When can extraneous solutions arise?

Answers

Answer 1

Extraneous solutions can arise when solving equations that involve operations such as square roots, logarithms, or inverse trigonometric functions. These operations have a restricted domain and when solving an equation using these operations, the solution may be valid within the domain of the operation but not within the domain of the original equation.

For example, when solving the equation √(x+4) = x, the solution is x= -4, but when we check it, we find that x+4 is negative and that the square root of a negative number is not defined, therefore x = -4 is an extraneous solution.

Extraneous solutions also appear when solving rational equations, it happens when solutions found make the denominator of the fraction equal to zero, which is not allowed.

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Related Questions

Which of the following is the solution set of the inequality x 4 5x 2/3 7x 3 5?

Answers

The solution set of the inequality x 4 5x 2/3 7x 3 5 is x> 7.

To solve this inequality, we need to isolate the x term on one side and all the other terms on the other side. We can do this by subtracting 4 from both sides of the equation, which gives us 5x 2/3 7x 3 - 4 < 0.

Next, we can add 7x 3 to both sides of the equation, giving us 5x 2/3 - 4 + 7x 3 < 7x 3.

Now, since the denominator of the first term is 3, we can multiply both sides of the equation by 3/3. This gives us 5x 2 - 4 3 + 21x 3 < 21x 3.

Next, we can subtract 21x 3 from both sides of the equation, which gives us 5x 2 - 4 3 < 0.

Now, we can divide both sides of the equation by 5, which gives us x 2 - 4/5 < 0.

Finally, we can take the square root of both sides of the equation, which gives us x - 2/√5 < 0.

This means that x > 2/√5 + 2. Since 2/√5 + 2 > 7, the solution of the inequality is x > 7.

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6. 0.3k - 4m
k = 20 and m = -2

Please help me and show the work please giving 20 points need in 20mins

Answers

The equation evaluates to -2 when k = 20 and m = -2

What is a linear equation in 2 variables?

A linear equation in two variables is an equation that can be written in the form of

ax + by = c

where a, b, and c are constants and x and y are the variables. This equation represents a straight line in a two-dimensional coordinate system where x and y are the coordinates. The values of x and y that satisfy the equation form the set of solutions of the equation.

0.3k - 4m = 0.3(20) - 4(-2) = 6 - 8 = -2

Explanation:

First substitute the given values of k and m into the equation: 0.3k - 4m = 0.3(20) - 4(-2)

Next, simplify the multiplication: 0.3(20) = 6

Now we can simplify the subtraction of a negative number: 4(-2) = 8

So we have: 6 - 8 = -2

As you can see, the equation evaluates to -2 when k = 20 and m = -2

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how to find the domain and range of a piecewise function

Answers

To find the domain and range of a piecewise function, you will need to consider each piece separately and then combine the results. First Identify the domain of each piece, then Identify the range of each piece and Combine the domains and ranges

Identify the domain of each piece: The domain of a piecewise function is the set of all input values (x-values) for which the function is defined. For example, if a piece of the function is defined for x > 2, then the domain for that piece would be all real numbers greater than 2.Identify the range of each piece: The range of a piecewise function is the set of all output values (y-values) that the function can produce. For example, if a piece of the function is defined as y = x^2, then the range for that piece would be all non-negative real numbers.Combine the domains and ranges: Once you have identified the domains and ranges for each piece, you can combine them to find the overall domain and range of the piecewise function. The overall domain is the set of all input values for which any piece of the function is defined, and the overall range is the set of all output values that any piece of the function can produce.

Note: Sometimes you may find that domain or range of a certain piece is not mentioned in the function, in that case, you can assume it to be the default domain or range of real numbers.

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can someone help me pls ​

Answers

Using trigonometric function concepts, it is found that:

The angle is of 45 degrees, or  radians.

Since pi radians is 180º, this angle is of 180/4 = 45 degrees.

From the tangent: cot∅ =1

For the cosecant and secant: csc∅=√2 :sec∅=√2.

What is trigonometric function?Trigonometric function, in mathematics, one of six functions (sine [sin], cosine [cos], tangent [tan], cotangent [cot], secant [sec], and cosecant [csc]) that represent ratios of sides of right triangles. These six trigonometric functions in relation to a right triangle are displayed in the figure. They are also known as the circular functions, since their values can be defined as ratios of the x and y coordinates (see coordinate system) of points on a circle of radius 1 that correspond to angles in standard positions. Trigonometry can be easily applied to surveying, engineering, and navigation problems in which one of a right triangle’s acute angles and the length of a side are known and the lengths of the other sides are to be found.

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A 3 yard roll of plastic wrap costs $0.99. What is the price per foot

Answers

$0.11 cents per ft
One yard is 3 ft so 3 yards would be 9ft
So .99 divided by 9 would be .11 per ft

Katja walks 5 blocks east. Then she walks straight back to where she started and walks 5 blocks west. Graph 5 on the number line to represent how far Katha walked east. Then graph Katha’s location after she walks west. Click on the number line to identify the locations of the points

Answers

Answer:

move 5 blocks to the west.

Step-by-step explanation:

A number line is just that – a straight, horizontal line with numbers placed at even increments along the length. The location of Katha after she walked west is -5 on the number line.

In a running competition, a bronze, silver and gold medal must be given to the top three girls and top three boys. If 10 boys and 11 girls are competing, how many different ways could the six medals possibly be given out?

Answers

The required number of ways the 6 medals can be distributed among boys and girls is 14400 ways.

What are permutation and combination?

In arithmetic, combination and permutation are two different ways of grouping elements of a set into subsets. In combination, the components of the subset can be recorded in any order. In a permutation, the components of the subset are listed in a distinctive order.

Here,
10 boys and 11 girls are competing, Number different ways, the six medals possibly be given out is,
= 10C3 × 10C3
= 14400

The required number of ways the 6 medals can be distributed among boys and girls is 14400 ways.

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Mybankaccount pays 4% interest at the end of each year. Suppose that I have $x in my bank account at the start of this year, and that I never add money to the account or take money out of the account. (a) Howmuchwill be in the account at the end this year (after the interest is paid)? (b) Howmuchwill be in the account at the end of next year (again, after the interest)? (c) Find an expression for the amount in the account after N years.

Answers

a. The amount in the account at the end this year after the interest is paid will be 1.04x

b. The amount in the account at the end this year after the interest is paid will be 1.08x.

c. The expression for the amount in the account after N years will be x + 0.04xn

How to calculate the interest?

From the information, Mybankaccount pays 4% interest at the end of each year. The person had $x in the bank account at the start of this year. The amount in the account at the end this year after the interest is paid will be:

= x + (x × 0.04 × 1)

= x + 0.04x

= 1.04x

The amount in the account at the end of 2 years after the interest is paid will be:

= x + (x × 0.04 × 2)

= x + 0.08x

= 1.08x

The expression for the amount in the account after N years will be:

x + (x × 0.04 × n)

x + 0.04xn

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Evaluate the expression for x = -8.x = 16, and x = 4.

X ÷ 4

Answers

Therefore , the solution of the given problem of expression comes out to be -2,4 and 1.

Expression is who or what?

It is important to multiply, divide, add, or delete in mathematics. The following is how an expression is put together: Number, expression, and mathematical operator The components of a mathematical statement, such as additions, subtraction, multiplication, or division, etc., include numbers, variables, and functions. It is possible to contrast expressions and phrases.

Here,

Given  :x = -8.x = 16, and x = 4.

The expression is:

=> X/4

For x =-8

=> -8/4 =-2

For x =16

=> 16/4 =4

For x=4

=>4/4 =1

Therefore , the solution of the given problem of expression comes out to be -2,4 and 1.

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Which of the angles below measures more than 90 degrees but less than 180 degrees?

A.Right Angle
B.Acute Angle
C.Obtuse Angle
D.Reflex Angle

Answers

C. Obtuse Angle measures more than 90 degrees but less than 180 degrees.

An obtuse angle is an angle that measures more than 90 degrees but less than 180 degrees. It is greater than a right angle (which measures exactly 90 degrees) but less than a straight angle (which measures exactly 180 degrees). Obtuse angles are often referred to as "greater than a right angle."

A Right angle is an angle which measures exactly 90 degrees.

An Acute angle is an angle which measures less than 90 degrees.

A Reflex angle is an angle which measures greater than 180 degrees but less than 360 degrees.

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Help me with this quick please

Answers

On solving the provided question, we can say that from provided graphs we say  [tex]x^2 = 4y + 7[/tex] =.> [tex]x^2 = 15 = > x = \sqrt{15}[/tex]

What is graphs?

Mathematicians use graphs to logically convey facts or values using visual representations or charts. A graph point will typically reflect a relationship between two or more things. Nodes, or vertices, and edges make form a graph, a non-linear data structure. Glue together the nodes, often referred to as vertices. This graph has vertices V=1, 2, 3, 5, and edges E=1, 2, 1, 3, 2, 4, and (2.5), (3.5). (4.5). Statistical charts (bar charts, pie charts, line charts, etc.) graphical representations of exponential growth. a logarithmic graph in the shape of a triangle

here,

from provided graphs we say

[tex]x^2 = 4y + 7[/tex]

y = 2

[tex]x^2 = 15 \\x = \sqrt{15}[/tex]

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Equation with variable on both sides

I need help today and I need it now just today

Answers

The solution to the equations are as follows:

x = 0

r = -4

b = -8

n = -3

How to solve equations?

Equations are mathematical statements containing two algebraic expressions on both sides of an 'equal to (=)' sign.

The equation can be solved by find the values of the variables as follows:

Hence,

5.

1 - 3x = 3x + 1

-3x - 3x = 1 - 1

-6x = 0

x = 0

6.

4r + 8 + 5 = -15 - 3r

4r + 13 = -15 -3r

13 + 15 = -3r - 4r

28 = -7r

r = 28 / -7

r = -4

7.

-12 - 4b = 4 - 2b

-12 - 4  = -2b + 4b

-16 = 2b

b = -8

8.

3n - 15 = 7n + n

3n - 7n - n = 15

-5n = 15

n = -3

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The population P (in millions) of Italy from 2003 through 2015 can be approximated by the model P = 57.59e0.0051t, where t represents the year, with t = 3 corresponding to 2003. (a) According to the model, is the population of Italy increasing or decreasing? Explain.

Answers

It can be concluded that the population of Italy is increasing.

What are functions?In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y.The set X is called the domain of the function and the set Y is called the codomain of the function.Functions whose domain are the non - negative integers, known as sequences, are often defined by recurrence relations.Given a function as [tex]${\displaystyle f\colon X\to Y}[/tex] its graph is, formally, the set -

        [tex]${\displaystyle G=\{(x,f(x))\mid x\in X\}.}[/tex]

Given is the population {P} (in millions) of Italy from 2003 through 2015 can be approximated by the model → {P} = 57.59[tex]$e^{0.0051t}[/tex], where {t} represents the year, with {t} = 3 corresponding to 2003.

The exponential function is a mathematical function denoted by : f(x) = eˣ.

We have the function -

{P} = 57.599[tex]$e^{0.0051t}[/tex]

For {t} = 1, we can find P(1) as -

P(1) = 57.79 x [tex]$e^{0.0051}[/tex]

P(1) = 57.79 x 1.00511302714

P(1) = 58.1

Now, P(3) > P(1), this means that the population of Italy is increasing.

Therefore, it can be concluded that the population of Italy is increasing.

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( –5g – 6) – (4g – 1)

Answers

Answer:

Simplify the following:

-5 g - 6 - (4 g - 1)

Factor -1 out of -5 g - 6 - (4 g - 1):

-(5 g + 4 g - 1 + 6)

Grouping like terms, 5 g + 6 - 1 + 4 g = (5 g + 4 g) + (6 - 1):

-((5 g + 4 g) + (6 - 1))

5 g + 4 g = 9 g:

-(9 g + (6 - 1))

6 - 1 = 5:

Answer: -(9 g + 5)

Step-by-step explanation:

Please answer I need help

Answers

Answer:

The correct answer is A. A trapezoid has only 1 pair of parallel sides, but a parallelogram has 2.

A parallelogram is a quadrilateral with opposite sides that are parallel and equal in length. A trapezoid is a quadrilateral with one pair of opposite sides that are parallel, known as the bases. The other two sides are called legs.

Both a parallelogram and a trapezoid are quadrilaterals that have four sides, but parallelogram has opposite sides parallel to each other, while trapezoid has only one pair of parallel sides, the bases. Also a parallelogram has opposite angles are equal while a trapezoid doesn't have this property.

Option B and D is incorrect, A parallelogram doesn't need to have 4 right angles, and a trapezoid doesn't need to have 4 right angles as well.

Option C is incorrect as well, A parallelogram has two pairs of parallel sides.

What does the factored form of a quadratic function tell you about the function's graph?

Answers

The factored form of the quadratic function is f(x) = a(bx + c) (dx + e)

The term quadratic function is defined as a polynomial function with one or more variables in which the highest exponent of the variable is two and the graph of the quadratic equation is always forms the parabola.

Here we have given that we have to find the  the factored form of a quadratic function

As we all know that the general form of the quadratic function is written as,

=>  f(x) = ax² + bx + c

While we factorize the general form then it can be expanded like the following,

=> f(x) = a(bx + c) (dx + e)

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[tex]\frac{6}{2k-6} =3[/tex]

Answers

[tex] \frac{6}{2k - 6} = 3 \\ \\ \longrightarrow \: 6 = 6k - 18 \\ \\\longrightarrow \: 6k = 18 + 6 \\ \\\longrightarrow \: 6k = 24 \\ \\ dividing \: both \: sides \: by \: 6 \: , \\ \\\longrightarrow \: \cancel \frac{6k}{6} = \cancel \frac{24}{6} \\ \\ \longrightarrow \: \boxed{k = 4}[/tex]

hope helpful! :)

How do you convert vertex form to factored form?

Answers

The way to convert the vertex form to factored form is explained below.

The term vertex is defined as  a special point of a mathematical object, and is usually a location where two or more lines or edges meet.

Here we have to write down the steps to convert the vertex form to factored form.

The very first steps is to expand the vertex form into standard quadratic form and then use the quadratic root formula to determine the roots.

Here as we all know that factored term is the product of a constant and two linear terms and they are the roots of the function.

So, now we have to use the quadratic formula to convert them into factor form.

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square root of 36 in simplest radical form

Answers

Answer: The square root of 36 is expressed by √36, where '√' is the radical symbol and 36 is the radical form.

Step-by-step explanation: √36 is equal to 6, which is a rational number. Since 36 is a perfect square, therefore, it becomes easy for us to find its square root.

Hoped It Helped

The square root of 36 is expressed by √36, where '√' is the radical symbol and 36 is the radical form is 6 .

What is square root ?

Its like abundance (quantity ) of which is called amplitude (quantity) of the particular square .

Give : √36

Now,

Few examples of square root,

√4 = 2

√9 = 3

√16 = 4

√25  = 5

√36 = 6

√36 is equal to 6, which is a rational number. Since 36 is a perfect square, therefore, it becomes easy for us to find its square root.

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What is AX² in quadratic equation?

Answers

AX² is the standard form of a quadratic equation, where A is a coefficient and X is a variable.

In mathematics, a quadratic equation is a polynomial equation of the form ax2 + bx + c = 0, where a, b, and c are real numbers and a is not equal to zero. A quadratic equation can have either one, two, or no real solutions, depending on the value of the discriminant, b2 - 4ac. The discriminant determines the nature of the roots of the equation: if it is positive, the equation has two distinct real solutions; if it is zero, the equation has one repeated real solution; and if it is negative, the equation has two complex conjugate solutions.

The standard form of a quadratic equation is ax2 + bx + c = 0, where a is the coefficient of the x2 term, b is the coefficient of the x term, and c is the constant or free term. AX2 is the standard form of the equation, where A is the coefficient of the x2 term and X is the variable. If the coefficient A is not equal to 1, the equation can be simplified by dividing both sides of the equation by A. Solving a quadratic equation requires the use of the quadratic formula, which is used to find the roots of the equation. The quadratic formula states that the roots of the equation are x = [-b ± √(b2 - 4ac)]/2a.

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10. When travel to visit my mom over Thanksgiving vacation I drive from my home to Grandpa Herman's
which is 16 miles away. That first part of my trip to Grandpa's takes 20 minutes with light traffic. We then leave
his house and travel 2 hours to Ganada and take a travel break. That second part of the trip is 185 miles.
Lastly, we drive the 145 miles from Ganada to my mom's home in Port O'Connor in 1 hour and 15 minutes.
What is our average speed of the trip in mi/min AND mi/hr?

Write the formula:
Fill in the formula:
Answer: _____mi/mins
Answer: _____ mi/hrs

Answers

The average speed of the trip will be 1.60 miles/mins and 96.64 miles/hrs.

What is an average?

An average of a list of data is a mathematical expression for the center value of a set of data. It is defined mathematically as the ratio of the total of all the data to the number of units in the list. In statistics, the average of a particular collection of numerical data is also known as the mean.

e.g.

The average of 2, 3, and 4 equals (2+3+4)/3 = 9/3 = 3. As a result, the central value of 2,3, and 4 is 3. Thus, the definition of average is to determine the mean value of a set of data.

Average speed:-

The formula for average speed is found by calculating the ratio of the total distance traveled by the body to the time taken to cover that distance.

average speed = total distance / total time

Now,

As given

Total distance covered = 16+185+145

                         =346 miles

Total time taken=20 + 120 + 75

                  =215 minute  (3.58 hours)

Average speed=346/215

                         =1.60 miles/mins

Average speed=346/3.58

                            =96.64 miles/hrs

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Triangle A B C. Angle C is 90 degrees. Hypotenuse side A B is c, C B is a, C A is b.
Solve the right triangle shown in the figure.
Angle A = 45 degrees, side A C = 14 kilometers, angle C = 90 degrees
a.
Side B C = 14 kilometers, angle B = 45 degrees, Side A B = 7 StartRoot 2 EndRoot kilometers
c.
Side B C = 14 kilometers, angle B = 45 degrees, Side A B = 14 StartRoot 2 EndRoot kilometers
b.
Side B C = 7 StartRoot 2 EndRoot kilometers, angle B = 45 degrees, Side A B = 14 kilometers
d.
Side B C = 14 StartRoot 2 EndRoot kilometers, angle B = 45 degrees, Side A B = 14 kilometers
See picture

Answers

The length of the side [tex]\overline{BC}[/tex], and [tex]\overline{AB}[/tex], found using Pythagorean Theorem, and the measures of the angle ∠B in the right triangle ΔABC are;

c. [tex]\overline{BC}[/tex] = 14km, ∠B = 45°, [tex]\overline{AB}[/tex] = 14·√2km

What is the Pythagorean Theorem?

The Pythagorean Theorem defines the relationship between the legs and the length of the hypotenuse side of a right triangle.

According to Pythagorean Theorem, the sum of the squares of the legs of a right triangle is the same magnitude as the square of the of the length of the hypotenuse side.

The parameters of the right triangle are;

∠A = 45°

[tex]\overline{AC}[/tex] = 14 km

∠C = 90°

The angle sum property can be used to find the measure of the angle ∠B as follows;

m∠A + m∠B + m∠C = 180° (Angle sum property of a triangle)

45° + m∠B + 90° = 180°

m∠B = 180° - (90° + 45°) = 45°

m∠B = 45°

Two angles in the triangle ΔABC are congruent, therefore, the triangle is an isosceles right triangle and the legs, [tex]\overline{AC}[/tex] and [tex]\overline{BC}[/tex] are congruent.

[tex]\overline{AC}[/tex] = [tex]\overline{BC}[/tex] (definition of congruency)

[tex]\overline{BC}[/tex] = [tex]\overline{AC}[/tex] = 14 kilometers

[tex]\overline{BC}[/tex] = 14 km

The hypotenuse side of the right triangle = [tex]\overline{AB}[/tex]

The legs of the right triangle are; [tex]\overline{AC}[/tex] and [tex]\overline{BC}[/tex]

Pythagorean Theorem, states that we get;

[tex]\overline{AB}[/tex]² = [tex]\overline{AC}[/tex]² + [tex]\overline{BC}[/tex]²

Therefore;

[tex]\overline{AB}[/tex]² = 14² + 14² = 14² × (1 + 1) = 14² × 2

[tex]\overline{AB}[/tex] = √((14)²×2) = 14·√2

[tex]\overline{AB}[/tex] = 14·√2 km

The correct option is option c.

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Point B is between A and C if A , B , and C are collinear and the equation AB + BC = AC is true, where AB , BC , and AC are the distances between points A and B , B and C , and A and C , respectively.

Answers

The statement AB + BC = AC is true according to the segment addition postulate.

What is the segment addition postulate?

The segment addition postulate is a geometry axiom that states that a line segment, divided into a number of smaller segments, has the length given by the sum of the lengths of the segments.

The smaller segments in this problem are given as follows:

AB and BC.

The large segment is given as follows:

AC.

Hence the equation is given as follows:

AB + BC = AC.

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<
Find the volume of the sphere.
22
Either enter an exact answer in terms of " or use
7
units3
for T.

Answers

If the radius of sphere is 7 units, the volume of the sphere is 1,437.33 units³

What is volume of a shape?

Each thing in three dimensions takes up some space. The volume of this area is what is being measured. The space occupied within an object's borders in three dimensions is referred to as its volume. It is sometimes referred to as the object's capacity.

Given that, the value of π = 22/7

The radius of sphere is 7 units

Volume of a sphere = 4/3πr³, where r is the radius of the sphere.

= 4/3 * 22/7 * 7³

= 4/3 * 22/7 * 7 * 7 * 7

= (4 * 22 * 7 * 7 * 7) / (3 * 7)

= 30,184/21

= 1437.3333333333 unit³

So that, the volume of a sphere = 1,437.33 unit³

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x - 5 (x - 1) = x -(2x - 3)

Answers

Answer:

Solve for X by simplifying both sides of the equation, then isolating the variable.

X = 4x − 2

Hope this helps!!

to write it in a slope-intercept form

Answers

y = mx+b



Explanation: The slope-intercept form is written as, where m is the slope and b is the y-intercept. It's usually easy to graph a line using y=mx+b.

What is the remainder when 2x² − 5x − 1 divided by X − 3?

Answers

The remainder when 2x² - 5x - 1 divided by x - 3 is (2x + 1)

Let p(x) represents a polynomial  2x² − 5x − 1 and m(x) represents a polynomial x - 3

i.e., p(x) =  2x² − 5x − 1

and m(x) = x - 3

we need to find the remainder when 2x² − 5x − 1 divided by x − 3

If we divide a polynomial  p(x) =  2x² − 5x − 1  by m(x) = x - 3 using polynomial long division method, the result is:

p(x) / m(x)

= ( 2x² - 5x - 1 ) ÷ (x - 3)

= [ ( 2x² - 5x - 1 ) × ( 2x + 1)] + 2

This means, the quotient is  ( 2x + 1) and the remainder is 2

(see in attachment)

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You went to Sweet Spot to get ome candy. The cot i $5. 00 per pound. You have a coupon that give you a dicount of $0. 20 off per pound. If you buy $4 pound of candy, what will your total cot be, excluding tax?

Answers

The total cost of the candy excluding tax is  $19.2

How to find the total cost of the candy excluding tax

From the question we have that

The cost of the candy per pound is $5.00

And a coupon that gives a discount of $0..20 per pound

hence 4 pounds of candy without discount is

= 4 * 5

= $20

the discount is solved to be

= 4 * 0.2

= 0.8

subtracting the discount

= 20 - 0.8

= $19.2

So the amount to be paid for the candy is $19.2

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What is the value of f(0)

Answers

Answer:

the absolute value of 0 is just 0

Step-by-step explanation:

A spinner with 4 equal sections is spun 20 times. The frequency of spinning each color is recorded in the table below.
Outcome Frequency
Pink
6
Blue
Orange
What statement best compares the theoretical and experimental probability of landing on orange?
O The theoretical probability of landing on orange is
5
the experimental probability is 20%.
O The theoretical probability of landing on orange is L, and the experimental probability is 20%.
O The theoretical probability of landing on orange
and the experimental probability is 30%.
The theoretical probability of landing on orange
and the experimental probability is 50%.

Answers

Answer:

The theoretical probability of landing on orange is one fourth, and the experimental probability is 20%.

Step-by-step explanation:

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