Solve for x. x2 + x - 6 = 0

Use the polynomial remainder theorem: the remainder upon dividing a polynomial [tex]p(x)[/tex] by [tex]x-c[/tex] is [tex]p(c)[/tex].

[tex](-b)^4+3(-b)^2-2(-b)+2=b^4+3b^2+2b+2[/tex]

[tex]((-b)^2-3)^2=b^4-6b^2+9[/tex]

Now

[tex]b^4+3b^2+2b+2=b^4-6b^2+9\implies9b^2+2b-7=0[/tex]

[tex]\implies(9b-7)(b+1)=0[/tex]

[tex]\implies b=\dfrac79\text{ or }b=-1[/tex]

Answer:

54 degrees

Step-by-step explanation:

I would turn this into a quadrilateral by connecting C to O and O to A.

The angles of a quadrilateral add up to 360 degrees.

So we have 90+90+126+angleABC=360

                            180+126+angleABC=360

                                  306+angleABC=360

                                           angleABC=360-306=54

Answer:

ABC = 54°

Step-by-step explanation:

Two tangent have been drawn from an external point B to the circle O.

These tangents touch the circle at points A and C.

Now as per Theorem of arcs and angles.

∠ABC = [tex]\frac{1}{2}[/tex]   [m(major arc AC) - m(minor arc AC)]

          = [tex]\frac{1}{2}[/tex]  [234 - 126]

         = [tex]\frac{1}{2}[/tex]  × (108)

         = 54°

Therefore, ABC = 54°

C

It's a reflection over the y-axis.

Answer:

Both!

Step-by-step explanation:

By both!

You have all 3 corresponding sides are proportional so you have SSS.  That is

24/8=15/5=18/6

You also have SAS because 15/5=18/6 and you have the angle between the sides that I'm referring to in that proportion.

The SSS (Side-Side-Side) theorem can prove triangle similarity if all sides of two triangles are proportional, whereas the SAS (Side-Angle-Side) can prove similarity if two sides and the included angle of one triangle are proportional to those of another triangle. However, these theorems are only applicable if these conditions are met.

In mathematics, particularly in geometry, we often deal with proving the similarity or congruence of shapes, especially triangles. The SSS (Side-Side-Side) and SAS (Side-Angle-Side) theorems are two of those methods used to establish similarity between triangles. However, these theorems need specific conditions to be applicable.

The SSS Similarity theorem states that if the three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar. In other words, if you can match up the sides of one triangle to the sides of the other in such a way that every pair of corresponding sides has the same ratio, then the triangles are similar.

On the other hand, the SAS Similarity theorem states that if an angle of one triangle is congruent to an angle of a second triangle, and the sides including these angles are in proportion, then the triangles are similar.

Given the exact measures of the sides and angles of the triangles, it's possible to use either the SSS or SAS theorem to establish similarity. If, however, these specifics are not available or do not meet the conditions required for SSS or SAS, we cannot use these theorems to prove similarity.

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The ratio of the number of mops to the number of brooms is,

⇒ Ratio = 3 : 4

A ratio indicates how many times one number contain in another number. The ratio of two number is written as x : y, which is equivalent to x/y.

Where, x and y are individual amount of two quantities.

And, Total quantity gives after combine as x + y.

Given that;

There are 8 brooms and 6 mops in a janitor's closet.

Hence, The ratio of the number of mops to the number of brooms is,

⇒ Ratio = number of mops / number of brooms

⇒ Ratio = 6 / 8

⇒ Ratio = 3 / 4

⇒ Ratio = 3 : 4

Thus, The ratio of the number of mops to the number of brooms is,

⇒ Ratio = 3 : 4

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Answer:

3,5

Step-by-step explanation:

Represent the width by W and the length by L.  L = W + 2 (because L and W are consecutive odd integers).

Then L * W = 15 units^2, and after substitution this becomes:

(W + 2) * W = 15, or W^2 + 2W - 15 = 0.

This factors as follows:  (W + 5)(W - 3) = 0, and the positive root is W = 3.

The length and width of the rectangle are 3 and 5 respectively.

Note that 3 and 5 are consecutive odd integers, and that 3 * 5 = 15 units^2.

Final answer:

The length and width of the rectangle are consecutive odd integers that equate to an area of 15 square units, which are 3 units and 5 units, respectively.

Explanation:

The question at hand involves finding the consecutive odd integers that represent the length and width of a rectangle with an area of 15 square units.

Since the area of a rectangle is the product of its length and width, and in this case both are odd integers, we can list the odd number pairs whose product is 15.

The only odd numbers that multiply together to equal 15 are 3 and 5.

Therefore, the dimensions of this rectangle with the consecutive odd integer sides are: length of 5 units and width of 3 units, or vice versa depending on the interpretation, but typically, length is considered to be greater than width in geometry.

slope is [tex]\frac{rise}{run}[/tex] or [tex]\frac{vertical}{horizontal}[/tex]

Look at the image below:

The red (4) is the rise and the blue (1) is the run

this means the slope is...

[tex]\frac{4}{1}[/tex]

4

Hope this helped!

~Just a girl in love with Shawn Mendes

Answer:

5

Step-by-step explanation:

We are given that there are two line segments, red and blue, that stretch from the center of the circle to a point on the circle.

Given that the length of the blue segment is 5, we are to determine the length of the red segment.

Since both the line segments stretch from the center of the circle to its circumference so they must be equal.

Therefore, length of the red line segment is 5.

Answer:

the answer is 5

Step-by-step explanation:

Answer:

8/9

Step-by-step explanation:

This is already in slope-intercept form.

Compare to y=mx+b where m is slope and b is y-intercept and you should see that 8/9 is in place of m and is therefore your slope

Answer:

A   12+6i

    ----------

     5

Step-by-step explanation:

6

-----------

2-i

Multiply by the complex conjugate

6               2+i

----------- *------------

2-i               2+i

12 +6i

-------------

4 -2i +2i -i^2

12+6i

----------

4 - (-1)

12+6i

--------------

5

The trigonometric expressions sin(π/6) is equivalent to the y coordinate of the terminal point (√3/2, 1/2) option (B) is correct.

The trigonometric ratio is defined as the ratio of the pair of a right-angled triangle.

We have:

y coordinate of the terminal point (√3/2, 1/2)

Terminal point = (√3/2, 1/2)

sin(a) = 1/2

a = sin⁻¹(1/2)

a = π/6

a is the angle.

y coordinate is sin(π/6)

Thus, the trigonometric expressions sin(π/6) is equivalent to the y coordinate of the terminal point (√3/2, 1/2) option (B) is correct.

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Final answer:

The trigonometric expression equivalent to the y-coordinate (1/2) of the given terminal point (√3/2, 1/2) is sin(30°) or sin(π/6).

Explanation:

The student is asking for the trigonometric expression equivalent to the y-coordinate of the terminal point (√3/2, 1/2). In trigonometry, the y-coordinate of a point on the unit circle corresponds to the sin of the angle from the positive x-axis to the radius that ends at the point. Since the terminal point given is in the first quadrant and matches the coordinates for a 30° (or π/6 radians) angle in standard position on the unit circle, the y-coordinate is equal to sin(30°) which is 1/2. Therefore, the trigonometric expression equivalent to the y-coordinate 1/2 is sin(30°) or sin(π/6).

Answer:

x² + 4x - 12, remainder - 60

Step-by-step explanation:

Using synthetic division to divide

Since dividing by x - 1, evaluate for x = 1

1 | 1  3  - 16  - 48

   ↓  1     4   - 12

  ---------------------------

   1   4  - 12  - 60 ← degree 2 polynomial

quotient = x² + 4x - 12, remainder = - 60

Hence

[tex]\frac{x^3+3x^2-16x-48}{x-1}[/tex] = x² + 4x - 12 - [tex]\frac{60}{x-1}[/tex]

The division of the given polynomials x³ + 3x² - 16x - 48 by x - 1 using synthetic division results in the polynomial x² + 4x - 12 - 60/(x - 1) with a remainder of -60.

The division of two polynomials can be performed using polynomial long division or synthetic division. In this case, we have a cubic polynomial divided by a linear polynomial: x³ + 3x² - 16x - 48 divided by x - 1.

We can use synthetic division to solve this. Place the coefficients of the dividend (1, 3, -16, -48) in a row and place the zero from the divisor (x - 1= 0, x = 1) to the left. Add down the columns and multiply the result by x in each row, placing the result in the next row.

Following these steps, the coefficients become 1, 4, -12, and -60 which translates to the polynomial x² + 4x - 12 - 60/(x - 1). The remainder (-60) divided by the divisor (x - 1) is added as the last term.

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Answer:

The missing term is 3b

Step-by-step explanation:

step 1

Find the coordinates of point F

Find the midpoint AB

[tex]F(\frac{-3a+3a}{2},\frac{b+b}{2})\\ \\F(0,b)[/tex]

step 2

Find the coordinates of point E

Find the midpoint AC

[tex]E(\frac{-3a-a}{2},\frac{b-5b}{2})\\ \\E(-2a,-2b)[/tex]

step 3

Find the distance EF

the formula to calculate the distance between two points is equal to

[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]

substitute

[tex]EF=\sqrt{(b+2b)^{2}+(0+2a)^{2}}[/tex]

[tex]EF=\sqrt{(3b)^{2}+(2a)^{2}}[/tex]

therefore

The missing term is 3b

Answer:

its 3b thats the answer

ANSWER

C. [tex]g(x) = {(x + 3)}^{2} - 2[/tex]

EXPLANATION

The blue graph is obtained by shifting the red to the left by 3 units and down 2 units.

The vertex is now at (-3,-2).

The base function has equation:

[tex]f(x) = {x}^{2} [/tex]

This implies that,

[tex]g(x) = f(x + 3) - 2[/tex]

[tex]g(x) = {(x + 3)}^{2} - 2[/tex]

The correct answer is C

First add 3 to both sides (what you do on one side you must do to the other). Since 3 is being subtracted, addition (the opposite of subtraction) will cancel it out (make it zero) from the right side and bring it over to the left side.

2 < u - 3

2 + 3 < u - 3 + 3

5 < u - 0

5 < u

For the graph will you have a empty or colored in circle?

If the symbol is ≥ or ≤ then the circle will be colored in. This represents that the number the circle is on is included.

If the symbol is > or < then the circle will be empty. This represents that the number the circle is on is NOT included.

Which direction will the ray go?

If the variable is LESS then the number then the arrow will go to the left of the circle.

If the variable is MORE then the number then the arrow will go to the right of the circle.

In this case your inequality is:

5 < u OR u > 5

aka 5 is less then u OR u is greater then 5

This means that the graph will have an empty circle and the arrow will go to the right of 5. Look at image below.

Hope this helped!

~Just a girl in love with Shawn Mendes

Answer:

correct answer options are ; C and D

Step-by-step explanation:

By definition, vertical angles are those that are opposite each other when two lines cross and are equal.

Observing the given options;

C. ∠LRE and ∠FRA

D.∠TRF and ∠NRL

In the other options given, it could have been correct if stated as;

A. ∠ARS and ∠ERO

B. ∠NRA and ∠TRE

Answer:

Option D (-1,-2)

Step-by-step explanation:

we know that

The ordered pair is 1 unit to the left of the origin and 2 units down from the origin

so

(x,y) ------> (0-1,0-2)

(x,y) ------> (-1,-2)

therefore

The coordinates of the ordered pair is (-1,-2)

Answer: c: (-1,-2)

Step-by-step explanation:

Just got it right on the quiz

The figures that are reflections of ABCD are KLMN, PQRS and WXYZ

The given parameters are

ABCD ≅ KLMN ≅ PQRS ≅ WXYZ

The above parameter can be rewritten as

ABCD ≅ KLMN

ABCD ≅ PQRS

ABCD  ≅ WXYZ

The above parameters mean that:

Using that as a guide, we understand that:

Congruent shapes can be reflections of one another

This means that the figures that are reflections of ABCD are the figures that are congruent to ABCD

Hence, the figures that are reflections of ABCD are KLMN, PQRS and WXYZ

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Answer: b,d

Step-by-step explanation: im smart

For this case we must indicate an expression equivalent to:

[tex](2 ^ 3) ^ {-5}[/tex]

By definition of power properties we have to:

[tex](a ^ n) ^ m = a ^ {n * m}[/tex]

We also have to:

[tex]a ^ {- 1} = \frac {1} {a ^ 1} = \frac {1} {a}[/tex]

Rewriting the expression we have:

[tex](2 ^ 3) ^ {-5} = 2 ^ {-15} = \frac {1} {2 ^ {15}}[/tex]

Answer:

Option A

Answer:

(3, 2)

Step-by-step explanation:

First, we have to eliminate a variable:

3(5x+2y = 19 )

2(4x-3y = 6)

------------------

15x+6y = 57

8x-6y = 12

Now we add the system of equations together:

15x+8x+6y-6y = 57+12

23x = 69

/23   /23

x = 3

Now we can plug this value back into one of the equations to get y:

5(3)+2y = 19

15+2y = 19

-15         -15

2y = 4

/2     /2

y = 2

Therefore, the solution to these system of equations is (3, 2)

Answer:

The solution of given system of equation is (3,2).

Step-by-step explanation:

The given system of equations is

[tex]5x+2y=19[/tex]            .... (1)

[tex]4x-3y=6[/tex]              .... (2)

Solve the system of equations using elimination method.

Multiply equation (1) by 3 and multiply equation (2) by 2.

[tex]15x+6y=57[/tex]            .... (3)

[tex]8x-6y=12[/tex]              .... (4)

Now, add the equations (3) and (4), to eliminate y.

[tex]15x+8x=57+12[/tex]

[tex]23x=69[/tex]

Divide both sides by 23.

[tex]x=3[/tex]

The value of x is 3. Substitute x=3 in equation (1), to find the value of y.

[tex]5(3)+2y=19[/tex]

[tex]15+2y=19[/tex]

Subtract 15 from both the sides.

[tex]2y=19-15[/tex]

[tex]2y=4[/tex]

Divide both sides by 2.

[tex]y=2[/tex]

The value of y is 2.

Therefore the solution of given system of equation is (3,2).

[tex]\dfrac{1}{4}+x=\dfrac{20}{3}\Big|\cdot12\\\\3+12x=80\\\\12x=77\\\\x=\dfrac{77}{12}[/tex]

Answer:

x = 77/12 or 6 5/12

Step-by-step explanation:

To solve this one step equation you have to isolate the variable. To isolate the variable you need to to do the inverse of addition which is subtraction.

So you have to subtract 1/4 by 1/4

And what you do to one side of the equation you have to do to the other side.

So 20/3 - 1/4 = 77/12

So x = 77/12 or change it to a mixed number 6 5/12.

Hope this helped you!!

Answer:

(x-3)(4x+5)

Step-by-step explanation:

4x2−12x+5x−15

=4x(x−3)+5(x−3)

=(x−3)(4x+5)

Answer:

B

Step-by-step explanation:

Given

4x² - 7x - 15

To factorise the quadratic

Consider the factors of the product of the x² term and the constant term which sum to give the coefficient of the x- term.

product = 4 × - 15 = - 60 and sum = - 7

The factors are - 12 and + 5

Use these factors to split the x- term

4x² - 12x + 5x - 15 ( factor the first/second and third/fourth terms )

= 4x(x - 3) + 5(x - 3) ← factor out (x - 3) from each term

= (x - 3)(4x + 5) ← in factored form

Hence 4x + 5 is a factor of the trinomial → B

Answer:

a)8

b)11

Step-by-step explanation:

given to 23-14

23-3=contains meat

a. Monica has [tex]8[/tex] choices to order from because she dislikes Ham.

b. Andrew has [tex]32[/tex] choices to order from because he likes meat but dislikes mixed meats.

a. Since Monica dislikes Ham, she cannot choose any of the 15 choices that contain Ham. She has [tex]\(23 - 15 = 8\)[/tex] choices left to order from.

b. Andrew likes meat but dislikes mixed meats, which means he can choose from the choices that contain only one type of meat (Ham or Chicken) or the vegetarian options. There are [tex]15[/tex] choices with Ham, [tex]14[/tex] choices with Chicken, and [tex]3[/tex] vegetarian choices.

[tex]\text{Total choices with only one type of meat}: \(15 + 14 = 29\)[/tex]

[tex]\text{Total choices with only one type of meat or vegetarian}: \(29 + 3 = 32\)[/tex]

Andrew has [tex]32[/tex] choices to order from.

Answer:

see explanation

Step-by-step explanation:

Calculate the distance (d) using the distance formula

d = √ (x₂ - x₁ )² + (y₂ - y₁ )²

with (x₁, y₁ ) = (3, 5) and (x₂, y₂ ) = (1, a)

d = [tex]\sqrt{(1-3)^2+(a-5)^2}[/tex]

  = [tex]\sqrt{(-2)^2+(a-5)^2}[/tex]

  = [tex]\sqrt{4+(a-5)^2}[/tex]

Equating d to [tex]\sqrt{8}[/tex], gives

[tex]\sqrt{4+(a-5)^2}[/tex] = [tex]\sqrt{8}[/tex]

Squaring both sides

4 + (a - 5)² = 8 ( subtract 4 from both sides )

(a - 5)² = 4 ( take the square root of both sides )

a - 5 = ± [tex]\sqrt{4}[/tex] = ± 2 ( add 5 to both sides )

a = 5 ± 2, that is

a = 5 - 2 = 3 OR a = 5 + 2 = 7

Answer:

Step-by-step explanation:

Here you estimate the proportion of people in the population that said did not have children under 18 living at home.It can also be given as a percentage.

The general expression to apply here is;

[tex]C.I=p+-z*\sqrt{\frac{p(1-p)}{n} }[/tex]

where ;

p=sample proportion

n=sample size

z*=value of z* from the standard normal distribution for 95% confidence level

Given;

n=5000

Find p

From the question 54% of people chosen said they did not have children under 18 living at home

[tex]\frac{54}{100} *5000 = 2700\\\\p=\frac{2700}{5000} =0.54[/tex]

To calculate the 95% confidence interval, follow the steps below;

The value of z* from the table is 1.96

The value of p=0.54 as calculated above

[tex]0.54(1-0.54)=0.2484[/tex]

Divide the value of p(1-p) with the sample size, n

[tex]\frac{0.2448}{5000} =0.00004968[/tex]

[tex]=\sqrt{0.00004968} =0.007048[/tex]

Here multiply the square-root of p(1-p)/n by the z*

[tex]=0.007048*1.96=0.0138[/tex]

The 95% confidence interval for the lower end value is p-margin of error

[tex]=0.54-0.0138=0.5262[/tex]

The 95% confidence interval for the upper end value is p+margin of error

[tex]0.54+0.0138=0.5538[/tex]

Answer: 0.5262 - 0.5538

Step-by-step explanation:

1) Find the standard deviation with the given information:

n=5000

p=54% ⇒ 0.54

1-p = 1 - 0.54 = 0.46

[tex]\sigma =\sqrt{\dfrac{p(1-p)}{n}}\\\\\\.\ =\sqrt{\dfrac{0.54(0.46)}{5000}}\\\\\\.\ =\sqrt{\dfrac{0.2484}{5000}}\\\\\\.\ =\sqrt{0.00004968}\\\\\\.\ =0.007048[/tex]

2) Find the margin of error (ME) with the given information:

C=95% ⇒ Z = 1.960

σ=0.007048

ME = Z × σ

     = 1.96 (0.007048)

     = 0.01381

3) Find the confidence interval with the given information:

p = 0.54

ME = 0.01381

C = p ± ME

    = 0.54 ± 0.01381

    = (0.5262, 0.5538)

Answer:

YZ = XZ

Step-by-step explanation:

Perpendicular Bisector:

A perpendicular bisector of a line segment 'l' is a line that is perpendicular to the line segment 'l' and cuts the line segment 'l' into two equal parts.

Given:

1. A triangle WXY.

2. A perpendicular bisector from vertex W that intersects XY at point Z.

Conclusion based on the drawing:

a. Z is the midpoint of the line segment XY because point Z lies on the perpendicular bisector of XY.

b. Hence, XZ = YZ.

Answer:

Option D.

Step-by-step explanation:

Given information: WXY is a triangle.

Steps of construction:

1. Draw a triangle WXY.

2. Constructs a perpendicular bisector from vertex W that intersects side XY at point Z.

If a line cuts a line segment exactly in half by a 90 degree angle, then it is called a perpendicular bisector.

WZ is a perpendicular bisector on XY. It means point Z divides the side XY in two equal parts.

[tex]YZ=XZ[/tex]

Therefore, the correct option is D.

Answer:

Step-by-step explanation:

[tex]\text{The equation of a horizontal line:}\ y=a.\\\\\text{The equation of a vertical  line:}\ x=a\\\\a-any\ real\ number\\\\\text{We have}\ x=-5.\ \text{It's a vertical line.}\ \text{The line that is perpendicular}\\\text{to the vertical line is a horizontal line.}\\\\\text{The line passes through the point}\ (1,\ \pi)\to x=1,\ y=\pi.\\\\\text{Therefore the equation is}\ y=\pi[/tex]

Answer:

(x) = 2(1/6)^x

Step-by-step explanation:

To easily solve this problem, we can graph each option using a graphing calculator, or any equation plotting tool.

Case 1

f(x) = 2(6)^x

Case 2

f(x) = 1/2*(6)^x

Case 3

f(x) = 2(1/6)^x

Case 4

f(x) = 1/2*(1/6)^x

By looking at the pictures below, we can tell that the correct option is

Case 3

f(x) = 2(1/6)^x

Since the stretch is done by a factor of 2

Answer:

Step-by-step explanation:

you need to double the radius which is 8 cm

then you do 10 * 8 * 3.14 = 251.2

Answer:

-3/4

Step-by-step explanation:

You could identify two points and used the slope formula given two points.

OR

You can just count from one point to another.

First step: Identify two points where the line crosses nicely.

I see one at the y-intercept (0,1) and another at (4,-2). So starting at the y-intercept, how much do we need to travel down to get to (4,-2).  Hopefully you say down 3, so the rise is -3.  Now how much to the right after traveling down 3 from (0,1) to we need to travel to get to (4,-2) .  Count the spaces...4 units right so the run is 4.

The slope is -3/4