solve for x in x²+8x+15=0 in factorisation method ​

[tex]\mathbf{u} =\langle -6, -10 \rangle \\ \\ \mathbf{v}=\langle 7,8 \rangle \\ \\ \mathbf{u} \cdot \mathbf{v}=(-6)(7)+(-10)(8)=-122 \\ \\ \therefore -3(\mathbf{u} \cdot \mathbf{v})=-3(-122)=\boxed{366}[/tex]

The value of -3(u•v) is 366 when vector u= -6i - 10j and vector v=-6i - 10j is given.

The dot product of two vectors is the sum of the products of their corresponding components.

Here, u= -6i - 10j, from the figure as u goes 6 coordinate left to the y axis and and 10 coordinates below the x axis.

v= 7i + 8j, from the figure as v corrdinates right to the y axis and 8 coordinats upwards to the x axis.

Therefore u.v= (-6i - 10j).(7i + 8j)

                      = -42 - 80

                      = -122

So, -3(u.v)= (-3)×(-122)

                =366

Hence, the value of -3(u•v) is 366

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Answer:

2nd option, x > 1.10

Step-by-step explanation:

[tex]7e^{2x}-5 > 58[/tex]

Add 5 to both sides,

[tex]7e^{2x}-5+5 > 58 +5\\7e^{2x} > 63[/tex]

Divided both sides by 7,

[tex]\frac{7e^{2x}}{7} > \frac{63}{7}[/tex]

[tex]e^{2x} > 9[/tex]

Apply Exponent Rule,

[tex]2x > 2ln(3)[/tex]

[tex]\frac{2x}{2} > \frac{2ln(3)}{2}[/tex]

x > ln(3) or x > 1.09861

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The composition of transformation that maps ABCD to EHGF is "(x, y) → (x', -y') → (x' + 6, y' + 3)".

The basic transformation rules are:

The given quadrilateral ABCD has vertices,

A(-5, 2), B(-3, 4), C(-2, 4), and D(-1, 2)

The transformed quadrilateral EHGF has vertices,

E(1, 1), H(3, -1), G(4, -1), and F(5, 1)

In the map ABCD to EHGF, the transformations that took place are:

1) Reflection over x-axis

2) Translation by (x + a, y + b)

Step 1: Reflection over x-axis; (x, y) → (x, -y)

A(-5, 2) → A'(-5, (-2)) = A'(-5, -2)

B(-3, 4) → B'(-3, (-4)) = B'(-3, -4)

C(-2, 4) → C'(-2, (-4)) = C'(-2, -4)

D(-1, 2) → D'(-1, (-2)) = D'(-1, -2)

So, the reflected quadrilateral is A'B'C'D'

Step 2: Translation by (x, y) → (x + a, y + b);

The reflected quadrilateral A'B'C'D' is now translated by

A'B'C'D' → EHGF

So, (for A'(-5, -2) and E(1, 1))

x + a = 1

⇒ -5 + a = 1

⇒ a = 1 + 5

∴ a = 6

and

y + b = 1

⇒ -2 + b = 1

⇒ b = 1 + 2

∴ b = 3

Thus, the translation is (x + 6, y + 3).

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The equation of the circle can be shown as, x² + y² = 9², or, x² + y² = 81.

The equation of a circle, with the center at the origin and the radius r units, is given as x² + y² = r².

In the question, we are asked to write the equation for a circle centered at the origin with x-intercepts of (-9, 0) and (9, 0).

We can find the radius of the circle using the distance formula,

D = √((x₂ - x₁)² + (y₂ - y₁)²), where (x₁, y₁) and (x₂, y₂) are the endpoints of a line segment.

Radius is the distance between the center and any point on the circle.

Thus, taking (x₁, y₁) as (0, 0), the origin, for the center, and (x₂, y₂) as (9, 0), for any point on the circle, we get the radius as:

r = √((9 - 0)² + (0 - 0)²),

or, r = √(9² + 0²),

or, r = √9² = 9.

Thus, the radius of the circle is r = 9 units.

Thus, the equation of the circle can be shown as, x² + y² = 9², or, x² + y² = 81.

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Answer:

See attached image.

Step-by-step explanation:

The image does not show the graph options.  The attachment sjhhows the graph of y < - 1/3 x + 1.

[tex](\stackrel{x_1}{-10}~,~\stackrel{y_1}{-7})\qquad (\stackrel{x_2}{-5}~,~\stackrel{y_2}{-9}) \\\\\\ \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{-9}-\stackrel{y1}{(-7)}}}{\underset{run} {\underset{x_2}{-5}-\underset{x_1}{(-10)}}} \implies \cfrac{-9 +7}{-5 +10} \implies \cfrac{ -2 }{ 5 }\implies -\cfrac{2}{5}[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{(-7)}=\stackrel{m}{-\cfrac{2}{5}}(x-\stackrel{x_1}{(-10)}) \\\\\\ y+7=-\cfrac{2}{5}(x+10)\implies y+7=-\cfrac{2}{5}x-4\implies y=-\cfrac{2}{5}x-11[/tex]

Step-by-step explanation:

what is the problem ?

the solution of such a system of 2 equations is the point, where they cross.

this point has the same coordinates for both equations, and that is called the solution.

and we can clearly see that point in the graph :

(-1, 1)

formally we can solve that way

y = 2x + 3

y = -x

therefore,

2x + 3 = -x

3x + 3 = 0

3x = -3

x = -1

y = -x = - -1 = 1

so, we get

(-1, 1) as solution.

Answer:

x - 4 = 2y

Step-by-step explanation:

"four less than a number x" therefore; x - 4

"is twice another number y" therefore; 2y

The radius of the sector is 10.27 miles

The area of a sector can be represented as follows;

area of a sector = ∅ / 360 × πr²

where

Therefore,

∅= 25 degrees

area of the sector  = 23 miles²

area of a sector = ∅ / 360 × πr²

23 = 25 / 360 × πr²

cross multiply

23 × 360  = 25πr²

8280 = 25πr²

divide both sides by 25

πr² =  8280 / 25

πr² =  331.2

r² =  331.2 / 3.14

r² = 105.477707006

r = √105.477707006

r = 10.2702336877 miles

Therefore, the radius of the sector is 10.27 miles

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The total cost of the glass for your window now  $174.8

The complete question is added as an attachment

From the attached figure, we have the following parameters:

The area of region 1 is calculated using:

Region 1 = Area of square - Area of circle B

This is calculated as

Region 1 = L^2 - π(d/2)^2

So, we have:

Region 1 = 24^2 - 3.14 * (20/2)^2

Evaluate the quotient

Region 1 = 24^2 - 3.14 * 10^2

Evaluate the exponent

Region 1 = 576 - 3.14 * 100

Evaluate the product

Region 1 = 576 - 314

Evaluate the difference

Region 1 = 262

The cost per square inch of this region is $0.10.

So, we have:

Total cost of region 1 = 262 * $0.10

Total cost of region 1 = $26.10

The area of region 2 is calculated using:

Region 2 = Area of circle B - Area of circle A

This is calculated as

Region 2 = π(D/2)^2 - π(d/2)^2

So, we have:

Region 2 = 3.14 * (20/2)^2 - 3.14 * (18/2)^2

Evaluate the quotient

Region 2 = 3.14 * 10^2 - 3.14 * 9^2

Evaluate the exponent

Region 2 = 3.14 * 100 - 3.14 * 81

Evaluate the product

Region 2 = 314 - 254.34

Evaluate the difference

Region 2 =  $59.66

The cost per square inch of this region is $0.15.

So, we have:

Total cost of region 2 = 59.66 * $0.15

Total cost of region 2 = $8.949

The circle B is divided into sectors.

And the angles of the sectors are 90 and 270 degrees

The area of each sector is

Area = ∅/360 * π(d/2)^2

So, we have:

Circle B = ∅/360 * π(d/2)^2 + ⊕/360 * π(d/2)^2

So, we have:

Circle B = 90/360 * 3.14 * (18/2)^2 + 270/360 * 3.14 * (18/2)^2

Evaluate the quotient

Circle B = 0.25 * 3.14 * 9^2 + 0.75 * 3.14 * 9^2

Evaluate the exponent

Circle B = 0.25 * 3.14 * 81 + 0.75 * 3.14 * 81

Evaluate the product

Circle B = 63.585 + 190.755

The cost per square inch of this region is $0.80 for 1 sector and $0.20 for the rest

So, we have:

Total cost of circle B = 63.585 * 0.80 + 190.755 * 0.20

Total cost of circle B = 89.019

Add the total costs

Total cost of the glass for your window = Region 1 + Region 2 + Circle B

Total cost of the glass for your window = $26.10 + $59.66 + $89.019

Evaluate the sum

Total cost of the glass for your window = $174.779

Approximate

Total cost of the glass for your window = $174.8

Hence, the total cost of the glass for your window now  $174.8

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Answer:

63:70

Step-by-step explanation:

The ratio of apartments to bathrooms in the building is 9:10.

Given that, a building has 63 apartments and 70 bathrooms.

The ratio is defined as the comparison of two quantities of the same units that indicates how much of one quantity is present in the other quantity.

The ratio of apartments to bathrooms is 63:70

= 9:10

Therefore, the ratio of apartments to bathrooms in the building is 9:10.

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Answer:

Option A - 32 cm³

Step-by-step explanation:

Volume of a pyramid is given by the formula :

V = [tex]\frac{lwh}{3}[/tex]

Substituting values given gives us :

V = (4)(4)(6)/3

V = (16)(6)/3

V = 96/3

V = 32

Units will be cm³ as this is volume

Hope this helped and have a good day

Answer:

A. 32 cm³

Step-by-step explanation:

V = (1/3)(area of base)(height)

The base is a square of side 4 cm.

area of base = side × side

V = (1/3)(4 cm × 4 cm)(6 cm)

V = 32 cm³

Number of ants = 6

Number of flies = 12

Number of buck = 1

Number of fawn = 1

Number of doe = 1

Number of squirrels = 3

Number of rabbit = 6

What is the number system?

A system of writing numbers is known as a number system. It is the mathematical notation for consistently employing digits or other symbols to represent the numbers in a particular set. It represents the arithmetic and algebraic structure of the numbers and gives each number a distinct representation.

A system for representing numbers is called a number system. It also defines a set of numbers to represent a quantity and is referred to as the system of numeration.

Given,

Total number of animals Cory saw = 27

Number of insects = [tex]27 \times \frac{2}{3}[/tex]

                               [tex]= 18[/tex]

Remaining number of animals = [tex]27-18[/tex]

                                                   = [tex]9[/tex]

From the insects-

Number of ants = [tex]18 \times \frac{1}{3}[/tex]

                          [tex]=6[/tex]

Number of flies = [tex]18-6[/tex]

                          = [tex]12[/tex]

Number of buck  = 1

Number of fawn = 1

Number of doe = 1

Let the number of squirrels be [tex]x[/tex]

Then, the number of rabbits = [tex]2x[/tex]

Remaining animals = [tex]2x+x[/tex]

⇒ [tex]9=3x[/tex]

⇒ [tex]x= 3[/tex]

Therefore,

Number of squirrels = 3

Number of rabbits = [tex]2x[/tex]

                               = 6

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The probability that the outcome is red then yellow is 0.036.

Given that in a jar there are 8 red marbles, 1 yellow marble and 6 green marbles.

We are required to find the probability of obtaining a red then yelow marble if two marbles are drawn and replacement is used.

Probability is the calculation of finding the chance of happening a event among all the events possible. It lies between 0 and 1.

Probability=Number of items/ Total items.

Number of red marbles=8

Number of yellow marble=1

Number of green marbles=6

Total marbles=15

Probability of obtaining red marble=8/15

Probability of obtaining yellow marble=1/15 (Because replacing is there so the number of total marbles do not decrease)

Required probability=8/15*1/15

=8/225

=0.035555

After rounding off we will get 0.036.

Hence the probability that the outcome is red then yellow is 0.036.

Question is incomplete. The following line should be included in question:

A jar contains 8 red marbles , a yellow marble and 6 green marbles.

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The surface area of the prism is 256in².

Given that the sides of the prism is 5in, 12in and 4in.

The total area occupied by the surfaces of an object is called the area.

Here, l=12in, b=5in and h=4in

Firstly, we will find the area of the base, we get

B=l×b

B=12×5

B=60in²

Now, we will find the perimeter of the base, we get

P=2(l+b)

P=2(12+5)

P=2×17

P=34in

Further, we will find the surface area of the prism by using the formula SA=2B+Ph.

Here, we will substitute the values from above, we get

SA=2×60in²+34in×4in

SA=120in²+136in²

SA=256in²

Hence, the surface area of the prism where the sides of the prism is 5in, 12in and 4in is 256in².

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[tex]x = \frac{\sqrt{5} + 1}{2} \approx 1.618033989[/tex]

Answer:

m=-3/7

Step-by-step explanation:

m=rise/run=Δy/Δx

m=y2−y1/x2−x1

m=10−7/−4−3

m=3/−7

m=−3/7

Answer:

Step-by-step explanation:

The first three terms of the arithmetic sequence for the number of words written are 10010, 9760, 9510. These have a common difference of 9760-10010 = -250. The first term and common difference can be used to make the explicit equation for the words written on the n-th day.

The explicit formula for the n-th term of an arithmetic sequence with first term a₁ and common difference d is ...

  [tex]a_n=a_1+d(n-1)[/tex]

For first term a₁ = 10010 and common difference d = -250, the explicit rule is ...

  [tex]a_n=10010-250(n-1)[/tex]

On day 14, n=14, and the number of words written is ...

  [tex]a_{14}=10010-250(14-1)\\\\a_{14}=10010-250(13)=10010-3250\\\\a_{14}=6760[/tex]

The writer would write 6760 words on the 14th day.

Based on the density and the given parameters, the density of the sheet of glass in kg per m cubed is 2500 kg/m cubed

The density of the sheet of glass  is given as

Density = 2.5g/cm cubed

Start by converting grams (g) to kilogram (kg)

This is done by dividing 2.5 by 1000

So, we have:

Density = 2.5/1000 kg/cm cubed

Evaluate the quotient

Density = 0.0025 kg/cm cubed

Next, we then convert cm cubed (cm^3) to meter cubed (m^3)

This is done by dividing the volume by 1000000

So, we have:

Density = 0.0025 kg/(1/1000000) m cubed

Express as product

Density = 1000000 * 0.0025 kg/m cubed

Evaluate the product

Density = 2500 kg/m cubed

Based on the density and the given parameters, the density of the sheet of glass in kg per m cubed is 2500 kg/m cubed

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Answer:

See below

Step-by-step explanation:

Are you wanting to convert density of   gm/ cm^3   to    kg/m^3 ??

    there are 100 x 100 x 100  cm^3 =  1,000,000  cm^3/ m^3

2.5 gm / cm^3   *  1 000 000 cm^3 / m^3   =  2, 500 000 gm / m^3

   now 1 kg is   1000 gm

2, 500 000 gm / m^3   *  1 kg / 1000 gm  = 2500 kg / m^3

The distance of the spaceship in discuss as in the task content given can be evaluated as; 800miles.

The distance travelled by the spaceship in discuss can be evaluated by means of the slope of the linear relationship as follows;

Hence it follows from ratios that by observation, the linear relationship has a slope of 200mi/min.

Consequently, we can evaluate the distance travelled after 4 minutes as;

Distance = 200 × 4 = 800mi.

Ultimately, the distance travelled per minute by the spaceship is; 800mi.

Remarks:

600 miles

520 miles

800 miles

1,080 miles

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Answer: C

Step-by-step explanation:

The red graph is the result of reflecting the graph of y=f(x) across the x-axis and then translating 1 unit to the right.

The time he saves when walking diagonally is 8 seconds

A rectangle has opposite sides equal to each other.

The angles are equal.

Therefore,

perimeter of a rectangle = 2l + 2w

where

Therefore,

distance of the diagonal  =√ 50² + 120²

distance of the diagonal  = √2500 + 14400

distance of the diagonal = √16900

distance of the diagonal = 130 feet

distance if he follows the length of the rectangle = 50 + 120 = 170 feet

Hence,

Time spent diagonally

5 = 130 / t

t = 130 / 5

t = 26 seconds

Time spent along the perimeter

5 = 170 / t

t = 170 / 5

t = 34 secods

Therefore,

Time he will save by walking diagonally = 34 - 26 = 8 seconds

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The trigonometric function that represents the curve seen in the picture is f(x) = 4.5 · sin (π · x / 2 - π) - 6.5.

In this problem we need to find a sinusoidal expression that models the curve seen in the picture. The most typical sinusoidal model is described below:

f(x) = a · sin (b · x + c) + d    (1)

Where:

Now we proceed to find the value of each variable:

Amplitude

a = - 2 - (-6.5)

a = 4.5

Angular frequency

b = 2π / T, where T is the period.

0.25 · T = 4 - 3

T = 4

b = 2π / 4

b = π / 2

Midpoint

d = - 6.5

Angular phase

- 2 = 4.5 · sin (π · 4/2 + c) - 6.5

4.5 = 4.5 · sin (π · 4/2 + c)

1 = sin (2π + c)

π = 2π + c

c = - π

The trigonometric function that represents the curve seen in the picture is f(x) = 4.5 · sin (π · x / 2 - π) - 6.5.

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Answer:

-9

Step-by-step explanation:

To find the y intercept, substitute 0 in for x.

So, the equation looks like this:

[tex]y=6(0-\frac{1}{2} )(0+3)[/tex]

Now, just solve for y and you will get y = -9

The y-intercept of the graph of the equation is y = -9 which is the correct answer that would be an option (A).

The general equation of a line is y = mx + c

where m is the slope of the line and c is the intercept.

A linear equation is defined as an equation in which the highest power of the variable is always one.

Given equation as:

y = 6 (x - 1/2)(x+3)

To determine the y-intercept of the graph of the equation

Substitute the value x = 0 in the equation,

y = 6 (0 - 1/2)(0+3)

Apply the arithmetic operation and solve for y

y = -9

Hence, the y-intercept of the graph of the equation is y = -9.

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