What are the solutions to the following quadratic?
5x² + 18x=25x+15+3x²
(x+5)(2x-3)
O (5,-3/2)
(x - 5)(2x+3)
(-5,3/2)

Answer:

[tex]x=\frac{5}{2},\:x=3[/tex]

Step-by-step explanation:

[tex]\mathrm{Factor\:}2x^2-11x+15[/tex]

[tex]\mathrm{Break\:the\:expression\:into\:groups}[/tex]

[tex]\left(2x^2-5x\right)+\left(-6x+15\right)[/tex]

[tex]\mathrm{Factor\:out}\:x\:\mathrm{from}\:2x^2-5x:\quad \:x\left(2x-5\right)[/tex][tex]\mathrm{Factor\:out}\:-3\:\mathrm{from}\:-6x+15:\quad \:-3\left(2x-5\right)[/tex]

[tex]x\left(2x-5\right)-3\left(2x-5\right)[/tex]

[tex]\mathrm{Factor\:out\:common\:term\:}2x-5[/tex]

[tex]\left(2x-5\right)\left(x-3\right)[/tex]

[tex]\mathrm{Using\:the\:Zero\:Factor\:Principle:\quad \:If}\:ab=0\:\mathrm{then}\:a=0\:\mathrm{or}\:b=0[/tex]

[tex]2x-5=0\quad \mathrm{or}\quad \:x-3=0[/tex]

[tex]\mathrm{Solve\:}\:2x-5=0:\quad x=\frac{5}{2}[/tex]

[tex]\mathrm{Solve\:}\:x-3=0:\quad x=3[/tex]

[tex]\mathrm{The\:solutions\:to\:the\:quadratic\:equation\:are:}[/tex]

[tex]x=\frac{5}{2},\:x=3[/tex]

The factors of the expression are (2x + 3) (2x - 9). The solution has been obtained by using factorization.

What is factorization?

Writing a number or other mathematical object as the result of numerous factors, typically smaller or simpler terms of the same kind, is known as factorization or factoring in mathematics.

We are given an expression as 4[tex]x^{2}[/tex] - 12x -27.

On factoring it, we get

⇒ 4[tex]x^{2}[/tex] - 12x -27

⇒ 4[tex]x^{2}[/tex] + 6x - 18x -27

⇒ 2x (2x + 3) - 9 (2x + 3)

⇒ (2x + 3) (2x - 9)

Hence, the factors of the expression are (2x + 3) (2x - 9).

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The constant of proportionality of the relation between the height on set and the actual height is given as follows:

0.55.

A proportional relationship is defined according to the equation presented as follows:

y = kx.

In which k is the constant of proportionality, representing the increase in the output variable y when the constant variable x is increased by one.

The variables for this problem are given as follows:

Considering the values of these variables for the door, the constant is obtained as follows:

k = 44/80

k = 0.55.

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If the force is 270 Newton and the area is 0.03 square meter, the pressure is equal to 9,000 N/m².

In Mathematics, pressure can be defined as a measure of the force exerted per unit area of an object or physical body. This ultimately implies that, it is usually measured in Newton per meter square.

Mathematically, the pressure of a physical object can be calculated by using this formula:

P = F/A

Where:

By substituting the given parameters, we have:

Pressure, P = 270/0.03

Pressure, P = 9,000 N/m².

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All four conditions have been met, the conditions for statistical inference for testing a difference in population proportions have been satisfied. The conditions for statistical inference for testing a difference in population proportions are:

Random samples: Both schools have selected simple random samples of their respective student populations, which satisfies this condition.

Large sample size: Both schools have over 2,000 students, which satisfies the large sample size condition.

Independent samples: The samples from schools A and B are independent, as the students from one school do not affect the students from the other school.

Success-failure condition: The sample sizes are large enough to satisfy the success-failure condition, which states that there should be at least 10 successes and 10 failures in each sample. Both schools have well over 10 successes and failures in their respective samples.

Since all four conditions have been met, the conditions for statistical inference for testing a difference in population proportions have been satisfied.

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Answer:

210

Step-by-step explanation:

Step-by-step explanation:

There are a couple of ways to solve this...this right triangle has two legs of length x  and hypotenuse of 9 units

Use Pythagorean theorem:

x^2 + x^2 = 9^2

x = 9/sqrt2   =  9 sqrt (2) / 2

Based on the information, the power company should hire the geologist.

It should be noted that to make a decision, we can use the Expected Value (EV) criterion. The EV of building at Diablo without hiring the geologist is:

EV(Diablo without geologist) = (0.2 * (-$10 million)) + (0.8 * $10 million) = $6 million

The EV of building at Roy Rogers City without hiring the geologist is:

EV(Roy Rogers without geologist) = $20 million

The probability of an earthquake occurring and the geologist predicting it is 0.95 * 0.2 = 0.19. The probability of an earthquake occurring and the geologist not predicting it is 0.05 * 0.2 = 0.01. The probability of an earthquake not occurring and the geologist predicting it is 0.1 * 0.8 = 0.08. The probability of an earthquake not occurring and the geologist not predicting it is 0.9 * 0.8 = 0.72.

The EV of hiring the geologist is:

EV(Diablo with geologist) = (0.19 * $0) + (0.01 * (-$10 million)) + (0.08 * (-$1 million)) + (0.72 * $10 million) = $6.8 million

Comparing the EVs, we can see that the company should build at Diablo with the geologist's help, as it yields a higher EV. The answer is B.

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The answer is not provided among the options given.

An equation known as proportion shows that the two ratios given are equal to one another. In other terms, the proportion declares that the two ratios or fractions are equal.

To solve for the value of b that makes the proportion true, we can use cross-multiplication:

12 = 21A

12b = 21A

b = 21A/12

To determine the specific value of b, we need to know the value of A. However, since A is not given in the question, we cannot determine the value of b. Therefore, the answer is not provided among the options given.

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a. The independent and dependent variables are:

b. Rate of change = $7.75

c. initial value = $20

d. The total charge for 17 people = $151.75

e. The number of people who could come for $500 is $62.

To determine the independent and dependent variables, we need to identify which variable depends on the other. In this case, the total charge depends on the number of people attending the party. Therefore, the number of people is the independent variable, and the total charge is the dependent variable.

To find the rate of change, we can use the slope formula:

slope = (change in y) / (change in x)

slope = (175 - 97.5) / (20 - 10)

slope = 7.75

Therefore, the rate of change is $7.75 per person.

To find the initial value, we can use either of the two data points. Let's use the data point where Jimmy and 10 people have a total charge of $97.50.

y = mx + b

97.5 = 7.75(10) + b

b = 20

Therefore, the initial value (the fixed fee for the party) is $20.

To find the number of people who could come for $500, we can rearrange the equation:

y = mx + b

500 = 7.75x + 20

480 = 7.75x

x ≈ 62

Therefore, Jimmy could invite 62 people (including himself) to the party for $500.

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To reach the revenue needed to cover expenses and begin turning a profit, or 120x = 120(92) = $11,040, the business must sell at least 92 guitars.

To start making a profit, the music company needs to generate enough revenue to cover its costs. The revenue function is given by r(x) = 120x, where x is the number of guitars manufactured and sold.

The cost function is given by c(x) = 100x + 1840, where 1840 represents the fixed cost (such as rent, salaries, etc.) associated with setting up the manufacturing process.

To determine the total revenue required to start making a profit, we need to find the break-even point where the revenue generated is equal to the total cost incurred:

r(x) = c(x)

Substituting the given functions, we get:

120x = 100x + 1840

Solving for x, we get:

20x = 1840

x = 92

Therefore, the company needs to sell at least 92 guitars to generate a revenue of 120x = 120(92) = $11,040, which is the amount required to cover its costs and start making a profit. If the company sells fewer than 92 guitars, it will not be able to cover its costs and will incur a loss.

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The distance after 79 minutes is 11 miles.

We know that the distance can be modeled with a linear equation like:

y = ax + b

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

We know that if the line passes through two points (x₁, y₁) and (x₂, y₂) then the slope can be written as:

a = (y₂ - y₁)/(x₂ - x₁)

Here we know the points (39, 45) and (61, 26.3)

Then we can write:

a = (26.3 - 45)/(61 - 39) = -0.85

So we can write:

y = -0.85*x + b

To find the value of b we can replace the values of one of the points, i will use (39, 45) to get.

45 = -0.85*39 + b

45 + 0.85*39 = b

78.15 = b

So the line is:

y =  -0.85*x + 78.15

The distance after 79 minutes is.

y =  -0.85*79 + 78.15

y = 11

The distance left is 11 miles.

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Line segments AB and A'B' are on the same line: False.

Line segments AC and A'C' are on parallel: False.

In Geometry, dilation can be defined as a type of transformation which typically changes the size of a geometric object, but not its shape.

Based on the transformation rule, line segment AB would move further away from point P because both coordinates A and B move away from point P. Therefore, line segments AB and A'B' cannot be on the same line because they are parallel.

Conversely, line segment AC and A'C' are located on the same line because points A and C would move away from P, but line segment AC passes through point P, and line segment A'C' would do the same. Therefore, line segments AC and A'C' are not parallel.

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Answer:

true false

Step-by-step explanation:

The equation that has the solutions shown in the table is:

y = x³ - 1

What is meant by an equation?

A mathematical equation is a formula that uses the equals sign to represent the equality of two expressions. Two expressions are combined in an equation using an equal symbol ("="). The "left-hand side" and "right-hand side" of the equation are the two expressions on either side of the equals sign. Typically, we consider an equation's right side to be zero. As we can balance this by deducting the right-side expression from both sides' expressions, this won't reduce the generality.

The values of x are constantly increasing by 1.

The difference in values of y are:

-1+2 = 1

0+1 = 1

7-0 = 7

This is not a quadratic equation because the difference of difference in y-values is not constant.

So we can eliminate the options c and d.

Now the sign of the input value is the same as the sign of the output value.

So the x³ term has to be positive.

Consider the equation y = x³ - 1

When x = -1

y = -1³ - 1 = -2

when x = 2

y = 2³ - 1 = 7

Therefore the equation that has the solutions shown in the table is:

y = x³ - 1

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The bond would pay the bondholder the following sum of $1,000 annually.

The bond being offered by Company ABC has a par value of $1,000, a fixed coupon rate of 5% per year, and a maturity of 10 years.

The coupon rate of 5% per year means that for each $1,000 bond, the bondholder will receive an annual interest payment of $50 (5% of $1,000).

Therefore, out of the given options, the correct amount that the bondholder would receive each year is $50.

It's important to note that this interest payment is fixed and does not change over the 10-year term of the bond. At maturity, the bondholder would receive the par value of $1,000 back from the company.

So, over the course of 10 years, the bondholder would receive a total of $500 ($50 per year) in interest payments, in addition to the return of the $1,000 par value at maturity.

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Answer:

(3,5)

Step-by-step explanation:

Twin primes are a pair of prime numbers that differ by 2. Let's consider a pair of twin primes, p and p + 2 (where p > 3). We want to prove that the number between them, p + 1, is divisible by 6.

We know that every prime number greater than 3 can be written in the form 6k ± 1 for some integer k. This is because any integer can be written in one of six possible forms: 6k, 6k + 1, 6k + 2, 6k + 3, 6k + 4, or 6k + 5. However, we can eliminate the forms that are divisible by 2 or 3 (except for 2 and 3 themselves), leaving only 6k ± 1 and 6k ± 5. Since twin primes differ by 2, they must both be of the form 6k ± 1.

Let's consider p, the smaller of the twin primes. We know that p is of the form 6k ± 1 for some integer k. If p is of the form 6k + 1, then p + 2 is of the form 6k + 3, which is not prime (since it is divisible by 3). Therefore, p must be of the form 6k - 1. Then, p + 1 is of the form 6k, which means it is divisible by 2 and by 3.

Similarly, if p + 2 is of the form 6k - 1, then p is of the form 6k - 3, which is not prime (since it is divisible by 3). Therefore, p + 2 must be of the form 6k + 1. Then, p + 1 is of the form 6k, which means it is divisible by 2 and by 3.

Therefore, in either case, the number between the twin primes, p + 1, is divisible by 6. We have shown that this is true for any pair of twin primes except for the pair (3, 5).

the quotient of 125 ÷ 25.1 is approximately equal to 5/1.004 or 4.98 (rounded to two decimal places).

Why it is and what is quotient?

We can rewrite the division 125 ÷ 25.1 as multiplication by the reciprocal of 25.1:

125 ÷ 25.1 = 125 × 1/25.1

Now, we can simplify this expression by multiplying the numerators and denominators:

125 × 1/25.1 = (125/1) × (1/25.1) = 125/25.1

Finally, we can simplify this fraction by dividing both the numerator and denominator by their greatest common factor (GCF), which is 5:

125/25.1 = (25 × 5) / (5 × 5.02) = 5/1.004

Therefore, the quotient of 125 ÷ 25.1 is approximately equal to 5/1.004 or 4.98 (rounded to two decimal places).

In mathematics, the quotient refers to the result obtained from dividing one quantity by another. It is the answer to a division problem, and is typically expressed as a fraction or decimal. For example, the quotient of 10 divided by 2 is 5, written as 10 ÷ 2 = 5.

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The length of PS is 34 inches. Points P,Q,S are points on a line, so that PQ is 6 in, The ratio PQ:QR is 3:4 and The ratio QR:RS is 2:5.

An equation is an expression containing numbers and variables linked together by mathematical operations such as addition, subtraction, division, multiplication and exponents.

The length of PQ is 6 in. The ratio PQ:QR is 3:4, hence:

PQ = PR * (3/7)

6 = PR * (3/7)

PR = 14 inches

QR = PR * (4/7)

QR = 14 * 4/7

QR = 8 inches

The ratio QR:RS is 2:5, hence:

QR = QS * (2/7)

8 = QS * 2/7

QS = 28 inches

Points P,Q,S appear in that order on a line. Therefore:

PS = PQ + QS

PS = 6 + 28

PS = 34 inches

The length of PS is 34 inches.

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The measure of ik in the given equation can be calculated by subtracting the sum of IR (which is equal to jkh) and mjk from 180e degrees.

In order to calculate the measure of ik in the given equation, we must use the formula for interior angles of a triangle which is the sum of all angles in a triangle is 180 degrees. In this equation, we are given the measures of two of the angles, IR and mjk. Therefore, the measure of ik can be calculated by subtracting the sum of IR and mjk from 180 degrees.

Given that IR = jk, we can calculate the measure of ik by first calculating the measure of jk. We are given that mik = (11x + 2), so we can solve for x and use that value to calculate jk. If we solve for x, we find that x = (mik-2)/11. We can then use that value to calculate jk, which equals mik-mjk. In this equation, we are given that mjk = (12x - 7), so jk = (11x + 2) - (12x - 7). Simplifying, we find that jk = 9x + 9.

Now that we have the measure of jk, we can calculate the measure of ik by subtracting the sum of jk and IR from 180 degrees. Since IR = jk, we can substitute jk in for IR, so ik = 180

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The LCM of A = a^3× b^2×c and B=a^5 ×b×c^4×d in exponential form is a^5 × b^2 × c^4 × d^1.

To find the LCM (Least Common Multiple) of two numbers, we need to find the greatest power of each prime factor that occurs in either number.

The prime factors in the given expressions are a, b, c, and d. For each prime factor, we need to find the greatest power that occurs in either A or B.

For a: The highest power of a that occurs in A is a^3, and the highest power that occurs in B is a^5. Therefore, the LCM must include a^5.

For b: The highest power of b that occurs in A is b^2, and the highest power that occurs in B is b^1. Therefore, the LCM must include b^2.

For c: The highest power of c that occurs in A is c^1, and the highest power that occurs in B is c^4. Therefore, the LCM must include c^4.

For d: The highest power of d that occurs in A is d^0, and the highest power that occurs in B is d^1. Therefore, the LCM must include d^1.

Putting all of this together, we have:

LCM(A,B) = a^5 × b^2 × c^4 × d^1

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hm, dunno.

2w^2(5w + 6)(-5w + 6)

Answer:

2w^2(6 + 5w)(6 - 5w)

Step-by-step explanation:

Answer:

30. Proofs attached to answer

Step-by-step explanation:

Proofs attached to answer

When a number is 12 more than another number and twice the sum of two numbers is four. Then two numbers are equals to the 7 and -5.

We have a number is 12 more than another number. Let the two numbers be 'a' and 'b' such that a> b. So, from the question, bigger number, a is 12 more than another number, b. In mathematics form, a = 12 + b --(1)

Also, twice the sum of two numbers is four, that is 2( a + b ) = 4 --(2)

We have to determine the values of two numbers. For this, solve the equation (1) and (2). Using the Substitution method,

Substitute, a = 12 + b from equation (1) to equation(2), 2( a + b ) = 4

=> 2( 12 + b + b) = 4

Simplify the expression

=> 24 + 4b = 4

=> 24 - 4 = -4b

=> 4b = - 20

dividing by 4 both sides

=> b = -5

Plugging the value of b = -5, in equation(1)

=> a = 12 + b = 12 + (-5)

=> a = 12 - 5 = 7

Hence required numbers are 7 and -5.

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The critical point of the function y = 3x² - 40x + 8 is at (6²/₃, -125¹/₃)

The critical points of a function are the point at which there is a maximum or minimum value of the function.

Since we have the function y = 3x² - 40x + 8, we desire to find its ctitical points. We proceed as follows.

To find the critical points of the function y = 3x² - 40x + 8, we differentiate with respect to x and equate to zero.

So, y = 3x² - 40x + 8,

dy/dx = d(3x² - 40x + 8)/dx

= d3x²/dx - d40x/dx + d8/dx

= 2 × 3x - 40 + 0

= 6x - 40

So, equating to zero, we have

dy/dx = 0

6x - 40 = 0

6x = 40

x = 40/6

To determine if this is a maximum or minimum point, we differentiate dy/dx.

so, d(dy/dx) = d(6x - 40)/dx

d²y/dx² = d6x/dx - d40/dx

= 6 - 0

= 6

Since d²y/dx² = 6 > 0. x = 40/6 is a minimum point

So, substituting x = 40/6 into the equation, we have

y = 3x² - 40x + 8

y = 3(40/6)² - 40(40/6) + 8

= 3 × 1600/36 - 1600/6 + 8

= 1600/12 - 1600/6 + 8

= 1600/6(1/2 - 1) + 8

= -1600/12 + 8

= (-1600 + 96)/12

= -1504/12

= -125¹/₃

So, the critical point is at (6²/₃, -125¹/₃)

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Answer:

A

Step-by-step explanation:

given the radius is x , then diameter = 2x and height = 2x - 1

the volume (V) of a cylinder is calculated as

V = πr²h ( r is the radius and h the height ) , then

V = πx²(2x - 1) ← distribute parenthesis by πx²

  = 2πx³ - πx² in³

Answer: the very top answer -> A

Answer:

Undefined

Step-by-step explanation:

The Xs are the same!

The point (3, -5π/3) can be represented by the formula x = 3, y = -5π/3.

The point (3, -5π/3) can be represented by the formula x = 3, y = -5π/3. To calculate this, we first find the x-coordinate, which is simply 3. To find the y-coordinate, we must first multiply 5 and π. 5π = 15.78. We then divide 15.78 by 3, which gives us 5.26. To make this a negative number, we add a negative sign, giving us -5.26. We can then convert this to fraction form by dividing the numerator and denominator by the greatest common factor, which is 1. This gives us -5π/3.

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Complete question:

What point represents (3, negative StartFraction 5 pi Over 3 EndFraction)?

Answer:

A

Step-by-step explanation:

The factored form of the quadratic function x² + 6x - 3 = 0 is given as follows:

x² + 6x - 3 = (x + 6.46)(x - 0.46).

The quadratic function for this problem is defined as follows:

x² + 6x - 3 = 0.

The coefficients are given as follows:

a = 1, b = 6, c = -3.

Hence the discriminant of the quadratic function is obtained as follows:

D = b² - 4ac

D = 6² - 4(1)(-3)

D = 48.

Then the first root is given as follows:

x = (-b - sqrt(D))/2a

x = (-6 - sqrt(48))/2

x = -6.46.

The second root is then obtained as follows:

x = (-b + sqrt(D))/2a

x = (-6 + sqrt(48))/2

x = 0.46.

Meaning that the factored form of the function is given as follows:

x² + 6x - 3 = (x + 6.46)(x - 0.46).

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