Which system of equations can you use to find the roots of the equation? x3 – 10x = x2 – 6 y = x3 – x2 + 10x + 6 y = 0 y = x3 – x2 + 10x y = 6 y = x3 – 10x y = x2 – 6.
From least to greatest, What are the x–coordinates of the three points where the graphs of the equations intersect? If approximate, enter values to the hundredths. , ,

The equation of the axis of symmetry is [tex]x = 1/2.[/tex]

The definition of the an equation in algebra is a logical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.

A mathematical equation, such as 6 × 4 = 12 x 2, states that two quantities or values are equal. A noun that counts. When two or more components must be taken into account together in order to comprehend or describe the whole situation, this is known as an equations.

To find the equation of the axis of symmetry for the parabola[tex]y = x^2 - x - 6,[/tex]  we first need to find the x-coordinate of the vertex.

We can use the formula [tex]x = -b/2a[/tex] to find the x-coordinate of the vertex. In this case, a = 1 and b = -1, so:

[tex]x = -(-1) / 2(1) = 1/2[/tex]

So the x-coordinate of the vertex is [tex]1/2[/tex].

To find the y-coordinate of the vertex,

x = 1/2

[tex]y = (1/2)^2 - (1/2) - 6 = -25/4[/tex]

So the coordinates of the vertex are [tex](1/2, -25/4).[/tex]

[tex]x = 1/2[/tex]

Therefore, the equation of the axis of symmetry is [tex]x = 1/2.[/tex]

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The required answers are a) 6.5 m, b) Dimensions 8 m, 5 m c) square has the greater area.

(a) Let s be the length of a side of the square. The perimeter of a square is given by 4s. Since the piece of wire used to make the square has length 26 m, we have:

[tex]$\begin{align*}4s &= 26 \s &= \frac{26}{4} = 6.5 \text{ m}\end{align*}[/tex]

Therefore, the length of a side of the square is 6.5 m.

(b) Let l be the length of the rectangle. Then the width of the rectangle is l-3. The perimeter of a rectangle is given by 2(l+w). Since the piece of wire used to make the rectangle has length 26 m, we have:

[tex]$\begin{align*}2(l + (l-3)) &= 26 \2(2l-3) &= 26 \4l - 6 &= 26 \4l &= 32 \l &= 8 \text{ m}\end{align*}[/tex]

Therefore, the length of the rectangle is 8 m and its width is 8-3=5 m.

(c) The area of the square is given by [tex]$s^2$[/tex], which in this case is [tex]$6.5^2 = 42.25$[/tex] square meters. The area of the rectangle is given by lw, which in this case is [tex]$8\times5 = 40$[/tex] square meters. Therefore, the square has the greater area.

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Complete question;

A workman has two pieces of worse, each of length 26 m. One piece is to be bent into a square. The other piece is to be bent into a rectangle whose width is 3 m shorter than its length.

(a) What is the length of a side of the square?

(b) What are the dimensions of the rectangle?

(c) Which shape has the greater area?

Answer:

Therefore, the depth of Challenger Deep to the nearest fathom is 6,466 fathoms, and it is greater than 7 miles.

Step-by-step explanation:

One fathom is equal to 6 feet. To convert 38,797 ft. to fathoms, we divide by 6:

38,797 ft. / 6 = 6,466.17 fathoms

Rounding this to the nearest fathom gives us 6,466 fathoms.

One mile is equal to 5,280 feet. Seven miles would be:

7 mi. x 5,280 ft/mi = 36,960 ft.

Since 38,797 ft. is greater than 36,960 ft., we can conclude that Challenger Deep's depth is greater than 7 miles.

Therefore, the depth of Challenger Deep to the nearest fathom is 6,466 fathoms, and it is greater than 7 miles.

Answer:

  -5

Step-by-step explanation:

You want the minimum value of f(x) = x² +6x +4.

The line of symmetry of a quadratic function f(x) = ax² +bx +c is given by ...

  x = -b/(2a)

The line of symmetry passes through the extreme point. When a > 0, that extreme point is the minimum.

Here, we have a=1, b=6, so the line of symmetry is ...

  x = -6/(2·1) = -3

The extreme point (minimum) is ...

  f(-3) = (-3)² +6(-3) +4 = (-3 +6)(-3) +4 = -9 +4

  f(-3) = -5

The minimum value of f(x) is -5.

As a result, 0.85 or 85% of vacationers are likely to visit either the Magic Kingdom or Epcot Center.

The measurement and study of random events are the focus of the mathematic branch known as probability. It is a means to express how likely an event or series of events are to occur. Probability is stated as a number between 0 and 1, where 0 denotes the impossibility of an event and 1 denotes its certainty. An occurrence that has a probability of 0.5 (or 50%) is equally likely to occur or not. In a variety of disciplines, including statistics, economics, engineering, and science, probability is used to evaluate and forecast the possibility of specific events in light of the information and presumptions at hand.

given

The formula for the union of two events must be used to determine the likelihood that a visitor will go to either the Magic Kingdom or Epcot Center:

P(A or B) equals P(A) + P(B) - P (A and B)

P(A) represents the likelihood that event A will occur, P(B) represents the likelihood that event B will occur, and P(A and B) represents the likelihood that both events will occur at the same time.

Let's describe event A in this scenario as a trip to the Magic Kingdom, and event B as a trip to the Epcot Center. We are aware of:

P(A) = 0.60 (the Magic Kingdom is visited by 60% of tourists)

P(B) = 0.50 (50 percent of tourists go to the Epcot Center)

P(A and B) = 0.25 (25% of tourists go to both places).

The result of the formula is:

P(A or B) is equal to P(A) + P(B) - P(A and B), which is 0.60 + 0.50 - 0.25 = 0.85.

As a result, 0.85 or 85% of vacationers are likely to visit either the Magic Kingdom or Epcot Center.

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The correct option is (d) i.e. by grouping x²(x + 11) -3(x +11).

What are Factors?

Factors are numbers or algebraic expressions that can be multiplied together to obtain a given product.

In algebra, factors are often used to factorize polynomials or other expressions, which involves expressing them as a product of simpler expressions.

One way to determine the factors of x³ + 11x² - 3x - 33 by grouping is:

x²(x + 11) - 3(x + 11)

We can see that the two terms have a common factor of (x + 11), so we can factor it out as follows:

 = x³ + 11x² - 3x - 33

 =x²(x + 11) -3(x +11)

 = (x² - 3) (x+11).

The quadratic factor can be factored further using the difference of squares formula:

(x + 11)(x - √3)(x + √3)

So the factors of x³ + 11x² - 3x - 33 are (x + 11)(x - √3)(x + √3).

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Histogram would be the best for the data representation of the girls by dietitian.

Histogram, it is an approximate representation of the distribution of numerical data. The most popular graph for showing frequency distributions is a histogram. Though it also closely resembles a bar chart, there are significant differences. One of the seven basic quality tools is this useful gathering and analysing information tool.

Variable = the height of 13-year-olds in the state of Texas.

Where the height of each bar represents the frequency of observations.

[tex](2 - 2.5) feet\\\\(2.6 - 3) feet\\(3.1 - 3.5) feet\\(4.1 - 4.5) feet[/tex]

So, The best graphical representation of the dietitian's data would be a histogram.

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The polynomial equation with the roots is P(x) = x^2 - 15x + 56

From the question, we have the following parameters that can be used in our computation:

Roots: x = 8 and x = 7

Using the above as a guide, we have the following:

The equation can be represented as

P(x) = (x - root)

Substitute the known values in the above equation, so, we have the following representation

P(x) = (x - 8)(x - 7)

Evaluate the product

P(x) = x^2 - 8x - 7x + 56

This gives

P(x) = x^2 - 15x + 56

Hence, the equation is P(x) = x^2 - 15x + 56

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Answer:

C: 54-x^2

Step-by-step explanation:

54(decreased=minus)-number^2

Answer:

Step-by-step explanation:

To fill in the table, we can use the following steps:

Calculate the interest for each month, which is the prior balance multiplied by the monthly interest rate (1.75%).

Subtract the monthly interest from the prior balance to get the new balance before the monthly payment.

Subtract the monthly payment from the new balance to get the ending balance.

Month Prior Balance 1.75% Interest on Prior Balance Monthly Payment Ending Balance

1 $5000 $87.50 $325 $4762.50

2 $4762.50 $83.39 $325 $4520.84

3 $4520.84 $79.06 $325 $4276.90

4 $4276.90 $74.50 $325 $4029.23

5 $4029.23 $69.69 $325 $3777.87

Therefore, after 5 months of making $325 payments, the remaining balance on the loan is $3777.87.

The angles measures are m∠1 is 103, m∠2  is 38, m∠3 is 77.

In geometry, an angle is a measure of the amount of turn between two lines or line segments that meet at a common endpoint, called the vertex of the angle. Angles are measured in degrees or radians.

A degree is a unit of measure for angles, where one full circle contains 360 degrees. A right angle, which forms a quarter of a circle, is 90 degrees. Angles that are smaller than a right angle are called acute angles, while angles that are larger than a right angle are called obtuse angles. Angles that add up to a straight line, which forms half of a circle, are called supplementary angles, while angles that add up to a full circle are called explementary angles.

A radian is another unit of measure for angles, where one full circle contains 2π radians. A right angle, which forms a quarter of a circle, is π/2 radians.

Here, m∠5 + m∠2 = 180   [ Supplementary Angles ]

142 + m∠2 = 180

m∠2 = 180 - 142 = 38

Now, m∠2 + m∠3 + m∠4 = 180   [ Angles in a Triangle ]

38 + m∠3 + 65 = 180

m∠3 = 180 - 103

m∠3 = 77

Again, m∠3 + m∠1 = 180   [ Supplementary Angles ]

m∠1 + 77 = 180

m∠1 = 180 - 77 = 103

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The complete question is:

Using the system of equations, we found that the amount to be invested in fund A is $180,000 and in fund B is $320,000.

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 amount in the IRA account = $500,000

Let the amount invested in fund A be x and fund B be y.

x + y = 500,000

The amount received annually for fund A = 0.052x

The amount received annually for fund B = 0.077y

This amount is given as:

0.052x + 0.077y = 34,000

Now we have a system of equations.

x + y = 500,000

0.052x + 0.077y = 34,000

Multiply the first equation by 0.052.

0.052x + 0.052y = 26,000

0.052x + 0.077y = 34,000

Subtracting,

-0.025y = -8000

y = 320,000

Then, x = 500,000 - y = 500,000 - 320,000 = 180,000

Therefore using the system of equations, we found that the amount to be invested in fund A is $180,000 and in fund B is $320,000.

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Answer: We are given a right triangle where the two legs have lengths 2 and 4, and we need to find the length of the hypotenuse (which is denoted by x in the question). We can use the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. That is,

hypotenuse^2 = leg1^2 + leg2^2

Plugging in the values we know, we get:

x^2 = 2^2 + 4^2

Simplifying the right-hand side, we get:

x^2 = 4 + 16

x^2 = 20

Taking the square root of both sides (and noting that x must be positive since it is a length), we get:

x = sqrt(20) = 2sqrt(5)

Therefore, the length of the hypotenuse is 2sqrt(5).

Step-by-step explanation:

Answer: 2252.25 - 1018.50 = P58.75

Step-by-step explanation:

The statement that is most likely true is: "The data in Sample B is clustered around the center where Sample A is more spread out."

A dataset is a collection of information, often represented in a tabular form, that contains one or more variables or pieces of data for a particular set of individuals or observations. Each individual or observation in the dataset is typically represented by a row in the table, while the variables are represented by columns. Datasets can be used to study a wide variety of phenomena, from social and economic trends to scientific experiments and medical studies. They can be collected through a variety of methods, including surveys, experiments, observations, and automated data collection systems, and can be analyzed using statistical methods to gain insights into the underlying patterns and relationships in the data.

Here,

This is because Sample A has a larger range than Sample B, indicating that the data points in Sample A are more spread out across the number line. Additionally, the median for both samples is the same (5), but the mean for Sample A is higher than the mean for Sample B, which could suggest that there are some larger values pulling the mean upward in Sample A. Taken together, these characteristics suggest that Sample A has more variability in its data points, while Sample B is more clustered around the center.

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The radian measure for 315° is (7/4)π radians.

What exactly is a radian measure?

Radian measure is a unit of measurement for angles, where one radian is defined as the central angle subtended by an arc of a circle with a length equal to the radius of the circle. Therefore, to convert degrees to radians, we need to multiply the degree measure by pi/180.

Now,

To convert 315 degrees to radians, we can use the formula:

radians = degrees x π/180

So, we have:

radians = 315 x π/180

Simplifying this expression, we get:

radians = (7/4) x π

Therefore,

             315 degrees is equal to (7/4) π radians.

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Answer:

To find the total interest paid by Brian, we can use the simple interest formula:

I = P * r * t

where I is the interest, P is the principal (the amount borrowed), r is the annual interest rate as a decimal, and t is the time period in years.

For Brian's loan, we have:

I = 1800 * 0.12 * 4 = $864

So Brian paid a total of $864 in interest over the 4-year term.

For Andy's loan, we can use the same formula:

I = 4200 * 0.10 * 2 = $840

So Andy paid a total of $840 in interest over the 2-year term.

To find the difference in interest paid, we subtract the smaller amount from the larger amount:

864 - 840 = $24

Therefore, Brian paid $24 more in interest than Andy. The answer is option D.

Answer: the answer is C. Brian paid $24 more in interest than Andy.

Step-by-step explanation: To find out how much interest each person paid, we can use the formula:

Interest = Principal x Rate x Time

where Principal is the amount borrowed, Rate is the annual interest rate (as a decimal), and Time is the length of the loan (in years).

For Brian's loan:

Principal = $1,800

Rate = 0.12

Time = 4 years

Interest = $1,800 x 0.12 x 4 = $864

For Andy's loan:

Principal = $4,200

Rate = 0.10

Time = 2 years

Interest = $4,200 x 0.10 x 2 = $840

So Andy paid $840 in interest, while Brian paid $864 in interest. To find out how much more Brian paid than Andy, we can subtract:

$864 - $840 = $24

Answer:

17,500

Step-by-step explanation:

Answer:

Let's first recall the Pythagorean identity:

sin²θ + cos²θ = 1

We can use this identity to find the value of sinθ, given that cosθ = 1/7 and θ is in the first quadrant. Since θ is in the first quadrant, both sinθ and cosθ are positive.

cosθ = 1/7

cos²θ = (1/7)² = 1/49

sin²θ + cos²θ = 1

sin²θ + 1/49 = 1

sin²θ = 1 - 1/49 = 48/49

sinθ = √(48/49) = (4/7)√3

Now we can use the definition of cotangent to find cotθ:

cotθ = cosθ/sinθ

Substituting the values we found for cosθ and sinθ, we get:

cotθ = (1/7)/[(4/7)√3] = √3/4

Therefore, cotθ = √3/4 when cosθ = 1/7 and θ is in the first quadrant.

Example of The Transitivity Property that follows

Premise:  If x = 3 and y = x + 2, then y = 5.

Premise: If y = 5 and z = y - 1, then z = 4.

Conclusion: If x = 3, then z = 4.

The transitivity property is a logical property that states that if a=b and b=c, then a=c. This property is commonly used in mathematics to make deductions and draw conclusions.

Example:

Premise: If x = 3 and y = x + 2, then y = 5.

Premise: If y = 5 and z = y - 1, then z = 4.

Conclusion: If x = 3, then z = 4.

Explanation: By the transitivity property, we can substitute the value of y in the second premise with the value of x + 2 from the first premise to get z = (x + 2) - 1, which simplifies to z = x + 1. Then, we can substitute x = 3 to get z = 4.

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Answer:

Listed below.

Step-by-step explanation:

(Let y be the length of the room and x be the width of the room.)

Because the width of the room is 5 feet less than the length of the room, we have:

The area of the room can be solved by multiplying its length and width, so we have x × y or xy = 750.

Substituting x for y - 5 into this equation:

If we expand the left side of the equation, we get:

Subtracting 750 from both sides, we get:

The equation [tex]y^{2}[/tex] - 5y - 750 = 0 can be factored into:

Therefore, 3 equations you can use are:

The best hypothesis based on the given information is Option C: Megan will clear 5'6" on 80% or more of her jumps.

A hypothesis is an educated guess or tentative explanation for a phenomenon or observation that can be tested through further investigation and experimentation. It is a proposed explanation or prediction for an event or set of events that is based on limited evidence or observations, and which serves as the starting point for scientific inquiry.

Based on the given information, Megan cleared 5'6" for 80% of her jumps. To create a hypothesis for the probability of future jumps cleared at 5'6", we need to make an educated guess based on this data.

Option A, stating that Megan will clear 5'6" for 100% of her jumps, is too optimistic and doesn't take into account the possibility of error or difficulty in jumping.

Option B, stating that Megan will clear 5'4" for 100% of her jumps, is not relevant to the given information about Megan's performance at 5'6".

Option D, stating that Megan will clear 5'8" on 80% or more of her jumps, is also not supported by the given information and is too high of a goal for Megan to realistically achieve.

Therefore, the best hypothesis based on the given information is Option C: Megan will clear 5'6" on 80% or more of her jumps. This hypothesis is supported by the fact that Megan has already achieved this level of performance in the past, and it allows for the possibility of some variability or error in future jumps.

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Answer: 48.3

Step-by-step explanation:

Answer:

48.3

Step-by-step explanation:

Answer:

10 miles

the first option

Work Shown:

[tex]17600 \text{ yards} = 17600 \text{ yards}\times\frac{3 \text{ feet}}{1 \text{ yard}}\times\frac{1 \text{ mile}}{5280 \text{ feet}}\\\\= \frac{17600 \times 3\times 1}{1 \times 5280\times 1} \text{ miles}\\\\= \frac{52800}{5280} \text{ miles}\\\\= 10 \text{ miles}\\\\[/tex]

Take notice in the jump from the 1st to 2nd lines, we have the "yards" and "feet" units cancel out.

Answer:

d ≈ 4.5

Step-by-step explanation:

calculate AB using the distance formula

d = [tex]\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }[/tex]

with (x₁, y₁ ) = A (1, 0 ) and (x₂, y₂ ) = B (5, 2 )

d = [tex]\sqrt{(5-1)^2+(2-0)^2}[/tex]

  = [tex]\sqrt{4^2+2^2}[/tex]

  = [tex]\sqrt{16+4}[/tex]

  = [tex]\sqrt{20}[/tex]

  ≈ 4.5 ( to the nearest tenth )

Answer:

x = -14

Step-by-step explanation:

(x+1)² + 120 = (x-3)²

expand each binomial:

x² + 2x + 1 + 120 = x² -6x + 9

combine 'like terms' to simplify:

8x = -112

x = -14