflection in the y-axis, the quadratic function would have the following formula:
y = -(2x² - 8x + 3)
= -2x² + 8x - 3
Therefore, the answer is option D: y=2x² +8x-3.
Step-by-step explanation:
show that log ab×log ba =1
Answer:
See below for proof.
Step-by-step explanation:
Given:
[tex]\log_ab \times \log_ba=1[/tex]
To prove the given equation, begin by changing the base of the two logs so that they have the same base.
[tex]\boxed{\textsf{Change of base}: \quad \log_pr=\dfrac{\log_xr}{\log_xp}}[/tex]
Change the base of both logs to x by using the change of base formula:
[tex]\implies \log_ab=\dfrac{\log_xb}{\log_xa}[/tex]
[tex]\implies \log_ba=\dfrac{\log_xa}{\log_xb}[/tex]
Substitute into the equation:
[tex]\implies \log_ab \times \log_ba=\dfrac{\log_xb}{\log_xa} \times \dfrac{\log_xa}{\log_xb}[/tex]
[tex]\textsf{Apply the fraction rule} \quad \dfrac{a}{c}\times \dfrac{b}{d}=\dfrac{a \times b}{c \times d}:[/tex]
[tex]\implies \dfrac{\log_xb \times \log_xa}{\log_xa \times \log_xb}[/tex]
Apply the commutative property of multiplication to the numerator:
[tex]\implies \dfrac{\log_xa \times \log_xb}{\log_xa \times \log_xb}[/tex]
Cancel the common factor logₓb:
[tex]\implies \dfrac{\log_xa}{\log_xa}[/tex]
[tex]\textsf{Apply the fraction rule} \quad \dfrac{n}{n}=1:[/tex]
[tex]\implies \dfrac{\log_xa}{\log_xa}=1[/tex]
Hence proving that:
[tex]\log_ab \times \log_ba=1[/tex]In one calculation:
[tex]\begin{aligned}\implies \log_ab \times \log_ba & =\dfrac{\log_xb}{\log_xa} \times \dfrac{\log_xa}{\log_xb}\\\\&=\dfrac{\log_xb \times \log_xa}{\log_xa \times \log_xb}\\\\&=\dfrac{\log_xa \times \log_xb}{\log_xa \times \log_xb}\\\\&=\dfrac{\log_xa}{\log_xa}\\\\&=1\end{aligned}[/tex]
if 8x ≤ g(x) ≤ 4x4 − 4x2 + 8 for all x, evaluate lim x→1 g(x).
If 8x ≤ g(x) ≤ 4x4 − 4x2 + 8 for all x, x will be evaluated as lim x→1 g(x), 1 is 8.
To evaluate the limit of g(x) as x approaches 1, we must first examine the given inequality. It states that if 8x is less than or equal to g(x), and g(x) is less than or equal to 4x4 - 4x2 + 8, then this is true for all x.
Calculating the limit of g(x) as x approaches 1, we start by substituting x = 1 into the given inequality. This gives us 8(1) ≤ g(1) ≤ 4(1)4 - 4(1)2 + 8. Simplifying, this gives us 8 ≤ g(1) ≤ 8, so g(1) must be equal to 8.
Therefore, the limit of g(x) as x approaches 1 is 8.
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100 POINTS Can someone factor these trinomials and show their work?
The polynomials in factor form are described below:
(x - 5) · (x + 4) (x - 8) · (x + 1) (a + 6) · (a - 4) (x + 10) · (x + 4) (n - 5) · (n - 9) (x + 16) · (x - 2) (x - 7) · (x + 4) (c + 20) · (c - 5) (x + 10) · (x - 5) (x + 3) · (x + 4) (x + 7) · (x - 4) (x - 9) · (x + 6) (x - 24) · (x - 2) (x + 12.761) · (x - 3.761) (x - 10) · (x + 1) How to factor a trinomial by algebra propertiesIn this problem we find fifteen cases of quadratic equations in standard form, that is, polynomial with grade 2 of the form:
y = a · x² + b · x + c = 0
Where a, b, c are real coefficients of the polynomial. Roots can be found by means of the following case for quadratic equations:
(x - r₁) · (x - r₂) = 0
x² - (r₁ + r₂) · x + r₁ · r₂ = 0
Where r₁ and r₂ are roots of the polynomial.
Then, the factor form of each case is described below:
Case 1
x² - x - 20
x² - (5 - 4) · x + 5 · (- 4)
(x - 5) · (x + 4)
Case 2
x² - 7 · x - 8
x² - (8 - 1) · x + 8 · (- 1)
(x - 8) · (x + 1)
Case 3
a² + 2 · a - 24
a² - (- 6 + 4) · a - 6 · 4
(a + 6) · (a - 4)
Case 4
x² + 14 · x + 40
x² - (- 10 - 4) · x + (- 10) · (- 4)
(x + 10) · (x + 4)
Case 5
n² - 14 · n + 45
n² - (5 + 9) · n + 5 ·9
(n - 5) · (n - 9)
Case 6
x² + 14 · x - 32
x² - (- 16 + 2) · x + (- 16) · 2
(x + 16) · (x - 2)
Case 7
x² - 3 · x - 28
x² - (7 - 4) · x + 7 · (- 4)
(x - 7) · (x + 4)
Case 8
c² + 15 · c - 100
c² - (- 20 + 5) · c - (- 20) · 5
(c + 20) · (c - 5)
Case 9
x² + 5 · x - 50
x² - (- 10 + 5) · x + (- 10) · 5
(x + 10) · (x - 5)
Case 10
x² + 7 · x + 12
x² - (- 3 - 4) · x + (- 3) · (- 4)
(x + 3) · (x + 4)
Case 11
x² + 3 · x - 28
x² - (- 7 + 4) · x + (- 7) · 4
(x + 7) · (x - 4)
Case 12
x² - 3 · x - 54
x² - (9 - 6) · x + 9 · (- 6)
(x - 9) · (x + 6)
Case 13
x² - 26 · x + 48
x² - (24 + 2) · x + 24 · 2
(x - 24) · (x - 2)
Case 14
x² - 9 · x - 48
x² - (-12.761 + 3.761) · x + (- 12.761 · 3.761)
(x + 12.761) · (x - 3.761)
Case 15
x² - 9 · x - 10
x² - (10 - 1) · x + 10 · (- 1)
(x - 10) · (x + 1)
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