Lesson Review

Directions: Factor the following trinomials containing negative numbers. Follow the rules for operations with signed numbers to identify the correct binomial factors.

1. x2 + 6x - 7

2. x2 - 5x + 6

3. x2 - 6x - 7

4. x2 + 5x - 6

5. x2 + x - 12

6. x2 - 2x - 8

7. x2 - 4x - 5

8. x2 + 2x - 3

9. x2 - 16

10. x2 - x - 12

11. x2 -9x + 18

12. x2 -5x - 14

13. x2 - 7x + 10

14. x2 -11x + 24

15. x2 - x - 30

Answer:

Step-by-step explanation:

To find the unit rate, we need to divide the distance traveled by the time taken.

In this case, the distance traveled is 9 miles, and the time taken is 36 minutes.

Unit rate = distance/time

Unit rate = 9 miles / 36 minutes

Simplifying the above expression by dividing both numerator and denominator by 9, we get:

Unit rate = 1 mile / 4 minutes

Therefore, the unit rate at which the go-kart is traveling is 1 mile per 4 minutes.

The translation used to create the image A'B'C'D', from the pre image ABCD is T(7, 0), which corresponds with the option;

7 units to the right

The coordinates of the vertices of the polygon ABCD are;

A(-2, -1), B(0, -4), C(4, -4), D(2, -1)

The coordinates of the vertices of the polygon A'B'C'D' are;

A'(5, -1), B'(7, -4), C'(11, -4), D'(9, -1)

The coordinates of the vertices of the image indicates;

The difference between the coordinates are;

A'(5, -1) - A(-2, -1) = (5 - (-2), -1 - (-1)) = (7, 0)

B'(7, -4) - B(0, -4) = (7 - 0, -4 - (-4)) = (7, 0)

C'(11, -4) - C(4, -4) = (11 - 4, -4 - (-4)) = (7, 0)

D'(9, -1) - D(2, -1) = (9 - 2, -1 - (-1)) = (7, 0)

The translation that is used to create the image is therefore;

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After considering the given interval we reach the conclusion that the coefficient of t² is 1/2(3w² - 1), under the condition that the given expression is [tex](1-2 t w+t^{2})^{-1/2}[/tex] with range of (t<<1)

To evaluate the value for the given expression we have to apply the principles of binomial theorem

Then

[tex](1-2 t w+t^{2})^{-1/2} = (1 + (-2 t w + t^{2})/2 + (-2 t w + t^{2})^{2/8} + (-2 t w + t^{2})^{3/16} + ...)[/tex]

= 1 - t w + 3/8 * t² * w² - 5/16 * t³ * w³ +

The coefficient of t is the coefficient of the first term with a power of t.

Therefore, the coefficient of t is 1/2(3w² - 1).

The binomial theorem refers to the statement regarding any positive integer n, the nth power of the sum of two numbers a and b could be expressed as the sum of n + 1 terms of the form. The binomial theorem is applied in algebra and probability theory.

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The following property must be given to prove that ΔCJG ~ ΔCEA: B. GJ║AE.

In Mathematics and Geometry, two triangles are said to be similar when the ratio of their corresponding side lengths are equal and their corresponding angles are congruent.

Additionally, the lengths of three (3) pairs of corresponding sides or corresponding side lengths are proportional to the lengths of corresponding altitudes when two (2) triangles are similar.

Based on the side, side, side (SSS) similarity theorem, we can logically deduce the following congruent and similar triangles:

ΔCJG ≅ ΔCEA (line segment GJ is parallel to line segment AE)

ΔBIJ ≅ ΔBDF (line segment IJ is parallel to line segment DF).

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The height of the trapezoid is approximately 13.043 mm.

To find the height of a trapezoid, you can use the formula:

Area = 1/2 * (base1 + base2) * height

Given:

Area = 150 square millimeters

base1 = 7 mm , w

base2 = 16 mm

150 = 1/2 * (7 + 16) * height

150 =1/2 * 23 * height

Make height the subject of the formula

300 = 23 * height

height = 300 / 23

height = 13.043 mm (rounded to three decimal places)

Therefore, the height of the trapezoid is approximately 13.043 mm.

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Answer:

The correct answer is (d): (x - 5) is a factor of f(x).

Step-by-step explanation:

Answer:

The other sides = 5 centimeters.

Step-by-step explanation:

Let's assume the length of the other two sides of the rectangle is "x" centimeters.

A rectangle has two pairs of equal sides. Therefore, we can set up the equation for the perimeter of the rectangle:

Perimeter = 2(length + width)

Given:

Length = 6 centimeters

Perimeter = 22 centimeters

Using the equation, we can substitute the given values:

22 = 2(6 + x)

Simplifying further:

22 = 12 + 2x

Subtracting 12 from both sides:

10 = 2x

Dividing both sides by 2:

5 = x

Therefore, the length of the other two sides of the rectangle is 5 centimeters.

Answer:

Step-by-step explanation:

area of a rectangular piece of cloth 3 1/4m by 25 cm

the area of ​​the rectangle is found by making the base by the height, we convert the meters into centimeters

3 1/4 m = 3.25m = 325 cm

we find the area

325 × 25 = 8125 cm² (or 0.8125 m²)

The difference in height is 400 metres.

Height refers to vertical distance from the top to the object's base. Occasionally, it is also labeled as an altitude which measures from down to top of a surface.

To get difference in height, we will subtract the height after the eruption from the height before the eruption:

Given:

Height before eruption = 2.95 kilometers

Height after eruption = 2.55 kilometers

Difference in height = Height before eruption - Height after eruption

Difference in height = 2.95 km - 2.55 km

Difference in height = 0.4 kilometers

0.4 kilometers to metres will be:

= 0.4 * 1000 metres

= 400 metres.

Full question:

Mount St. Helens, located in Washington, erupted on May 18, 1980. Before the eruption, the volcano was 2.95 kilometers high. After the eruption, the volcano was 2.55 kilometers high. Find the difference in height of Mount St. Helens before and after the eruption.

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To find the measure of angle AB, we can use the fact that the sum of the angles in a triangle is 180 degrees.

Given:

Angle A = 65 degrees

Angle B = 91 degrees

We can subtract the sum of these two angles from 180 degrees to find angle AB:

Angle AB = 180 - (Angle A + Angle B)

Angle AB = 180 - (65 + 91)

Angle AB = 180 - 156

Angle AB = 24 degrees

Therefore, the measure of angle AB is 24 degrees.

Sue's total pay for the year based on the values will be £38110

Sue's total pay of year will have three parts

Fixed Salary for 12 months

 one month salary = £1410

 so 12 month salary = £1410 x 12 = £16920

26% of total profit

  Total cost to the company- £473,500

  Total income for the company - £549,000

  Profit =  Total income for the company - Total cost to the company

            =  £549,000 - £473,500 = £75500

   Sues income from profit = 26% of £75500 = (26 × 75500)/100 = £19630

Bonus if Sue sells at least 16 cars

  Given number of months when sue solds atleast 16 cars = 4

  So bonus income = 4 × 390 = £1560

Adding the three above parts

Sue's total pay for the year = £16920 + £19630 + £1560 = £38110

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Complete question

Ace autos sell cars and pay sue each month as follows

-salary of £1410

-26% of the total profit the company makes from her sales

- a £390 bonus if she sells at least 16 cars

The table shows information about the cars she sold last year

Total cost to the company- £473,500

Total income for the company - £549,000

Number of months when she sold at least 16 cars - 4

work out sues total pay for the year

Answer:

48 prime factors are: 2 x 2 x 2 x 2 x 3

The expression that equals D + 3C in standard form is 2w² + 8w - 25.

To find an expression that equals D + 3C in standard form, we first need to simplify D and C.

Starting with D = 2w² - w - 7, we can rearrange the terms to put it in standard form:

D = 2w² - w - 7

D = 2w² - 2w + w - 7

D = 2w(w - 1) + (w - 7)

Next, simplifying C = 3w - 6:

C = 3w - 6

Now, we can substitute these expressions into D + 3C:

D + 3C = (2w² - w - 7) + 3(3w - 6)

Expanding and simplifying:

D + 3C = 2w² - w - 7 + 9w - 18

D + 3C = 2w² + 8w - 25

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Answer:

  B, C, D

Step-by-step explanation:

You want to identify the shapes that are similar to polygon K.

Polygon K is a right triangle that has leg lengths in the ratio 4:2 = 2:1.

Polygons B, C, and D are right triangles similar to polygon K:

Polygons A and E are also right triangles, but have different ratios of leg lengths. In A they are 1/1, and in E they are 4/3. These triangles are not similar to triangle K.

__

Additional comment

Polygons similar to the same polygon are similar to each other. Rigid transformations such as reflection and rotation result in figures congruent to the original. Congruent figures are also similar.

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Answer: 360 degrees.

Step-by-step explanation: The sum of the interior angles in a quadrilateral is always 360.

(you can check using the formula 180(n-2), where n is the number of sides.)

1. All angles of a rectangle are right angles (90 degrees), while the angles of a parallelogram can be any angle other than 90 degrees.

2. The opposite sides of a rectangle are equal in length, whereas in a parallelogram, opposite sides are of equal length as well, but they may not be perpendicular to each other.

The questions are a bit unclear and have some spelling errors, but I will try my best to provide an answer:

- Mention the transformation of the brick when it is flipped or turned sideways:

If the brick is flipped over, the transformation it undergoes is called a reflection. This can also be thought of as a mirror image of the original brick. If the brick is turned sideways, the transformation it undergoes is called a rotation. In a rotation, the object is turned about a point, and the image of the object appears to have been rotated around that point.

- When the key is pushing the brick, without using or flipping the brick, what... (I'm not sure what the question is asking for, as it seems incomplete or there is some text missing.)

- What are the two differences between a rectangle and a parallelogram?

  In summary, the transformation of a flipped brick is a reflection, while a turned sideways brick undergoes a rotation. The differences between a rectangle and a parallelogram include the angles (all right angles for a rectangle and any angle other than 90 degrees for a parallelogram) and the opposite sides (equal length and perpendicular for a rectangle, equal length but not necessarily perpendicular for a parallelogram).

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The graph of the function is attached and the amplitude and the period are 5 and 2π

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

f(x) = 5cos(θ)

A sinusoidal function is represented as

f(x) = Acos(B(x + C)) + D

Where

So, we have

A = 5

Period = 2π/1

Evaluate

A = 5

Period = 2π

Hence, the amplitude is 5 and the period is 2π

The graph of the function is attached

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The values of the composite functions are

(1) (f + g)(x) = 4x² + 2x - 2

(2) (fg)(x) = 8x³ - 4x² - 2x + 1

(3) (fⁿ)(x) = (4x² - 1)ⁿ

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

f(x) = 4x² - 1

g(x) = 2x - 1

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

(f + g)(x) = f(x) + g(x)

(f + g)(x) = 4x² - 1 + 2x - 1

Evaluate the like terms

(f + g)(x) = 4x² + 2x - 2

(fg)(x) = f(x) * g(x)

(fg)(x) = (4x² - 1) * (2x - 1)

Evaluate the products

(fg)(x) = 8x³ - 4x² - 2x + 1

(fⁿ)(x) = (f(x))ⁿ

So, we have

(fⁿ)(x) = (4x² - 1)ⁿ

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The probability that either event will occur is 0.62

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

Event A = 6 + 6 = 12

Event B = 20 + 6 = 26

Both A and B = 6

Other Events = 20

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

Total = A + B + C + Others - Both

So, we have

Total = 12 + 26 - 6 + 20

Evaluate

Total = 52

So, we have

P(A) = 12/52

P(B) = 26/52

Both A and B = 6/52

For either events, we have

P(A or B) = (12 + 26 - 6)/52

Evaluate

P(A or B) = 0.62

Hence, the probability that either event will occur is 0.62

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Answer:

  See attached

Step-by-step explanation:

You want the given numbers classified as to Natural, Whole, Integer, Rational, Irrational, and/or Real.

All the given numbers are real.

The number √2 is irrational. All the others are rational.

The values √16 = 4, 0, and -5 are integer values. Whole numbers are non-negative integers, so exclude -5. Natural numbers are positive integers, so exclude 0 and -5.

The Xs indicate the categories each number belongs to.

Answer:

Step-by-step explanation:

148 people

The equation in slope-intercept form is,

y = - 2x + 2

We have to given that;

The point-slope equation is,

y - 6 = - 2 (x + 2)

Now, We can rewrite the equation in slope-intercept form as;

y - 6 = - 2 (x + 2)

y - 6 = - 2x - 4

y = - 2x - 4 + 6

y = - 2x + 2

Thus, the equation in slope-intercept form is,

y = - 2x + 2

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The distance that the plane traveled from point A to point B is given as follows:

10,750 ft.

The three trigonometric ratios are the sine, the cosine and the tangent, and they are defined as follows:

The altitude of 5375 feet is the opposite angle, hence the position A is obtained as follows:

tan(15º) = 5375/A

A = 5375/tangent of 15 degrees

A = 20060 ft.

The position B is obtained as follows:

B = 5375/tangent of 30 degrees

B = 9310 ft.

Hence the distance is of:

20060 - 9310 = 10,750 ft.

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The minimum grade needed to have overall grade of 85% in the class is 90%

A percentage is a number or ratio expressed as a fraction of 100. It is represented by by the symbol %.

Percentage can also be said as per 100

For example if there are total of 100 students in a class and their are 40 girls, the percentage of girls in the class is

40/100 × 100

= 40%

For the student to have a overall grade of 85%, the minimum grade in exam is calculated as

(100+70+80+x)/4 = 85

250+x = 340

x = 340 - 250

x = 90%

Therefore the minimum graded needed is 90%

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The equation of the graph in the required form is y = 5|x| - 5

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

The graph (see attachment)

The graph is an absolute value graph

And it can be expressed in the form of y = a|x – h| + k

Where

Vertex = (h, k)

The vertex of the graph is (0, -5)

So, we have

y = a|x| - 5

Using the point on the graph, we have

a|1| - 5 = 0

So, we have

a = 5

This means that the equation of the graph is y = 5|x| - 5

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g is a function because each student is uniquely mapped to each course the student takes in a school year.

In Mathematics and Geometry, a function refers to any mathematical equation which is typically used to define and represent a relationship that exists between two or more variables such as an ordered pair, points on a graph or table.

Based on the description of g, we can reasonably infer and logically deduce that the relation g represent a function because the input values are uniquely mapped to the output values.

In this context, we can conclude that the description of g represents a function because a student represent the input values that is being to each course (output value) the student takes in a school year.

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Given statement solution is :- Regardless of the values of T, Q, and S, the whole Valid Statements & Implications is always true.

Let's analyze the two given statements:

¬{[Q → (¬S ∧ R)] ∨ ¬P} → (¬P ↔ P) ∴ ¬(Q ∧ P) ∨ (T ∧ R)

To prove this statement, we can use logical equivalences and deductions:

¬{[Q → (¬S ∧ R)] ∨ ¬P} → (¬P ↔ P) (Given)

¬{[¬Q ∨ (¬S ∧ R)] ∨ ¬P} → (¬P ↔ P) (Implication equivalence)

¬{[(¬Q ∨ ¬S) ∧ (¬Q ∨ R)] ∨ ¬P} → (¬P ↔ P) (Implication equivalence)

¬{(¬Q ∨ ¬S ∨ ¬P) ∧ (¬Q ∨ R ∨ ¬P)} → (¬P ↔ P) (De Morgan's Law)

(Q ∧ S ∧ P) ∨ (¬Q ∧ P) → (¬P ↔ P) (De Morgan's Law)

At this point, the formula ¬P ↔ P is a contradiction. The left side (¬P) states that P is false, while the right side (P) states that P is true. Therefore, the whole statement is always true, regardless of the values of Q, S, and P.

Since the statement is always true, we can conclude ¬(Q ∧ P) ∨ (T ∧ R) is true as well, making the second part of the question valid.

[(T ∨ Q) → (T ∨ S)] ∨ [(T ∨ Q) → (T ∨ S)]

In this statement, we have two identical sub-statements connected by a logical OR operator.

If we assume that (T ∨ Q) → (T ∨ S) is true, then the whole statement is true, regardless of the values of T, Q, and S.

If we assume that (T ∨ Q) → (T ∨ S) is false, then we need to check the other part of the statement.

Assuming (T ∨ Q) → (T ∨ S) is false means that (T ∨ Q) is true, but (T ∨ S) is false. In this case, the second part of the statement [(T ∨ Q) → (T ∨ S)] is true.

Therefore, regardless of the values of T, Q, and S, the whole Valid Statements & Implications is always true.

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The integers that satisfy the inequality are -1, 0, 1, and 2.

To find the integer values that satisfy the inequality x < 6/(x - 1), we can start by analyzing the domain of the expression.

First, note that the denominator (x - 1) cannot be equal to zero since division by zero is undefined. Therefore, we must exclude x = 1 from the domain.

Next, consider the sign of the expression 6/(x - 1). When the denominator (x - 1) is positive, the expression will be positive. Similarly, when the denominator is negative, the expression will be negative.

Considering the inequality x < 6/(x - 1), we can divide the problem into two cases based on the sign of (x - 1):

Case 1: (x - 1) > 0

In this case, the expression 6/(x - 1) is positive. To satisfy the inequality x < 6/(x - 1), x must be less than the positive value of 6/(x - 1). Since the denominator is positive, we can drop the absolute value sign. Therefore, the inequality becomes:

x < 6/(x - 1)

Case 2: (x - 1) < 0

In this case, the expression 6/(x - 1) is negative. To satisfy the inequality x < 6/(x - 1), x must be greater than the negative value of 6/(x - 1). Since the denominator is negative, we need to flip the inequality sign and change the direction. Therefore, the inequality becomes:

x > 6/(x - 1)

Now, let's solve each case separately:

Case 1: (x - 1) > 0

Since the denominator (x - 1) is positive, we can multiply both sides of the inequality by (x - 1) without changing the direction of the inequality. This gives:

x(x - 1) < 6

Expanding the left side:

x² - x < 6

Rearranging the inequality:

x² - x - 6 < 0

Now we can factorize the quadratic:

(x - 3)(x + 2) < 0

To determine the solution, we need to consider the sign of the expression for different intervals:

When x < -2, both factors are negative, so the expression is positive.

When -2 < x < 3, the first factor (x - 3) is negative, and the second factor (x + 2) is positive, so the expression is negative.

When x > 3, both factors are positive, so the expression is positive.

Therefore, the solution to Case 1 is:

-2 < x < 3

Case 2: (x - 1) < 0

Since the denominator (x - 1) is negative, we need to flip the inequality sign. Multiplying both sides of the inequality by (x - 1) changes the direction of the inequality. This gives:

x(x - 1) > 6

Expanding the left side:

x² - x > 6

Rearranging the inequality:

x² - x - 6 > 0

Factoring the quadratic:

(x - 3)(x + 2) > 0

Considering the sign of the expression for different intervals:

When x < -2, both factors are negative, so the expression is positive.

When -2 < x < 3, the first factor (x - 3) is negative, and the second factor (x + 2) is positive, so the expression is negative.

When x > 3, both factors are positive, so the expression is positive.

Therefore, there are no solutions in Case 2.

Combining the solutions from both cases, we find that the integer values satisfying the inequality x < 6/(x - 1) are:

-2 < x < 3

In other words, the integers that satisfy the inequality are -1, 0, 1, and 2.

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Answer: To determine which statement is true about lines r and s, we need to compare their slopes.

The slope of a line can be calculated using the formula:

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

For line r, using the points (-5, 2) and (-3, 8):

slope_r = (8 - 2) / (-3 - (-5))

= 6 / 2

= 3

For line s, using the points (-6, 4) and (2, 12):

slope_s = (12 - 4) / (2 - (-6))

= 8 / 8

= 1

Now, let's analyze the statements:

Line r is steeper than line s. (False)

Since the slope of line r (3) is greater than the slope of line s (1), this statement is true.

Line r is steeper than line (Incomplete statement)

This statement is incomplete.

Line s has a negative slope. (False)

The slope of line s (1) is positive, not negative. So, this statement is false.

Line s is steeper than line r. (False)

Since the slope of line s (1) is less than the slope of line r (3), this statement is false.

Line r has a negative slope. (False)

The slope of line r (3) is positive, not negative. So, this statement is false.

Based on the analysis, the true statement about lines r and s is:

Line r is steeper than line s.