To prove that if x ≠ 3, then x^2 - 2x + 3 ≠ 0, we can use a proof by contradiction. We assume that x ≠ 3 and x^2 - 2x + 3 = 0, and then show that this leads to a contradiction.
Assume that x ≠ 3 and x^2 - 2x + 3 = 0. We can rewrite the equation as x^2 - 2x = -3.
Now, let's factor the left side of the equation: x(x - 2) = -3.
Since x(x - 2) is the product of two factors, at least one of them must be non-zero. If x = 0, then x(x - 2) = 0, which contradicts the assumption that x^2 - 2x + 3 = 0. Therefore, we can conclude that x - 2 ≠ 0.
Dividing both sides of the equation x(x - 2) = -3 by (x - 2), we get x = -3/(x - 2).
Now, if x - 2 = 0, then the right side of the equation becomes undefined, which contradicts the assumption that x ≠ 3. Therefore, we can conclude that x - 2 ≠ 0.
Since both x - 2 ≠ 0 and x ≠ 3, we can cancel out (x - 2) from both sides of the equation, yielding x = -3/(x - 2).
However, this equation implies that x can be equal to 3, which contradicts the initial assumption that x ≠ 3. Therefore, our assumption that x ≠ 3 and x^2 - 2x + 3 = 0 leads to a contradiction.
Hence, we can conclude that if x ≠ 3, then x^2 - 2x + 3 ≠ 0.
Regarding the second part of the question, if we take x ∈ C (the set of complex numbers), the result may not hold true. This is because in the complex number system, there exist values of x for which x^2 - 2x + 3 = 0, even if x ≠ 3. In the complex number system, the equation may have complex roots that satisfy the equation. Therefore, the result stated above is valid only when x is restricted to the real number system.
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Do sides 5 cm 12cm and 13cm form a right angled triangle give reason?
Yes, the mentioned sides of triangle will be right angles triangle because they follow Pythagoras theorem.
A right angled triangle is characterized by presence of hypotenuse, perpendicular and base. The sides of right angled triangle are related through Pythagoras theorem, which is -
Perpendicular² = Base² + Hypotenuse²
As we know that hypotenuse is the longest side of triangle, hence, will be the 13 cm.
So, keeping the values in formula -
13² = 5² + 12²
Taking square on both the sides of equation
169 = 25 + 144
Performing addition on Right Hand Side of the equation
169 = 169
Since, LHS is equal to RHS, hence, it is right angled triangle.
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Xin has $126 to purchase balloons for her friend's birthday. Each balloon costs $6. The function p= 126-6m represents the amount of
money Xin has after purchasing n balloons. What is the zero of the function?
So, on solving the question we can say that equation p= 126-6m ; p = 2 = -124 = -6m => m = 20.6666666667
What is equation?An equation is a mathematical formula that connects two assertions using the equal sign (=) to denote equivalence. In algebra, an equation is a mathematical statement that establishes the equivalence of two mathematical expressions. For instance, an equal sign separates the components 3x + 5 and 14 in the equation 3x + 5 = 14. A mathematical formula is used to explain the connection between two sentences on either side of a letter. Frequently, there is just one variable, which is also the symbol. for example, 2x - 4 = 2.
the equation is p= 126-6m
p = 2
-124 = -6m
m = 20.6666666667
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Rhind Rhind Papyrus Problem 27
A quantity ,and its ⅕ added together becomes 21. What is the quantity
which means : x + x ⅕ = 21
Answer:x=35/2 or 17.5
Step-by-step explanation:
x + .2x = 21
1.2x = 21
1.2x/1.2 = 21/1.2
x = 17.5
answer a i just did it
Step-by-step explanation:
The upfront cost of a Porsche is 64K. At the end of 12 Years, the car is worth only $4000, Write a linear equation to represent the value of the car in terms of the year.
The linear equation to represent the value of the car in terms of the year will be c = 64000 - 5000y
How to illustrate the equation?An equation simply has to do with the statement that illustrates the variables given. In this case, it is vital to note that two or more components are considered in order to be able to describe the scenario.
In this situation, the upfront cost of a Porsche is 64K and at the end of 12 Years, the car is worth only $4000.
The difference in amount will be:
= $64000 - $4000
= $60000
Number of years = 12
Amount in depreciation per year = $60000 / 12
= $5000 per year
The linear equation to represent the value of the car in terms of the year will be:
c = 64000 - 5000y
where c = value
y = number of years
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What is called a complex number?
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, which satisfies the equation i2 = -1.
What is complex number?
The number a is the real part of the complex number and b is the imaginary part. A complex number is said to be a sum of a real number and an imaginary number. Complex numbers are used in many branches of mathematics, including calculus, linear algebra, quantum mechanics, and electrical engineering.
They are also used in engineering, physics, chemistry, and other scientific fields. Complex numbers are useful in many real-world applications, such as finding the roots of a polynomial equation or solving systems of linear equations. Complex numbers can also be used to represent points in the plane and can be used to model phenomena in the physical world, such as electrical circuits and waveforms.
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Every month, the library holds a book sale. All books are $1.50. Choose the statement below that is true. A The price of a book depends on what kind of books you buy. B The kind of book you buy depends on the price. C The total cost of your books depends on how many books you buy. D The number of books you buy depends on the total cost.
The only true statement among the options given is C, The total cost of your books depends on how many books you buy.
What is the direct proportion?
Direct proportion, also known as direct variation, is a relationship between two variables in which the ratio of the variables is always constant. This means that if one variable increases, the other variable will also increase, and if one variable decreases, the other variable will also decrease.
According to the information given, all books are $1.50 each, so the price does not depend on what kind of book you buy. The statement A is false.
Also, the kind of book you buy does not depend on the price, so statement B is false.
The number of books you buy does depend on the total cost, so statement C is true.
Therefore, the only true statement among the options given is C, The total cost of your books depends on how many books you buy.
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At a local pizza shop customers pay a set price for a large pizza plus an additional per topping ordered. A large pizza with 2 toppings would cost $13. 50, and a large pizza with 5 toppings would cost $15. 75
The cost price of large pizza is $12 and that of toppings is 0.75.
An equation is a mathematical statement with an 'equal to' symbol between two expressions that have equal values. For example, 3x + 5 = 15. There are different types of equations like linear, quadratic, cubic, etc.
In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal. For instance, 3x + 5 = 14 is an equation, in which 3x + 5 and 14 are two expressions separated by an 'equal' sign.
Let the cost of large pizza be x and that of toppings be y.
x + 2y = 13.50
x = 13.50 - 2y - (1)
x + 5y = 15.75 - (2)
substitute equation (1) in equation (2),
13.50 - 2y + 5y = 15.75
3y = 15.75 - 13.50
3y = 2.25
y = 2.25 / 3
y = 0.75
now, put the value of y = 0.75 in equation (1),
x = 13.50 - 2 × 0.75
x = 13.50 - 1.5
x = 12
Therefore, the cost price of large pizza is $12 and that of toppings is 0.75.
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]What is the first step in solving the quadratic equation -5x2+8=133?
taking the square root of both sides of the equation
subtracting 8 from both sides of the equation
squaring both sides of the equation
adding 8 to both sides of the equation
The first step in solving the quadratic equation is subtracting 8 from both sides.
What is an equation?An equation is a mathematical statement that is made up of two expressions connected by an equal sign.
Example:
2x + 4 = 9 is an equation.
We have,
5x² + 8 = 133
Subtract 8 into both sides.
5x² = 133 - 8
5x² = 125
Divide both sides by 5.
x² = 125/5
x² = 25
x = √25
x = 5
Thus,
The root of the equation is 5.
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Jean’s highest bowling score last season was 237 out of 300. What percent is Jean’s high score?
Answer:
79%
Step-by-step explanation:
imma human calculator and no one can stop me
A car traveled a distance of 192 miles from New York go a small town in Pennsylvania in 3 hours. Find the speed of the car.
Answer: 64 miles per hour
Step-by-step explanation: 192 divided by 3= 64
Create math problems with the words : spent, difference, decreased, and dropped (temp) for subtraction
The math word problems with the given words is A has 52 marbles, B has 105 marbles, find the difference between the number of marbles between A and B.
What are word problems?A math word problem is a question that is written as one or more sentences and asks students to use their mathematical understanding to solve an issue from "real world."
1. A has 52 marbles, B has 105 marbles, find the difference between the number of marbles between A and B.
2. The temperature of city A is 35 degrees, on the second day the temperature decreased to 15 degrees. How much did the dropped (temp)?
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Can somebody help ASAP? I will give lots of points
Three-part question:
Answer:
A- NQ,
B- 24,
C- 14+10=24
In January, Jamie pays $2.00 for a tube of toothpaste. When she buys the same brand and size of toothpaste in February, she pays $2.25. What is the percent of increase in the price of the toothpaste from January to February? Round to the nearest tenth of a percent.
Answer:
12.5%
Step-by-step explanation:
0.125 * 2 = .25
2+.25 =2.25
Evaluate the expression 9x for x = 2
The value of expression 9x for x= 2 is 18.
what are expressions?
Expressions in math are mathematical statements that have a minimum of two terms containing numbers or variables, or both, connected by an operator in between. The mathematical operators can be of addition, subtraction, multiplication, or division. For example, x + y is an expression, where x and y are terms having an addition operator in between. In math, there are two types of expressions, numerical expressions - that contain only numbers; and algebraic expressions- that contain both numbers and variables.
e.g. A number is 6 more than half the other number, and the other number is x. This statement is written as x/2 + 6 in a mathematical expression. Mathematical expressions are used to solve complicated puzzles.
Now,
Given expression = 9x
For x=2
Value of expression=9*2
=18
Hence,
After evaluating 9x for x=2 value will be 18.
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Write the letter of each answer
How do you graph 2 3 on a number line?
We have plotted the point (2/3) on a number line and attached and explained it below.
A number line is one of the methods of visually representing any point that is a real number on a graduated straight line. This is made by marking graduations on a simple normal line and then marking whatever point we need on it.
The point (2/3) has been given in the problem which we need o plot on the number line. Since this point is a fraction, we need to divide/solve it first and then see where it will fit. On dividing the fraction, we find that the point is 0.66. Now, we find the graduations 0 and 1 since it lies between them. The image for this has been attached below.
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Multiple Choice: Triangle ABC has been reduced to create triangle XYZ. By what ratio has it been reduced?
A
12
3/4
2/3
1/2
1.5
18
15
B
80
Y
XL
12
10
Answer:
2/3
Step-by-step explanation:
Choose any of the similar sides and set up the ratio, then reduce to the simplest fraction:
12/18 = 4/6 = 2/3
8/12 = 4/6 = 2/3
10/5 = 2/3
PLEASE HELP!!! I NEED HELP WITH MY HOMEWORK!!!!
(GEOMETRY)
WILL MARK BRAINLIEST!!!!
The right triangle in marked question number 3 have the value of u = 2 and v = √2 using the trigonometric ratio of sine and cosine of the angle 45°.
What are trigonometric ratiosThe trigonometric ratios involves the relationship of an angle of a right-angled triangle to ratios of two side lengths. Basic trigonometric ratios includes; sine cosine and tangent.
Considering the angle 45° for the right triangle, recall that sin 45° and cos 45° are generally both equal to √2/2
sin 45° = √2/u {opposite/hypotenuse}
√2/2 = √2/u
u = √2/√2 × 2 {cross multiplication}
u = 2
cos 45° = v/u {adjacent/hypotenuse}
√2/2 = v/2
v = 2/2 × √2
v = √2
Therefore, the value of u = 2 and that of n = 10, using the trigonometric ratio of sine and cosine of the angle 45° for the right triangle.
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What is dilated by a factor of 3?
Factor of scale (Dilation) Factor of scale (Dilation) How much larger or smaller the image will be depends on the scale factor in the dilation of a mathematical entity (compared to the original object).
What does it mean to be "dilated by a factor"?Factor of scale (Dilation) Factor of scale (Dilation) How much larger or smaller the image will be depends on the scale factor in the dilation of a mathematical entity (compared to the original object). An expansion takes place when the scale factor's absolute value is higher than one.The most important thing to remember is that the dilation value influences the distance between two places.A is initially one unit below P and two units to P's left, just like in the first example (dilation by a factor of three). Since 1 + 3 = 3, the dilated point, A', should be three units below P. Since 2*3 = 6, A' ought to be six units to the left of P.To learn more about dilation refer to:
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How to find gradient
The difference between the y coordinates and the difference between the x coordinates then divide them
What is a gradient in math?Gradient is a differential operator used in mathematics to transform a three-dimensional vector-valued function into a vector whose three elements are the partial derivatives of the function with respect to its three variables. The gradient sign is ∇
As a result, f = ifx + jfy + kfz, where fx, fy, and fz are the first partial derivatives of f and I j, and k are the unit vectors of the vector space, is the gradient of a function, also known as grad f or f. f is the direction of the heat-flow vector in a temperature field, where f is a temperature field in physics that indicates the temperature at every point in a space.
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Find the length of side
x to the nearest tenth.
Answer:
Step-by-step explanation:
Isosceles triangle (Same angle) means both sides are the same length, triangle is also right meaning you can plug values into Pythagorean theorem ([tex]a^2 + b^2 = c^2[/tex])
[tex]x^2+x^2 = 1^2\\2x^2=1\\x = \sqrt{\frac{1}{2} }\\x = 0.707...\\x=0.7[/tex]
Assuming all my math is right, this is your answer
Latisha is compiling a program for a video game.
For one part of the program they use the rule (x,y) → (x + 10, y - 6) to move a character on the screen.
(a) What output does the rule give when the input is (10,-5)? Show your work.
(b) What output does the rule give when the input is (-4,8)? Show your work.
The shifted coordinates for each of the following are (20,-11) and (6,2).
Coordinates are a pair of integers (Cartesian coordinates), or sporadically a letter and a number, that identify a certain place on a grid, also referred to as a coordinate plane. The x axis (horizontal) and y axis are the two axes that make up a coordinate plane (vertical).
Given, (x,y) → (x + 10, y - 6)
a) when the input is (10,-5) -
10 + 10 = 20
- 5 - 6 = - 11
(20,-11)
b) when the input is (-4,8) -
-4 + 10 = 6
8 - 6 = 2
(6,2)
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Anya is 3 years older than her brother,Cole. In 11 years,Cole will be twice Anya’s current age. Find their current age
By solving these equations, Cole is currently 27 years old, and Anya is 30 years old.
How does math work out age?A person's date of birth is compared to the date on which their age needs to be calculated as part of the age calculation process. The person's age is determined by deducting the person's birthdate from the given date. Age equals Given Date - Birthdate
Let x stand in for Anya's current age and y for Cole's present age.
Cole's older brother, Anya, is three years older. Hence:
x = y + 3
x - y = 3 (1) (1)
The total of their ages will be 79 in 11 years. Hence:
(x + 11) + (y + 11) = 79
x + y = 57 (2) (2)
Equations 1 and 2 are simultaneously solved to produce:
x = 30, y = 27
Cole is currently 27 years old, and Anya is 30 years old.
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ABDE is a cyclic quadrilateral.
ABC and EDC are straight lines.
Angle DBC = 60°.........................................................
The measurement of angle BCD is 30°
What is a cyclic quadrilateral?A cyclic quadrilateral is a four sided shape that can be inscribed into a circle. Each vertex of the quadrilateral lies on the circumference of the circle and is connected by four chords. The opposite angles of a cyclic quadrilateral have a total of 180°.
Given that, ABDE is a cyclic quadrilateral. ABC and EDC are straight lines. Angle DBC = 60°
According to the figure, BD║ AE
Therefore,
∠ DBC ≅ ∠ EAB (definition of corresponding angles)
∠ EAB = 60°
Since, ∠ EAB : ∠ BCD = 2:1
Therefore, ∠ BCD = 60 × 1/2 = 30°
Hence, the measurement of angle BCD is 30°
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Daniel and Elizabeth had to choose between making a 600-mile round trip by car or plane. Round-trip airfare was $260 for each of them and would take 2 hours. Round-trip cab fare to the airport would add $15. Driving their car would take 2 days each way and would add the cost of gas, food, and lodging. Elizabeth estimated that the whole trip would need $60 for gas, their combined food bill of $25 per day (4 days) , and that a room to share would be $48 for each night of the 4 total nights they would stay in a motel. Find the average cost per passenger-mile of driving.
The average cost per passenger - mile of driving is $0.503 per mile.
What is Average Cost?Average cost is calculated by dividing the total cost by the total output quantity.
There are two options for Daniel and Elizabeth, either by driving car or by plane.
We have to calculate the average cost per passenger - mile of driving in car.
Cost of gas = $60
Food bill for a day for two of them = $25
Food bill for a day for one = 25 / 2 = $12.5
Food bill for one passenger for 4 days = $12.5 × 4 = $50
Cost of room for 1 night = $48
Cost of room for 4 nights = $192
Total cost for 1 passenger = $60 + $50 + $192 = $302
Average cost per passenger - mile of driving = 302 / 600 = $0.503 per mile.
Hence the average cost per mile per passenger is $0.503 per mile.
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How do I dilate a shape on a grid using a scale factor of 1 2?
Every coordinate point of the original triangle has its scale factor doubled by one-half if the scale factor is one. multiply by half after dilation with a scale factor of two.
How do you dilate by scale factor?We locate the dilation's center point and measure the distances between it and other points on the preimage and the image in order to get the scale factor for the dilation. As correctly stated by Math Bits Notebook, the scale factor is determined by the ratio of these distances.Keep in mind that each coordinate must be multiplied by the scale factor in order to enlarge an object in the coordinate plane. The mapping will be (x, y) (1.5 x, 1.5 y) with this dilation. The original is represented by the blue quadrilateral in the graph above, while the dilatation is shown by the red picture. Connect the image's vertices to the centre of dilation by drawing straight lines.To learn more about multiply refer to:
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Every coordinate point of the original triangle has its scale factor doubled by one-half if the scale factor is one. multiply by half after dilation with a scale factor of two.
How do you dilate by scale factor?We locate the dilation's center point and measure the distances between it and other points on the preimage and the image in order to get the scale factor for the dilation. As correctly stated by Math Bits Notebook, the scale factor is determined by the ratio of these distances.
Keep in mind that each coordinate must be multiplied by the scale factor in order to enlarge an object in the coordinate plane. The mapping will be (x, y) (1.5 x, 1.5 y) with this dilation. The original is represented by the blue quadrilateral in the graph above, while the dilatation is shown by the red picture. Connect the image's vertices to the center of dilation by drawing straight lines.
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HELP PLS WHATS THE VALUE
Applying the triangle sum theorem, the value of x in the triangle is: C. 36.
What is the Triangle Sum Theorem?The Triangle Sum Theorem states that the sum of the measures of the interior angles of a triangle is 180 degrees. This theorem applies to all triangles, regardless of their shape or size.
The theorem can be proven using the fact that the sum of the measures of the interior angles of any polygon with n sides is (n-2)x180. Since a triangle has three sides, the sum of its interior angles is 180 degrees.
Therefore:
2x + 2x + x = 180
Combine like terms
5x = 180
Divide both sides by 5
5x/5 = 180/5
x = 36
The value of x is: C. 36.
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ken added 5/8 cups more walnuts to the trail mix how many cups of Tremors does he have
Answer:
Please provide more information. Since you don't provide the original amount of walnuts used, we can't solve the problem.
Pls the area of the figure.
Answer:
36
Step-by-step explanation:
4 x 4 = 16
2 x 10 = 20
You would have to separate the square and the rectangle.
The way you do this is by cut of the number and subtracting them, like 6 and 2 to get the square.
Does 5 5 and 7 make a right triangle?
The base of the triangle 7 can be divided in half by a line of symmetry of the isosceles triangle, which will bisect the vertex angle. This creates two right triangles: Each with a base of 3.5 and a hypotenuse of 5.
What is meant by isosceles triangle?
A triangle with two equal sides is known as an isosceles triangle in geometry. Both having exactly two sides of equal length and having at least two sides of equal length are acceptable specifications, with the latter version containing the equilateral triangle as an exception.The faces of bipyramids, the golden triangle, the isosceles right triangle, and some Catalan solids are all examples of isosceles triangles.Isosceles triangles have been studied mathematically since the Babylonian and Egyptian periods.Isosceles triangles have been utilized as ornamentation since the dawn of time and are widely seen in architecture and design, such as in building pediments and gables.To learn more about isosceles triangle refer to
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The base of triangle 7 can be bisected by the line of symmetry of the isosceles triangle that bisects the apex angle. This creates two right triangles.
Each has a base of 3.5 and a hypotenuse of 5.
What does isosceles triangle mean?A triangle with two equal sides is called an isosceles triangle in geometry.
Both exactly 2 equal sides and at least 2 equal sides are permissible specifications, with the exception of equilateral triangles in the latter version. The bipyramid, the golden triangle, the isosceles right triangle, and the face of the Catalan solid are examples of isosceles triangles.
Isosceles triangles have been studied mathematically since Babylonian and Egyptian times.
Used as an ornament since the dawn of time, isosceles triangles are widely used in architecture and design, including in the construction of pediments and pediments.
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