Solve each equation by the quadratic formula: 8x^2+15x=7x-4

Always double-check your graph by entering various x values and ensuring that the associated y values fall along the curve you drew.

A graph is a picture of data that illustrates the connection between two or more factors. A graph in mathematics is typically a two-dimensional coordinate system with a horizontal x-axis and vertical y-axis (vertical). Data points are represented on a graph by points that are drawn with their x- and y-coordinates. The graph can be used to display trends, correlations, and temporal shifts as well as other patterns and connections between the variables. Graphs are used to aid in the interpretation and communication of data in a wide range of disciplines, including mathematics, science, economics, and social studies.

given

Locate the vertex: The vertex of a quadratic function is the lowest or highest spot on the graph. For the x-coordinate of the vertex, use the expression x = -b / (2a), and for the y-coordinate, use y = f(x), where f(x) is the quadratic function.

The places where the graph crosses the x-axis and the y-axis are known as the intercepts. If x = 0, then calculate f to determine the y-intercept (0).

Additional information: Plug a few x values into the function to determine the associated y values. You'll have more data points to display on the graph as a result.

Always double-check your graph by entering various x values and ensuring that the associated y values fall along the curve you drew.

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Ben's ball lands approximately 2.5 seconds after Andrew's ball.

Since the value of parameter a is -5 for both balls, the height of each ball follows the equation:

h(t) = -5t² + vt + h0

where;

Let's assume that Andrew's ball is hit with an initial velocity of v1, and Ben's ball is hit with an initial velocity of v₂. We also know that Ben's ball reaches a maximum height 50% greater than Andrew's, which means that:

h_max2 = 1.5h_max1

At the maximum height, the velocity of the ball is zero, so we can find the time it takes for each ball to reach the maximum height by setting v = 0 in the equation for h(t):

t_max1 = v₁ / (2 x 5)

t_max2 = v₂ / (2 x 5)

Since Ben's ball reaches a maximum height that is 50% greater than Andrew's, we can write:

h_max2 = 1.5h_max1

-5(t_max2)² + v₂t_max2 + h0 = 1.5(-5(t_max1)² + v1 * t_max1 + h0)

Simplifying this equation, we get:

-5(t_max2)² + v₂t_max2 = -7.5(t_max1)² + 1.5v₁t_max1

We also know that Andrew's ball lands after 4 seconds, which means that h(4) = 0:

h(4) = -5(4)² + v1 * 4 = 0

-80 + 4v1 = 0

v1 = 80/4

v1 = 20 m/s

Solving these equations for t_max2 and v2, we get:

t_max1 = v1 / (2 x 5)

t_max1 = 20 / (2 x 5)  = 2 s

t_max2 = 1.5 * t_max1 = 3 s

v2 = 1.5 * v1 = 30 m/s

To find the time it takes for Ben's ball to land, we need to find the time t2 when h(t2) = 0.

We can use the equation for h(t) with v = v2, h0 = 0, and solve for t:

-5t² + v₂t = 0

-5t² + 30 = 0

5t² = 30

t² = 30/5

t² = 6

t = √6

t = 2.5 s

Therefore, Ben's ball lands approximately 2.5 seconds after Andrew's ball.

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Hence, we can learn vital information about the behaviour and shape of a polynomial function's graph by scrutinising its factors and other characteristics.

A polynomial function's factors include crucial details about the function graph. The factors specifically determine the locations where the graph intersects the x-axis, known as the function's roots or x-intercepts. By setting each factor to zero and calculating the associated value of x, these roots can be discovered.

The behaviour and structure of the graph are also influenced by the degree of the polynomial function, the signs of its leading coefficient, and other coefficients. The number of turning points in the graph, for instance, is determined by the degree of the polynomial function, whereas the concavity or convexity of the graph is determined by the leading coefficient and other coefficients' signs.

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Answer:

she added the denominator

Step-by-step explanation:

Answer:

She added the numerators and denominators. This is not correct.

Step-by-step explanation:

[tex]x(\frac{2}{3} +\frac{1}{4} )=x(\frac{8+3}{12} )[/tex]

[tex]=\frac{11}{12} x[/tex]

Hope this helps.

The length of segment CB in trapezoid CBAD is 11.

In a trapezoid, the midsegment is the line segment that joins the midpoints of the two non-parallel sides. In this case, KJ is the midsegment of trapezoid CBAD, which means that it is parallel to both CB and AD, and its length is equal to the average of the lengths of CB and AD.

We are given that;

CB=4x-13

KJ=6x-18

DA=25

we can use the formula for the midsegment of a trapezoid to set up an equation and solve for x:

KJ = (CB + AD) / 2

6x - 18 = (4x - 13 + 25) / 2

6x - 18 = (4x + 12) / 2

6x - 18 = 2x + 6

4x = 24

x = 6

Now that we have found the value of x, we can substitute it back into the expression for CB to find its length:

CB = 4x - 13 = 4(6) - 13 = 11

Therefore, the answer of the given trapezoid will be 11.

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The value of x = 44.

Interior angles are those found within a polygon. For instance, a triangle has three interior angles. The phrase "angles limited in the interior area of two parallel lines when intersected by a transversal are known as interior angles" is the third definition of internal angles.

We must use the congruence of the alternate interior angles created by transversal t and parallel lines p and q in order to determine the value of x. Hence, we can construct the equation shown below:

180 - 2x = 3x - 40

When we simplify this equation, we obtain:

5x = 220

x thus equals 44.

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Answer: 0.125 and 12.5%

Step-by-step explanation:

To convert any fraction to decimal form, we just need to divide its numerator by the denominator.

Here, the fraction is 1/8 which means we need to perform 1 ÷ 8.

This gives the answer as 0.125. So, 1/8 as a decimal is 0.125

To get its percentage form, you simply multiply the obtained decimal value by 100:

0.125 x 100 = 12.5

The solution is, volume of cone is  =113.10.

The formula for the volume of a cone is ⅓ r^2h cubic units, where r is the radius of the circular base and h is the height of the cone.

here, given that,

a cone having a radius of 3 inches

and the height of 12 inches.

now, we have to find the volume of the cone having a radius of 3 inches and the height of 12 inches.

we know that,

Volume =  ⅓ r²h

now, substituting the values of r = 3 & h = 12, in formula of volume,

we get,

            V =1/3  9 * 12

             =113.10

Hence, volume of cone is  =113.10

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Answer:

29 months

Step-by-step explanation:
if the trumpet is 360 and he has 285, we need to find out how much he needs to gain to get the trumpet.

360-285=80

If the Intrest is 1% per month, he’s gaining $2.85 per month.
So we divide 80 by 2.85, and get 28.07.

We have to go to the next number because it gets the money at the end of the month.
29 months

Answer:

Step-by-step explanation:

Let's use the variable "a" to represent the number of stamps Angela has.

From the problem, we know that Nicolas has 48 stamps, and that this is 2 times 5 less than the number of stamps Angela has. "2 times 5 less than the number of stamps Angela has" can be written as:

2(a - 5)

So the equation that relates the number of stamps Nicolas and Angela have is:

48 = 2(a - 5)

We can simplify this equation by first distributing the 2:

48 = 2a - 10

Then, we can add 10 to both sides of the equation:

58 = 2a

Finally, we can solve for a by dividing both sides by 2:

a = 29

Therefore, the equation we can use to find the number of stamps Angela has is:

2(a - 5) = 48

or, simplified:

a = 29

The constant of proportionality for the relationship between number of laps (x) and minutes swimming (y) is 5x-2y = 0.

A proportionality graph, also known as a direct proportionality graph, is a graph that represents the relationship between two variables that are directly proportional to each other.

In a proportionality graph, the x-axis represents the independent variable, while the y-axis represents the dependent variable. When the two variables are directly proportional, the graph will form a straight line passing through the origin (0,0). This means that as the independent variable increases, the dependent variable also increases in proportion to it.

Here we by using the graph the relationship is 5x = 2y

Let x is a number of laps and y is the minutes swimming then the constant of proportionality is 5x-2y = 0.

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The length of the segment indicated is 8.

In geometry, two figures or objects are said to be congruent if their shapes and sizes match, or if one is the mirror image of the other.

Here, two right-angled triangles are given. In a right-angled triangle, the sides are the same, and it can be proved by congruence triangle rules.

The two angles are congruent if they are complements of the same angle (also known as congruent angles). Congruent angles are all just angles.

So, the sides' hypotenuse is 8 which is equal to x. So the value of x is 8.

Thus, the length of the segment indicated is 8.

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Answer: 0.72

Step-by-step explanation:

0.62 + 0.10 = 0.72

If a poisson process rate was 1.5 (15 events in 10 min) with a mean of 0.387, then

a) p(no events for 3 min) - 0.0498

b) p(1 event in 1 min) - 0.3347

c) p(>= 1 event in 1 min) - 0.7769

d) uncertainty for a, b, c - For a), σ = 1.732 and for b) and c), σ = 1.225

A Poisson process is a stochastic process that counts the number of events in a given time interval. It is characterized by two parameters: the rate, λ, which is the average number of events in a given time interval, and the mean, μ, which is the average number of events in the entire process.

a) The probability of no events in 3 minutes is given by the Poisson probability mass function:

P(X = 0) = (λt)^0 * e^(-λt) / 0! = e^(-λt)

Where t is the time interval (3 minutes), and λ is the rate (1.5 events per minute). Plugging in the values gives:

P(X = 0) = e^(-1.5 * 3) = 0.0498

b) The probability of 1 event in 1 minute is given by the Poisson probability mass function:

P(X = 1) = (λt)^1 * e^(-λt) / 1! = λt * e^(-λt)

Where t is the time interval (1 minute), and λ is the rate (1.5 events per minute). Plugging in the values gives:

P(X = 1) = 1.5 * e^(-1.5) = 0.3347

c) The probability of at least 1 event in 1 minute is given by the complement of the probability of no events:

P(X >= 1) = 1 - P(X = 0) = 1 - e^(-λt)

Where t is the time interval (1 minute), and λ is the rate (1.5 events per minute). Plugging in the values gives:

P(X >= 1) = 1 - e^(-1.5) = 0.7769

d) The uncertainty for each of the probabilities is given by the standard deviation of the Poisson distribution:

σ = sqrt(λt)

For a), the standard deviation is:

σ = sqrt(1.5 * 3) = 1.732

For b) and c), the standard deviation is:

σ = sqrt(1.5 * 1) = 1.225

Therefore, the uncertainty for each of the probabilities is given by the standard deviation.

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The correlation coefficient relating is 0.76 thath mean  the accompaniment rates for wives and sons is the accompaniment rate for wives increases, the accompaniment rate for sons also tends to increase.

To find the correlation coefficient relating the accompaniment rates for wives and sons, we can use the formula:

r = (nΣxy - (Σx)(Σy)) / √[(nΣx^2 - (Σx)^2)(nΣy^2 - (Σy)^2)]

Where:
- r is the correlation coefficient
- n is the number of observations (in this case, 9)
- x and y are the accompaniment rates for wives and sons, respectively
- Σxy is the sum of the products of x and y
- Σx and Σy are the sums of x and y, respectively
- Σx^2 and Σy^2 are the sums of x squared and y squared, respectively

Using the data from the table, we can calculate the following:

Σx = 27.2 + 40.1 + 35.7 + 37.8 + 38 + 38.4 + 38.7 + 29.8 + 23.8 = 309.5
Σy = 2.2 + 1.5 + 0.3 + 3.5 + 5.4 + 11 + 11.9 + 8.6 + 7.4 = 51.8
Σxy = (27.2)(2.2) + (40.1)(1.5) + (35.7)(0.3) + (37.8)(3.5) + (38)(5.4) + (38.4)(11) + (38.7)(11.9) + (29.8)(8.6) + (23.8)(7.4) = 1164.41
Σx^2 = 27.2^2 + 40.1^2 + 35.7^2 + 37.8^2 + 38^2 + 38.4^2 + 38.7^2 + 29.8^2 + 23.8^2 = 11491.89
Σy^2 = 2.2^2 + 1.5^2 + 0.3^2 + 3.5^2 + 5.4^2 + 11^2 + 11.9^2 + 8.6^2 + 7.4^2 = 349.98

Plugging these values into the formula, we get:

r = (9)(1164.41) - (309.5)(51.8) / √[(9)(11491.89) - (309.5)^2][(9)(349.98) - (51.8)^2] = 0.76

This value of r indicates a strong positive correlation between the accompaniment rates for wives and sons. This means that as the accompaniment rate for wives increases, the accompaniment rate for sons also tends to increase.

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Answer:

x > 5

Step-by-step explanation:

-2x - 2 < 8

-2x < 10

x > 5

So, the answer is x > 5

4/11

The probability of choosing a green marble followed by a brown marble is 4/11. Simple probability is the likelihood that an event occurs, and in this case the likelihood is 4/11. This is calculated by taking the total number of green marbles (4) divided by the total number of marbles in the jar (11).

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The graph and the solution to the system of inequalities are added as attachments

System 1

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

x ≤ -3

y < 5/3x + 2

The above system is a system of inequalities that contain two linear inequalities

Next, we plot the graph

One of the points in the shaded region is (-4, -5)

System 2

Here, we have the following system

y ≤ -5/2x - 2

y < -1/2x + 2

The above system is a system of inequalities that contain two linear inequalities

Next, we plot the graph

One of the points in the shaded region is also (-4, -5)

System 3

Here, we have the following system

y ≥ 2/3x + 3

y > -4/3x - 3

The above system is a system of inequalities that contain two linear inequalities

Next, we plot the graph

One of the points in the shaded region is also (4, 7)

See attachment for the graph of the inequalities

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Answer:

d>500

t>130

Step-by-step explanation:

For the first problem:

we must set up an inequality to answer :D soooo:

d>500

the same thing goes for the second one:

t>130

Hope this is right!

In a right-angled triangle, the sum of two acute angles is equal to 90°.

So if α and β are the two acute angles of a right-angled triangle, then α + β = 90°.Using the formula for sin of the difference of two angles, we can write:sin2α = sin(90° − β) = sin90°cosβ − cos90°sinβ = cosβsin2β = sin(90° − α) = sin90°cosα − cos90°sinα = cosαSince sin2α = cosβ and sin2β = cosα, we can conclude that sin2α = sin2β.Now, let's write 3sinx − 5cosx in the form kcos(x+a). Using the formula for cos of the sum of two angles, we can write:3sinx − 5cosx = kcos(x+a) = k(cosxcos a − sinxsin a)Equating the coefficients of sinx and cosx, we get:−5 = kcos a3 = −ksin aSquaring and adding these equations, we get:25 + 9 = k^2(cos^2a + sin^2a) = k^2So k = √34.Taking the ratio of the two equations, we get:−5/3 = cos a/sin a = 1/tan aSo tan a = −3/5.Using the inverse tangent function, we get:a = tan^−1(−3/5)So the final answer is:kcos(x+a) = √34cos(x + tan^−1(−3/5)).

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The required length of side CD is 22 units.

Since the perimeter of the rectangle is 80 units, we can use the formula for the perimeter of a rectangle, which is:

perimeter = 2(length + width)

In this case, we know that the perimeter is 80 units, so we can write:

80 = 2(length + width)

We also know that CD is a side of the rectangle, so it must be either the length or the width. Let's assume that CD is the length, so we can write:

CD = 4z + 2

Substituting this expression for the length into the formula for the perimeter, we get:

80 = 2(4z + 2 + width)

Simplifying the right-hand side, we get:

80 = 8z + 4 + 2width

Subtracting 4 from both sides, we get:

76 = 8z + 2width

Dividing both sides by 2, we get:

38 = 4z + width

Now we can use the fact that CD is a side of the rectangle to substitute for width. Since the opposite side of the rectangle must also have length 4z + 2, we can write:

width = 3z + 3

Substituting this into the previous equation, we get:

38 = 4z + 3z + 3

Simplifying the right-hand side, we get:

38 = 7z + 3

Subtracting 3 from both sides, we get:

35 = 7z

Dividing both sides by 7, we get:

5 = z

Now we can substitute this value of z into the expression for CD:

CD = 4z + 2 = 4(5) + 2 = 22

Therefore, the length of side CD is 22 units.

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Answer:

Not a function

Step-by-step explanation:

For this to be a function there can not be 2 of the same x values.

Our points for this are (−2, −5), (−1, −3), (−2, 6), (5, 7)

Your x value is the number on the left side of the point=

When we look at our points we see that -2 is repeating for (-2,-5) and (-2,6)

If we plot these points on a graph and use the vertical line test we see that our line passes thru the 2 points.

So this is not a function  

Answer:

[tex]x = \frac{y}{3} - 1[/tex]

Step-by-step explanation:

[tex]1. \: 4y - 8 - 4 = 12x \\ 2. \: 4y - 12 = 12x \\ 3. \: \frac{4y - 12}{12} = x \\ 4. \: \frac{4(y - 3)}{12} = x \\ 5. \: \frac{y - 3}{3} = x \\ 6. \: - 1 + \frac{y}{3} = x \\ 7. \: \frac{y}{3} -1 = x \\ 8. \: x = \frac{y}{3}-1[/tex]

Answer:

Step-by-step explanation:

To find the length of the square side lengths and the diameter of the circle, we can divide the perimeter by 4:

79.04/4 = 19.76

This is the diameter of the circle. The radius is half the diameter:

19.76/2 = 9.88

Hope this helps!

Answer: 9.88 cm

Step-by-step explanation:

To find the radius, we need to find half the length of one side of the square.

The perimeter of the square is all the sides of the square added together.

Squares have 4 sides so divide the perimeter by 4 to find the length of 1 side.

79.04 / 4 = 19.76 is the length of one side.

Half of 19.76 is 9.88 so the radius of the circle is 9.88 cm.

Hope this helps!

(a) After 2 seconds, the ant's position in the  (Euclidean) coordinates is (4,0)

(b) To pass the point (0.999, 0), the ant needs 0.4995 seconds.

The ant is traveling along the positive x-axis at a speed of 2 Poincare units of length per second. This means that its position at any given time can be determined using the equation:

x = 2t

where x is the ant's position along the x-axis, and t is the time in seconds.

(a) To find the ant's position after 2 seconds, we can simply plug in t = 2 into the equation:

x = 2(2) = 4

So the ant's position after 2 seconds is (4, 0).

(b) To find how long it will take the ant to pass the point (0.999, 0), we can set x = 0.999 and solve for t:

0.999 = 2t

t = 0.999/2 = 0.4995

So it will take the ant 0.4995 seconds to pass the point (0.999, 0).

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Step-by-step explanation:

so, let me retype this.

horizontal asymptote : y = 2

that means lim x going to ±infinity f(x) = 2.

vertical asymptotes :

x = 3

x = -4

that means the function must have these 2 points, where the expression leads to a division by 0 or something similar that would make the result undefined.

we got 4 functions :

A) y = x²/(x² + x - 12)

B) y = x²/(x² - x - 12)

C) y = 2x²/(x² + x - 12)

D) y = 2x²/(x² - x - 12)

so, for which ones we have y = 2 as limit when x goes against + or - infinity ?

that would be C and D.

A and B lead to x²/x² = 1 as limit for gigantic numbers.

C and D lead to 2x²/x² = 2 as limit.

remember, when x gets really, really big, the "±x - 12" part becomes irrelevant.

so, we look at C and D.

which one lead to a division by 0 at x = 3 and x = -4 ?

that would be C.

for x = 3

x² + x - 12 = 3² + 3 - 12 = 9 + 3 - 12 = 12 - 12 = 0

for x = -4

x² + x - 12 = (-4)² - 4 - 12 = 16 - 4 - 12 = 12 - 12 = 0

D with x² - x - 12 would have x = -3 and x = 4 as zeroes.

these are different asymptotes than requested.

so, C is the right answer.

Answer: Hayley is hosting a party. She has 50 pieces of candy she wants to give to her 10 friends. How many pieces should each friend get so everyone receives the same amount?

Answer:

4y^2 - 12y + 9

Step-by-step explanation:

(3-2y)^2 = (3-2y) (3-2y)

Distribute:

3(3) + (3)(-2y) + (-2y)(3) + (-2y)(-2y)

= 9 -6y -6y +4y^2

Combine like terms:

4y^2 -12y + 9

I hope this helps!

Answer:

9 - 12y + 4y²

Step-by-step explanation:

(3 - 2y)²

= (3 - 2y)(3 - 2y)

each term in the second factor is multiplied by each term in the irst factor, that is

3(3 - 2y) - 2y(3 - 2y) ← distribute parenthesis

= 9 - 6y - 6y + 4y² ← collect like terms

= 9 - 12y + 4y²

[tex]\qquad \qquad \textit{direct proportional variation} \\\\ \textit{\underline{y} varies directly with \underline{x}}\qquad \qquad \stackrel{\textit{constant of variation}}{y=\stackrel{\downarrow }{k}x~\hfill } \\\\ \textit{\underline{x} varies directly with }\underline{z^5}\qquad \qquad \stackrel{\textit{constant of variation}}{x=\stackrel{\downarrow }{k}z^5~\hfill } \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{\textit{"y" varies directly with "x"}}{y = k(x)}\hspace{5em}\textit{we also know that} \begin{cases} x=400\\ y=10 \end{cases} \\\\\\ 10=k(400)\implies \cfrac{10}{400}=k\implies \cfrac{1}{40}=k\hspace{5em}\boxed{y=\cfrac{1}{40}x} \\\\\\ \textit{when x = 450, what's "y"?}\qquad y=\cfrac{1}{40}(450)\implies y=\cfrac{45}{4}[/tex]