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The Equation is biquadratic , the four solutions of the equation  x⁴ - 5x² - 14 = 0 are  x = +√7 , x = -√7 ,  x = +√2i and x = -√2i   .

The Biquadratic equation is given as :  x⁴ - 5x² - 14 = 0 ;

to change the given equation to a quadratic equation ,

we substitute x⁴ = y² and x² = y .....equation(1)

the quadratic equation becomes as :  y² - 5y - 14 = 0;

Now, we observe that to factor this expression, we find the two number whose product is 14 and whose difference is 5, such numbers are 7 and 2.

that means ⇒  y² - 5y - 14 = (y - 7)(y + 2);

from equation(1) ,

we get ; ⇒  x⁴ - 5x² - 14 = (x² - 7)(x² + 2) = 0;

Solving the two factors ,

we get ;

⇒ x² - 7 = 0 ⇒ x² = 7 ⇒  x = +√7 and x = -√7 ;  and

⇒ x² + 2 = 0  ⇒ x² = -2 ⇒  x = +√2i and x = -√2i .

Therefore , the given equation has two Real Solutions and two Complex Solutions .

The given question is incomplete , the complete question is

What are the solutions of the equation  x⁴ - 5x² - 14 = 0   ?

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Using exponential function, The value after 10 will be P(10) = 2125.0.05¹⁰.

A mathematical function with the following form is an exponential function: f ( x ) = an x. where an is a constant known as the function's base and x is a variable. The transcendental number e, or roughly 2.71828, is the exponential-function base that is most frequently encountered.

This is an exponential behavior expressed as:

P(t) = Abⁿ

P(t) = Ae^(n ln b)

Where P(t) is the  worth after 10 years,

A is the original investment,

and b is the percentage (87% in this case).

From the problem, we identify:

A = 2125

b = 0.05

The final model is:

P(t) = 2125.0.05¹⁰

After 10 years (t = 10),

the investment will be (to the nearest whole number):

P(10) = 2125 ×0.05¹⁰

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Amelia cut a 168 inch long ribbon into 6 equal part. Patrick cut a 14 foot long ribbon into 7 equal piece.

Amelia's ribbon pieces were 28 inches long each. Patrick's ribbon pieces were 2 feet long each.

Amelia's ribbon:

To calculate how long each piece of ribbon is, we need to divide 168 inches (the total length of the ribbon) by 6 (the number of pieces).

168 ÷ 6 = 28

Therefore, Amelia's ribbon pieces were 28 inches long each.

Patrick's ribbon:

To calculate how long each piece of ribbon is, we need to divide 14 feet (the total length of the ribbon) by 7 (the number of pieces).

14 ÷ 7 = 2

Since 1 foot is equal to 12 inches, we can convert 2 feet to 24 inches.

Therefore, Patrick's ribbon pieces were 2 feet (or 24 inches) long each.

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Answer:

[tex]\boxed{a = 4}[/tex]

Step-by-step explanation:

Note:

[tex]\fbox{\parbox{\textwidth}{%\textrm{\textsf{\textbf{Note:}}\\\textsf{This question is stated in a confusing manner and since I cannot see\\part a I am taking whatever I see on the screen as gospel truth\\\\From the question it appears that we are given gf(a) =3 and asked to solve for a.\\\\I am proceeding on that basis}}}[/tex]

We have the following functions

[tex]f(x) = \dfrac{2}{x}\\\\g(x) = \dfrac{x + 1}{x}\\[/tex]

[tex]gf(x) = g(f(x))[/tex]

[tex]\textrm{To find g(f(x)) wherever you see an x in gx), substitute the expression for f(x)}\\[/tex]

[tex]Since f(x) = \dfrac{2}{x}\\\\\textrm and}\\\\g(x) = \dfrac{x + 1}{x}\\[/tex]

[tex]\dfrac{\dfrac{{2}}{x}+1}{\dfrac{2}{x}}\\\\\\\\\left(\dfrac{{2}}{x}+1}\right) \div \dfrac{2}{x}\\\\= \left(\dfrac{{2}}{x}+1} \right) \times \dfrac{x}{2}\\\\[/tex]

[tex]\dfrac{2}{x} + 1} = \dfrac{2 + x}{x}\\[/tex]

[tex]\left(\dfrac{{2}}{x}+1} \right) \times \dfrac{x}{2}\\\\\\= \dfrac{2 + x}{x} \times \dfrac{x}{2}\\\\\\[/tex]

[tex]\textrm{The x's cancel out giving $\dfrac{2+x}{2}$}= 1 + \dfrac{x}{2}[/tex]

[tex]\textrm{For a specific value, a, if $g(f(a)) = 1 + \dfrac{a}{2}$ = 3}}[/tex]

[tex]\textrm{Then $\dfrac{a}{2}$ = 3-1 = 2}[/tex]

[tex]\boxed{a = 4}[/tex]

The value of Expression √x + 1/√x=(4√5+2√2)/3.

Algebraic expressions called polynomials include coefficients and variables. Indeterminates are another name for variables. For polynomial expressions, we can do mathematical operations like addition, subtraction, multiplication, and positive integer exponents but not division by variables. x2+x-12 is an illustration of a polynomial with a single variable.

There are three terms in this example: x2, x, and -12.

Given:

x=7+√40

=7+2√10

=5+2+2√10

=(√5)²+2.√5.√2+(√2)²

=(√5+√2)²

∴√x=√5+√2 (neglecting the negative sign)

1/√x=1/(√5+√2)

=(√5-√2)/(√5+√2)(√5-√2)

=(√5-√2)/(5-2)

=(√5-√2)/3

√x+1/√x

=(√5+√2)+(√5-√2)/3

=(3√5+3√2+√5-√2)/3

=(4√5+2√2)/3

=(4√5+2√2)/3

√x + 1/√x=(4√5+2√2)/3.

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The correct answer is "Figure A’B’C’D’E’F’ is smaller than figure ABCDEF, because the scale factor is less than 1".

The Scale Factor is used to scale shapes in various dimensions. Geometry teaches us about various geometrical shapes in both two and three dimensions.

The scale factor is a measure for figures with similar appearances but differing scales or measures. Assume two circles appear identical but have different radii.

A scale factor is defined as the ratio of the scale of a given original object to the scale of a new object that is its representation but of a different size (bigger or smaller).

Thus, the correct answer is "Figure A’B’C’D’E’F’ is smaller than figure ABCDEF, because the scale factor is less than 1"

Complete question:

Figure ABCDEF is shown on a coordinate grid. Figure ABCDEF is dilated with the origin as the center of the dilation to create figure A’B’C’D’E’F’ using the rule (x, y)—›(0.95x, 0.95y). Which statement is true?

Answer choices:

Figure A’B’C’D’E’F’ is smaller than figure ABCDEF, because the scale factor is greater than 1.

Figure A’B’C’D’E’F’ is larger than figure ABCDEF, because the scale factor is less than 1.

Figure A’B’C’D’E’F’ is smaller than figure ABCDEF, because the scale factor is less than 1

Figure A’B’C’D’E’F’ is larger than figure ABCDEF, because the scale factor is greater than 1.

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A’B’C’D’E’F’ is smaller than figure ABCDEF, because the scale factor is greater than 1.

A’B’C’D’E’F’ is larger than figure ABCDEF, because the scale factor is less than 1.

A’B’C’D’E’F’ is smaller than figure ABCDEF, because the scale factor is less than 1

A’B’C’D’E’F’ is larger than figure ABCDEF, because the scale factor is greater than 1.

x/sin 45 = 4√22/sin 90

CB = 4√22/sin 90 × sin 45

CB = 4√11

or

CB = √(4√22)²-(4√11)²

= 4√11

Answer:

CB = AC = 4√11

Step-by-step explanation:

We know that the sum of the angles of a triangle is 180°. If angle A = 45° and C = 90°, then angle B = 180° - 90° - 45° = 45°. This shows us that the triangle is isosceles and the leg AC is equal to the leg CB.

Consequently: CB = AC = 4√11

If we didn't know the length of AC, then it can be calculated from the hypotenuse:

AC² + CB² = AB²
2 * CB² = (4√22)²
CB² = (4√22)² / 2
CB² = 16 * 22 / 2
CB = √(16 * 11)
CB = 4√11

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Step-by-step explanation:

The answer is gotten by 5% of 70

5/100 × 70/1

= 35/10

= 3.5

Answer: $73.50

Step-by-step explanation:

First, we need to find what is 5% worth of 70 bucks. We would divide 5 by 100 since percents are based on 100 which gets us 0.05. Next do 0.05 * 70 = 3.5. Means that she'll pay $3.50 on top of the 70 bucks, so add the two together, common sense: $73.50. Have a good day ;)

Answer:

x = 79

Step-by-step explanation:

Since PQ and RS are most likely parallel lines, we can consider the orange angles (image provided below) to be similar due to them being alternate interior angles.

Now that we know that one angle of the triangle given (orange) is 46 degrees, we can solve for the other angle of the triangle. A triangle adds up to 180 degrees, so we can find the missing angle by subtracting 180 from the given angles: 180-33-46 = 101 (marked in red).

With the red angle, we can find x by subtracting 180 from 101. Why you may ask? Because RS is parallel to PQ, so the degree will be 180. Therefore, the missing angle, x, will be 79.

What is the price of an item after a 30% discount if the original price of the item is 56 dollars?

Answer: The Declaration of Independence, adopted by the Continental Congress on July 4, 1776, is a reflection of the colonists' motivations in several ways. Firstly, the Declaration states that the colonists believed that all men are created equal and have the right to life, liberty, and the pursuit of happiness. This reflects the colonists' belief in the Enlightenment principles of natural rights and individual liberty, which were influential in shaping their motivations for seeking independence from British rule.

Secondly, the Declaration cites a long list of grievances against King George III, accusing him of violating the colonists' rights and abusing his power. This reflects the colonists' growing frustration and anger towards British rule, which had been building for many years prior to the drafting of the Declaration.

Thirdly, the Declaration states that the colonists had attempted to resolve their differences with the British government through peaceful means, but that these efforts had been unsuccessful. This reflects the colonists' desire for a peaceful resolution to their issues, but their willingness to fight for their rights if necessary.

In summary, the Declaration of Independence reflects the colonists' belief in natural rights and individual liberty, their growing frustration and anger towards British rule, and their desire for a peaceful resolution to their issues but their willingness to fight for their rights if necessary.

Step-by-step explanation:

The value of n in the equation n/5 + 0.6 = 2 is 7.

It should be noted that 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.

It is important to note that an equation is the mathematical statement which can be made up of two expressions which are connected by an equal sign.

In this case, n/5 + 0.6 = 2

0.2n + 0.6 = 2

Collect the like terms

0.2n = 2 - 0.6

0.2n = 1.4

Divide

n = 1.4 / 0.2

n = 7

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Complete question

Solve for n

n/5 + 0.6 = 2

The length of EF, to the nearest tenth of a foot is found to be 14.3 ft.

In rDEF, DE = 90°, DF = 81°, and DE = 14. 6 ft,

Because the given triangle is a right angled triangle, we can use the concepts of trigonometry in order to solve the given problem.

Using the trigonometric relation,

sin A = P/H

Where,

P is perpendicular and H is the hypotenuse.

We have, A = 81°,

It is given that the value of DE is 14.6 ft.

We know, sin(81°) = 0.98

So, the length of EF, to the nearest tenth of a foot is,

0.98 = EF/14.6

EF = 14.3 ft.

So, the length of the EF is 14.3 ft.

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Complete Question - In ΔDEF, ∠E = 90°, ∠F = 81° and DE = 14.6ft. Find the length of EF, to the nearest tenth of a foot.

The distance between two points (2, 3) and (5, 7) is 5 units.

Therefore the answer is 5.

To find the distance between the points (2, 3) and (5, 7), we can use the distance formula:

d = √((x₂ - x₁)² + (y₂ - y₁)²)

where (x₁, y₁) = (2, 3) and (x₂, y₂) = (5, 7)

Substituting the coordinates of the two points into the formula:

d = √((5 - 2)² + (7 - 3)²)

d = √((3)² + (4)²)

d = √(9 + 16)

d = √(25)

d = 5

So the distance between the points (2, 3) as well as (5, 7) will be  5 units by solving the above expression.

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A fraction is said to be in standard form when both the numerator and denominator are co-prime numbers.

A well-known form of a range in Maths is largely cited for the illustration of large numbers or small numbers. We use exponents to symbolize such numbers in general shape. Any number that we can write as a decimal range, between 1. zero and 10.0, multiplied by means of a power of 10, is stated to be in general shape.

“ Standard form” is likewise known as “clinical Notation” depending on the mathematical lingo of the u. s. you are in. within the UK and the international locations that comply with the same conventions as the UK, the time period “medical Notation” is most typically used while in nations that comply with the usa conventions majorly talk to this shape of writing as the “fashionable form”.

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Yes 360 degrees is full turn calling it a full rotation.

A degree is the name given to the mathematical unit of measurement for an angle. A protractor is a device used to determine the degree of an angle. Angles can be measured at different angles with different degrees, such as 30°, 45°, 60°, and so on. A complete circle rotates at 360°.

Each of the 360 equal segments that make up a rotation is referred to as a degree. We use a circle ° to represent a degree. 180°, for instance, denotes 180 degrees. An angle in degrees is an angle whose measure is expressed in degrees.

A 360 degree is also called as a Complete angle because it complete the whole round making the 180 degree angle twice giving us 360 degree.

A 360 when written as radian can be given by 2π ( where π= 180 degree)

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The graph's shape changes depending on the base. The graph can be reflected across the y-axis by replacing x with x, and across the x-axis by replacing y with y.

How to graph an exponential function with the notation f (x) = bx f (x) = b x 1

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Answer:

190,217

Step-by-step explanation:

Let x = number of doctors.

27.6% of x is 52,500

0.276x = 52,500

x = 52,500/0.276

x = 190,217

Answer:

The equation x2+6x+9 = 0 has roots of 3+i and 3-i. By the fundamental theorem of algebra, this equation has three roots, two of which may be equal. The standard quadratic equation using the given set of solutions {3+i,3-i} is y=x2+6x+9.

RS is 75 units when T and q are the midpoints of legs SU and RV of a trapezoid.

The bases are in step with one another. The median runs perpendicular to each bases, and its length will be determined by averaging the bases' lengths. 360° is the sum of all of the angles in a trapezoid. A aspect's angles upload as much as an entire one hundred eighty stages.

So, imagine cutting the trapezoid from half and and rotating and join it to became a long parallelogram

Now,

VR = UV + RS = 2QT

⇒ 35 + RS = 2 × 55

⇒  RS = 110 - 35

⇒  RS = 75

Thus, RS is 75 units when T and q are the midpoints of legs SU and RV of a trapezoid.

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The number of trees more than 10m tall but not more than 20m tall is 18 trees.

0 < h ≤ 5 = 5

height greater than 0m less than or equal to 5m

5 < h ≤ 10 = 9

height greater than 5m less than or equal to 10m

10 < h ≤ 15 = 13

height greater than 10m less than or equal to 15m

15 < h ≤ 20 = 5

height greater than 15m less than or equal to 20m

20 < h ≤ 25 = 1

height greater than 20m less than or equal to 25m

The number of trees that are more than 10m tall but not more than 20m tall are;

10 < h ≤ 15 = 13

15 < h ≤ 20 = 5

So,

13 + 5 = 18 trees

Therefore, the total number of trees which are 10m tall but not more than 20m tall is 18 trees.

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The inequality to find the number of fruit snacks that were in the bag is

x >= 15

What is Algebraic expression ?

Algebraic expression can be defined as the combination of variables and constants.

Given ,

You had a bag of fruit snacks that you shared with 2 friends. Each of you got no fewer than 6 fruit snacks.

Given expression , x/3  >=  5

Multiply 3 on both sides

we get,

x/3  *  3  >=  5*3

x >= 15

So the inequality holds x >= 15 .

Hence, The inequality to find the number of fruit snacks that were in the bag is x >= 15

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The value of x in equation 2x+3y=36, when y=6 is 9.

The replacement operation can be used in a variety of situations involving formal objects that contain symbols; it involves systematically replacing occurrences of a symbol with a particular value.

Substitution is a fundamental computer algebra operation.

Given equation 2x+3y=36 ; y=6

Solve the equation for given y.

Plugging y=6 in the given equation

We get 2x + 3(6) = 36

2x +18 = 36

2x = 36 - 18

2x = 18

x = 9

Therefore, the value of x in equation 2x+3y=36, when y=6 is 9.

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D) The term "imaginary roots" describes the roots of the equation 3x2 - 4x + 5 = 0 the best.

The formula is the quadratic equation ax2 + bx + c = 0 in standard form.

x = [-b ± √(b2 - 4ac)]

/ 2a

It follows that

3x2 - 4x + 5 = 0

Here, an equals 3, b equals -4, and c equals 5.

Using these numbers as replacements in the formula

x = [-(-4) ± √((-4)2 - 4 × 3 × 5)]

/ (2 × 3)

By even more simplifying

x = [4 ± √(16 - 60)]

/6

x = [4 ± √-44]/ 6

As a result,

x = [4 ± 2√-11]/ 6

x = [2 ± I √11]/ 3

Additionally, b2 - 4ac = -44 0 is the discriminant. So, we have invented roots.

Therefore, the term "imaginary roots" is the best description of the roots of the equation 3x2 - 4x + 5 = 0.

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Full Question = Choose the best description of the roots of the equation 3x2 - 4x + 5 = 0.

Double root

Real and rational roots

Real and irrational roots

Imaginary roots

Answer:

[tex]y=-\frac{6}{5} x[/tex]

Step-by-step explanation:

[tex]\frac{y_{2} -y_{1} }{x_{2} -x_{1} }[/tex]

[tex]\frac{7-(-5)}{-5-5}[/tex]

[tex]\frac{7+5}{-10}[/tex]

[tex]-\frac{12}{10}[/tex]

[tex]y=-\frac{6}{5} +b[/tex]

[tex]-5=-\frac{6}{5} (5)+b[/tex]

- 5 = - 5 + b

b = 0

[tex]y=-\frac{6}{5} x[/tex]

The following triangles are congruent.

Yes AAA

Bob spends 9 minutes running for every 36 minutes that he spends practicing dribbling.

What is ratio ?

A ratio is a mathematical comparison of two or more values or quantities. It is a way of expressing the relationship between different amounts or measurements.

The ratio of running to dribbling is 5 minutes of running to 20 minutes of dribbling. We can use this ratio to find out how many minutes Bob spends running if he spends 36 minutes practicing dribbling.

First, we can set up a proportion with the known ratio and the unknown value ,

5/20 = x/36

To find x we can cross-multiply to get ,

20x = 180

then divide both sides by 20 ,

x = 9

So, Bob spends 9 minutes running for every 36 minutes that he spends practicing dribbling.

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5/20 = x/36

To find x we can cross-multiply to get ,

20x = 180

then divide both sides by 20 ,

x = 9

So, Bob spends 9 minutes running for every 36 minutes that he spends practicing dribbling.

The linear equation perpendicular to the line y = 1/2x - 4 is:

y = -2x + 25

A general linear equation can be written as:

y = m*x + b

Where m is the slope and b is the y-intercept.

And two lines are perpendicular if the product between the slopes is -1, then if we want to find a line perpendicular to:

y = 1/2x - 4

Then the slope m must be sch that:

m*(1/2) = -1

m = -2

Then the line is like:

y = -2*x + b

And we know that this passes through (10, 5), replacing these values:

5 = -2*10 + b

5 = -20 + b

5 + 20 = b

25 = b

So the linear equation is:

y = -2x + 25

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There are 364 solutions to the given inequality: x1 + x2 + x3 <= 11.

To get the solution of the inequality x1 + x2 + x3 <= 11, we first need to include an additional variable in the inequation to write in as an equation.

Therefore, let the additional variable be x4.

Thus, equation is x1 + x2 + x3 + x4 = 11

The number of solutions of an equation can be calculated by the formula,

[tex]( \left \ {{n+ r - 1} \atop {n}} \right.)[/tex], where n is the constant value of the equation and r is the number of variables in the equation.

In this case, n = 11 and r = 4

Therefore,

[tex]( \left \ {{n+ r - 1} \atop {n}} \right.)[/tex] = [tex]( \left \ {{11+ 4 - 1} \atop {11}} \right.)[/tex] = [tex]( \left \ {{14} \atop {3}} \right.)[/tex]

= 14 * 13 * 12 * 11!/(3 * 2 * 1 * 11!)

=2184/6 = 364

Therefore, there are 364 possible solutions of the inequality.

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The ant population increases by factor of 5 every week. After 3 weeks, there will be 800 ants in the farm.

The problem can be solved by considering the ant population growth as a geometric sequence.

In a geometric sequence, a term is obtained by multiplying the previous term by a constant, which is called a common ratio. We can consider this common ratio as a growth factor.

In this case, the common ratio = 5

week 1 = 32 ants

week 2 = 32 x 5 ants = 160 ants

week 3 = 160 x 5 ants = 800 ants

We can also find the number of ants using the geometric sequence formula for the nth term:

a(n) = a . rⁿ⁻¹

Where:

a = first term = 32

r = common ratio = 5

Hence,

a(3) = 32 x 5² = 800 ants

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