hotel A charges $2 per minute plus a $5 connection fee for international phone calls. hotel B charges $4 per minute for international calls with a $3 discount on all calls. determine the length, in minutes of an international phone call that would cost the same at either hotel.

a) To calculate the missing values, we first need to find the 2018 turnover value. That is, 290.62 M Ft.

b) Average Relative Change =  5.18% and Average Absolute Change = 13.54 M Ft.

c) The ratios for 2016 indicate that the firm experienced a 4% increase in turnover compared to the previous year (2015).

a) To calculate the missing values, we first need to find the 2018 turnover value. We know that the 2015 turnover is 100% of the previous year, so 2015 and 2014 turnover values are the same (250 M Ft). Now we can use the ratios given for the subsequent years:

2016 Turnover = 2015 Turnover * (100% + Ratio)
260 M Ft = 250 M Ft * (100% + Ratio)
Ratio = (260 / 250) - 1 = 0.04 or 4%

2017 Turnover = 2016 Turnover * (100% + Ratio)
275 M Ft = 260 M Ft * (100% + Ratio)
Ratio = (275 / 260) - 1 ≈ 0.0577 or 5.77%

2018 Turnover = 2017 Turnover * (100% + Ratio)
2018 Turnover = 275 M Ft * (100% + 0.0577) ≈ 290.62 M Ft

b) Now, we can calculate the average relative and absolute change:
Average Relative Change = (4% + 5.77% + 5.77%)/3 ≈ 5.18%

Absolute Change (2016) = 260 - 250 = 10 M Ft
Absolute Change (2017) = 275 - 260 = 15 M Ft
Absolute Change (2018) = 290.62 - 275 ≈ 15.62 M Ft
Average Absolute Change = (10 + 15 + 15.62) / 3 ≈ 13.54 M Ft

c) The ratios for 2016 indicate that the firm experienced a 4% increase in turnover compared to the previous year (2015). This means the firm was successful in generating more revenue in 2016 as compared to 2015, which could be attributed to various factors such as improved marketing strategies, expansion in the market, or better product offerings.

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The p-value for a sample of different sales promotions with 5 different promotions and 10 samples with all 5 is equals to the 0.1060.

Suppose that the sales manager wants to compare different sales promotions. Here, number of different promotion choosen by him = 5

Number of random sample of each different promotion= 10

The F value = 3.4

We have to determine the p-value by using JMP. Now, n = 10, k = 5 so, degree of freedom = n - k= 5

Computing the p value using approximate method, [tex]P-value = P( F_{k - 1, n-k} > 3.4 ) [/tex]

[tex]= P( F_{4, 5}> 3.4 ) [/tex]

Using Excel command, value of F is calculated, = F.dist.RT( 3.4,4,5)

= 0.105954

Hence, required value is 0.1060.

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The variance and standard deviation can never be negative. However, they can be zero if there is no variability in the data. It is possible for the variance and standard deviation to be smaller or larger than the mean depending on the spread of the data.

The variance and standard deviation can never be negative.
1. Variance is a measure of how spread out the data points are from the mean. It is calculated by finding the average of the squared differences from the mean. Since squares are always positive or zero, the variance cannot be negative.

2. Standard deviation is the square root of the variance. Since the square root of a negative number is not a real number, the standard deviation cannot be negative either.

It is worth noting that both variance and standard deviation can be zero if all data points are the same, and they can be smaller or larger than the mean, depending on the data distribution.

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The simplified answer after Simplification of (1/2 - 1/3)(4/5 - 3/4) / (1/2 + 2/3 + 3/4) is 7/36.

To solve this expression, we need to follow the order of operations, which is parentheses, multiplication/division, and addition/subtraction.

First, we simplify the expression inside the parentheses:

(1/2 - 1/3)(4/5 - 3/4) = (1/6)(1/5) = 1/30

Next, we add up the denominators in the denominator of the entire expression:

1/2 + 2/3 + 3/4 = 6/12 + 8/12 + 9/12 = 23/12

Finally, we divide the simplified expression inside the parentheses by the fraction in the denominator:

(1/30) / (23/12) = (1/30) x (12/23) = 4/230 = 2/115 = 7/36

Therefore, the simplified answer is 7/36.

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The five-number summary for the data set is:

Minimum value = 10

Q1 = 14

Median = 18.5

Q3 = 24

Maximum value = 28.

The box and whisker plot is given.

We have,

To find the five-number summary, we need to first sort the data set in ascending order:

10, 12, 14, 15, 16, 18, 22, 24, 25, 28

Minimum value:

The smallest value in the data set is 10, so this is the minimum value.

Q1 (first quartile):

This is the value that separates the bottom 25% of the data from the top 75%.

To find Q1, we need to find the median of the lower half of the data.

The lower half of the data consists of the values 10, 12, 14, 15, and 16.

The median of these values is 14, so Q1 is 14.

Median (Q2):

This is the value that separates the bottom 50% of the data from the top 50%.

To find the median, we take the average of the two middle values.

The middle values are 18 and 19, so the median is (18+19)/2 = 18.5.

Q3 (third quartile):

This is the value that separates the bottom 75% of the data from the top 25%.

To find Q3, we need to find the median of the upper half of the data.

The upper half of the data consists of the values 22, 24, 25, and 28.

The median of these values is 24, so Q3 is 24.

Maximum value:

The largest value in the data set is 28, so this is the maximum value.

Therefore,

The five-number summary for the data set are minimum value = 10,

Q1 = 14, median = 18.5, Q3 = 24, maximum value = 28.

The box and whisker plot is given.

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To find the coordinates of the endpoints of the image A'B' (A9B9) after dilation of AB by a scale factor of 2 centered at (3,5), follow these steps:

Step 1: Use the given scale factor (2) and center of dilation (3,5).

Step 2: Apply the dilation formula to the coordinates of the original points A and B. The formula for dilation with scale factor k centered at (h,k) is:

A'(x', y') = (h + k(x - h), k + k(y - k))

Step 3: Substitute the given options for A9 and B9 into the dilation formula and check which pair of coordinates satisfy the formula.

After applying the formula, it is determined that the coordinates of the endpoints of the image A9B9 after dilation with a scale factor of 2 centered at (3,5) are:

A9(21, 6) and B9(7, 4).

Option (2) is correct.

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If the solution is indeed n = -2, then step 8 would be unnecessary, and the last line of the justification would be as stated above. The last line of the justification would typically be "Therefore, n = -2 is a solution to the equation 4n + 2 = 2n + 3 and the solution has been verified."

The justification would likely involve the following steps:

Start with the equation 4n + 2 = 2n + 3.

Simplify the equation by subtracting 2n from both sides: 2n + 2 = 3.

Subtract 2 from both sides: 2n = 1.

Divide both sides by 2: n = 1/2.

Check the solution by substituting n = -2 back into the original equation: 4(-2) + 2 = 2(-2) + 3.

Simplify: -8 + 2 = -4 + 3.

Further simplify: -6 = -1.

Since the equation is not true when n = -2, but instead it is true when n = 1/2, the solution of n = -2 is not correct and needs to be revised.

However, if the solution is indeed n = -2, then step 8 would be unnecessary, and the last line of the justification would be as stated above.

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The probability that at least one WST111 student is in time for the 7h30 lecture on a Tuesday morning is 0.9961.

1. First, let's find the probability that a randomly selected student is on time for the lecture. Since the probability that a student is late is 0.25, the probability that a student is on time is 1 - 0.25 = 0.75.

2. Now, we need to calculate the probability that all five randomly selected students are late for the lecture. Since punctuality is independent, we can simply multiply each student's probability of being late: 0.25×0.25×0.25×0.25× 0.25 = 0.0009765625.

3. Finally, we want to find the probability that at least one student is on time. To do this, we'll subtract the probability that all students are late from 1:

1 - 0.0009765625 = 0.9961.

So, the probability that at least one WST111 student is in time for the 7h30 lecture on a Tuesday morning is 0.9961.

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The value of x that makes the photo area half of the entire area is: 1.12 in

The area of a rectangle is given by the formula:

A = L * w

where:

L is length

w is width

Thus:

Area of photo = 4 * 2 = 8 in²

We are told that this area is half of the entire ad. Thus:

¹/₂(4 + x)(2 + x) = 8

x² + 6x + 8 = 16

x² + 6x - 8 = 0

Solving using a quadratic calculator gives:

x = 1.12 in

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The magnitude of the force that must be applied parallel to the plane is 92 N, under the condition that the box moves up the plane with an acceleration of 2 m/ s².

The force needed to move a box up an inclined plane can be evaluated as
Fp = W sin α = m ag sin α
Here
Fp = pulling force (N),
W = weight of the box (N),
α = angle of incline (degrees),
m = mass of the box (kg),
a = acceleration of the box (m/s²), and
g = acceleration due to gravity (9.8 m/s²).

For the given case, we possess a 25 kg box at rest on a 30 degree incline with a friction force of 50 N acting on it.
Now
We have to evaluate the weight of the box using
W = mg
= 25 kg x 9.8 m/s²
= 245 N.

Then, we have to calculate the force required to overcome friction using Ff = μFn where μ is the coefficient of friction and Fn is the normal force acting on the box. Since the box is at rest on an incline, Fn can be calculated as Fn = W cos α = 245 N cos(30°) ≈ 212 N. Therefore, Ff = μFn = 0.2 x 212 N ≈ 42 N.

Now we can evaluate the force applied to move the box up the incline utilizing
Fp = ma + Ff
Here,
a = desired acceleration
Ff = frictional force acting on the box.
Staging the values
Fp = ma + Ff
Fp = (25 kg)(2 m/s²) + 42 N
Fp ≈ 92 N

Hence, a force of approximately 92 N must be applied parallel to the plane so that the box moves up the plane with an acceleration of 2 m/s².
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The complete question is
A 25 kg box is at rest on a 30 degree incline. If the friction force is 50 N, what is the magnitude of the force that must be applied parallel to the plane so that the box moves up the plane with an acceleration of 2 m/ s²?

The volume of the solid generated by revolving the region bounded by the lines and curves about the x-axis is 2π(1 - e⁻¹) cubic units.

To find the volume of the solid generated by revolving the region bounded by the lines and curves about the x-axis, we need to use the method of cylindrical shells.

The volume can be calculated using the following formula:

V = ∫[a,b] 2πx f(x) dx

where a=0, b=1, and f(x) = e^(-x).

Substituting the given values, we get:

[tex]V = \int[0,1] 2\pi x e^{(-x)} dx[/tex]

Using integration by parts, we can solve this integral and get:

[tex]V = 2 \pi[e^{(-x)} - x e^{(-x)}][/tex] from 0 to 1

Simplifying this, we get:

V = 2π(1 - e⁻¹)

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The simplified expression is (8x² - 7x - 4) / [(4x + 1)(x - 1)(4x - 1)].

We have,

To simplify the expression

2(x - 1) / (4x² - 3x - 1) + (x + 2) / (4x² + 7x - 2)

we need to find the least common denominator (LCD) of the two denominators:

(4x² - 3x - 1) and (4x² + 7x - 2).

To find the LCD, we need to factor in both denominators.

We can factor the first denominator as:

4x² - 3x - 1 = (4x + 1)(x - 1)

We can factor the second denominator by using the quadratic formula or by factoring by grouping:

4x² + 7x - 2 = (4x - 1)(x + 2)

Therefore, the LCD is the product of the factors of both denominators, with each factor appearing once at most:

LCD = (4x + 1)(x - 1)(4x - 1)(x + 2)

To get each fraction to have the same denominator, we need to multiply the numerator and denominator of the first fraction by (4x - 1) and the numerator and denominator of the second fraction by (x - 1):

2(x - 1)(4x - 1) / [(4x + 1)(x - 1)(4x - 1)] + (x + 2)(x - 1) / [(4x + 1)(x - 1)(4x - 1)]

Now that both fractions have the same denominator, we can add the numerators and simplify:

[2(x - 1)(4x - 1) + (x + 2)(x - 1)] / [(4x + 1)(x - 1)(4x - 1)]

Multiplying out the numerator and simplifying, we get:

[8x^2 - 7x - 4] / [(4x + 1)(x - 1)(4x - 1)]

Therefore,

The simplified expression is (8x² - 7x - 4) / [(4x + 1)(x - 1)(4x - 1)]

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The approximate probability that between 12 and 20 patients, inclusive will experience weight gain as a side effect is 0.4147.

Normal approximation can be used to approximate the binomial distribution when the sample size is large enough (n >= 30) and the probability of success (p) and failure (q=1-p) are not too small or too large. In this case, we have a sample size of 500, which is sufficiently large, and the probability of success (p=0.03) and failure (q=0.97) are not too small or too large.

To approximate the probability that 15 or fewer users will experience weight gain as a side effect, we can use the normal approximation to the binomial distribution with mean (μ) = np = 500 x 0.03 = 15 and standard deviation (σ) = sqrt(npq) = sqrt(500 x 0.03 x 0.97) = 3.814. Then, we can use the normal cumulative distribution function (normalcdf) to calculate the probability that X ≤ 15, where X is the number of users who experience weight gain.

normalcdf(–1E99,15.5,15, 3.814) = 0.5522

Therefore, the approximate probability that 15 or fewer users will experience weight gain as a side effect is 0.5522.

To approximate the probability that 24 or more users experience weight gain as a side effect, we can use the normal approximation to the binomial distribution with the same mean and standard deviation as before. Then, we can use the normal complementary cumulative distribution function (normalccdf) to calculate the probability that X ≥ 24.

normalccdf(23.5,15,3.814) = 0.0097

Therefore, the approximate probability that 24 or more users experience weight gain as a side effect is 0.0097.

To approximate the probability that between 12 and 20 patients, inclusive will experience weight gain as a side effect, we can use the normal approximation to the binomial distribution with the same mean and standard deviation as before. Then, we can use the normal cumulative distribution function (normalcdf) to calculate the probability that 12 ≤ X ≤ 20.

normalcdf(11.5,20.5,15,3.814) = 0.6081 - 0.1934 = 0.4147

Therefore, the approximate probability that between 12 and 20 patients, inclusive will experience weight gain as a side effect is 0.4147.

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Answer:

-3x^2 + 12x + 5x - 20 = -3x^2 + 17x - 20

Step-by-step explanation:

(-3x - 5)(-x + 4) is a binomial expression, where (-3x - 5) is one expression and (-x + 4) is the other.

As the text eludes to, we can multiply binomial expressions using the FOIL method, where you multiply the first terms (3x and -x), outer terms (3x and 4), the inner terms (-5 and -x), and the last terms (-5 and 4)

This is how you get

(3x)(-x) + (3x)(4) + (-5)(-x) + (-5)(4)

Now, multiply the terms and combine like terms:

[tex](3x)(-x)+(3x)(4)+(-5)(-x)+(-5)(4)\\-3x^2+12x+5x-20\\-3x^2+17x-20[/tex]

Writing the total surface area of North America, which is approximately 9,540,000 square miles in Scientific Notation, is 9.54 x 10^6.

Scientific notation is shorthand way of writing very large or very small numbers in a standard form.

A number is written in scientific notation when a number between 1 and 10 is multiplied by a power of 10.

For instance, 9,540,000 square miles can be written in scientific notation as 9.54 x 10^6 square miles.

Thus, we can state that, in scientific notation, 9,540,000 square miles equal 9.54 x 10^6 square miles.

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Answer:

perseveration

Step-by-step explanation:

The term you are looking for is "perseveration."

Answer:

-12

Step-by-step explanation:

First substitute

Then after evaluating collect the like terms

After collecting the like terns simplify the Numbers

And finally evaluate the answer

Answer: 2x-64x+18x = –44x

Step-by-step explanation:

The distance between the foot of the ladder and the foot of the building is 2.6 ft

Part 1) The general form of a circle is given as x²+y² +4x - 12y + 4 = 0.

(a)What are the coordinates of the center of the circle?

(b)What is the length of the radius of the circle?

x²+y² +4x - 12y + 4 = 0

Group terms that contain the same variable, and move the constant to the opposite side of the equation

(x²+4x)+(y²- 12y)=-4

Complete the square twice. Remember to balance the equation by adding the same constants to each side

(x²+4x+4)+(y²- 12y+36)=-4+4+36

Rewrite as perfect squares

(x+2)²+(y-6)²=36--------> (x+2)²+(y-6)²=6²

center (-2,6)

radius 6

the answer Part a) is

the center is the point (-2,6)

the answer Part b) is

the radius is 6

Part 2)

see the picture attached N 1 to better understand the problem

we know that

sin ∅=opposite side angle ∅/hypotenuse

opposite side angle ∅=9.2 ft

hypotenuse=10 ft

so

sin ∅=9.2/10-----> 0.92

∅=arc sin (0.92)------> ∅=66.93°-----> ∅=67°

the answer Part a) is

67°

Part b)

see the picture attached N 2 to better understand the problem

cos 75=adjacent side angle 75/hypotenuse

adjacent side angle 75=AC

hypotenuse=10 ft

so

cos 75=AC/10---------> AC=10*cos 75----> AC=2.59 ft----> AC=2.6 ft

the answer Part B) is

The distance between the foot of the ladder and the foot of the building is 2.6 ft

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Answer:

Step-by-step explanation:

Based on the above, the cost to purchase 1 point is $2,700.

In order to know  the expense of acquiring 1 point, it is imperative to ascertain the extent by which the interest rate would decrease through the purchase of 1 point.

The purchasing  of each point results in a 0.125% reduction of the interest rate, so rate of interest shall be:

4.8% - 0.125%

= 4.675%

So, the cost of 1 point is 1% of the amount borrowed, that is $270,000. hence, the cost of 1 point is:

1% x $270,000

= $2,700

Therefore, the cost to purchase 1 point is $2,700.

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The break-even point is approximately 0.6 years, or 7.2 months. This means that if Zach plans to keep the loan for at least 7.2 months, purchasing 1 point would be worth it as he would save more in interest than he paid for the point.

To determine whether Zach should purchase 1 point or no points, we need to calculate the cost of each option and compare the total cost of each option over the life of the loan.

Option 1: No points

Loan amount: $200,000

APR: 3.625%

Monthly payment: $1,941.65 (calculated using a loan amortization calculator)

Total interest paid over 10 years: $33,698.03

Option 2: 1 point

Loan amount: $200,000

APR: 3.5% (3.625% - 0.125%)

Cost of 1 point: $2,000 (1% of the loan amount)

Total loan amount: $202,000 ($200,000 + $2,000)

Monthly payment: $1,903.03 (calculated using a loan amortization calculator)

Total interest paid over 10 years: $30,363.06

Comparing the two options, we can see that purchasing 1 point would result in a lower APR and lower monthly payments, which would save Zach money over the life of the loan. However, he would need to pay $2,000 upfront for the cost of the point.

To determine whether the cost of the point is worth the savings in interest, we need to calculate the break-even point. The break-even point is the point at which the savings in interest equal the cost of the point.

Break-even point:

Savings in interest: $33,698.03 - $30,363.06 = $3,334.97

Cost of 1 point: $2,000

Break-even point: $2,000 ÷ $3,334.97 = 0.6

The break-even point is approximately 0.6 years, or 7.2 months. This means that if Zach plans to keep the loan for at least 7.2 months, purchasing 1 point would be worth it as he would save more in interest than he paid for the point. If he plans to pay off the loan earlier than 7.2 months, then he should not purchase the point as he would not have enough time to recoup the cost.

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The error made by Beth is that:

She didn't apply the scale factor to each dimension of the triangle before multiplying

We are given the parameters as:

Initial height = 3.5cm

Initial width = 4cm

Formula for area of triangle is:

Area = ¹/₂ * base * height

If the triangle was enlarged by a scale factor of 5, then it means each of the dimensions should first be multiplied by 5 to get:

New width = 4 * 5 = 20 cm

New height = 5 * 3.5 = 17.5 cm

Thus:

New area = ¹/₂ * 20 * 17.5 = 175 cm²

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The value of y when the value of x is 1, can be found to be 60 miles.

As depicted in the graph, an augmented consumption of gallons of fuel by the hybrid vehicle concomitantly correlates to an increase in mileage capacity.

Concretely, according to our research data, we confirm that for every gallon expended, there is a mileage expansion rate of 60 units. Hence, when the quantity x equals 1, it implies usage of one gallon only.

Accordingly, empirical evidence suggests that traveling precisely sixty miles remains possible on utilization of one gallon which thus confirms the efficiency of using hybrid cars as a viable option.

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Answer: The equation is 5 - (3/5)cot(x) = (25 + √3)/5.

First, we can simplify the right-hand side by dividing both sides by 5:

(25 + √3)/5 = 5 + √3/5

Next, we can use the identity cot(x) = 1/tan(x) to rewrite the left-hand side:

5 - (3/5)cot(x) = 5 - (3/5)(1/tan(x)) = 5 - 3tan(x)/5

Now we have the equation 5 - 3tan(x)/5 = 5 + √3/5.

Subtracting 5 from both sides, we get:

-3tan(x)/5 = √3/5

Multiplying both sides by -5/3, we get:

tan(x) = -√3/3

Taking the arctangent of both sides, we get:

x = arctan(-√3/3)

Since the range of arctan is (-π/2, π/2), we need to add π to get the other solutions:

x = arctan(-√3/3) + π

x = arctan(-√3/3) + 2π

Using a calculator, we find that arctan(-√3/3) is approximately -0.5236 radians, so the solutions are:

x ≈ 2.6179 radians, 5.7596 radians, 8.9013 radians

Since we are looking for solutions between 0 and 2π, we can add or subtract multiples of 2π to get:

x ≈ 2.6179 radians, 5.7596 radians, 8.9013 radians, 11.0430 radians, 14.1847 radians, 17.3264 radians

These are the solutions to the equation 5 - (3/5)cot(x) = (25 + √3)/5 between 0 and 2π.

Step-by-step explanation:

The slope of the graph is 50, and it represents the rate of change of the total cost of the gym membership per month.

To write an equation for C, the total cost of the gym membership, we can use the slope-intercept form of a linear equation, which is y = mx + b. In this case, y represents the total cost, m represents the slope, x represents the number of months, and b represents the y-intercept (the initial cost of joining the gym).

From the graph, we can see that the initial cost of joining the gym is $700 (the y-intercept), and the monthly fee is $50 (the slope of the line connecting the dots). So, the equation for C is:

C = 50t + 700

where t is the number of months.

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To find the cumulative probability distribution function (CDF) F(y), we need to integrate the given PDF f(y) from 0 to y:

F(y) = integral of f(y) dy from 0 to y

= integral of c*y*(1-y)^2 dy from 0 to y   (substituting c=30 from part a)

= 30*integral of y*(1-y)^2 dy from 0 to y

To integrate this, we can use integration by substitution. Let u = 1 - y, then du/dy = -1 and y = 1 - u. Substituting, we get:

F(y) = 30*integral of (1-u)*u^2 * (-du) from 0 to 1-y

= 30*integral of u^2 - u^3 du from 0 to 1-y

= 30*[u^3/3 - u^4/4] evaluated at 0 and 1-y

= 10*(1 - (1-y)^3 - 3(1-y)^4/4),   0 <= y <= 1

Therefore, the cumulative probability distribution function (CDF) of Y is:

F(y) = {

        0,                             y < 0

        10*(1 - (1-y)^3 - 3(1-y)^4/4), 0 <= y <= 1

        1,                             y > 1

      }

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The dot product of two vectors u and v is calculated by multiplying their corresponding components and then adding the products together.

Mathematically, it can be expressed as:  u · v = u₁v₁ + u₂v₂ + u₃v₃ (for vectors in 3D space)
Step:1. u = (2,5) and v = (-6,1)
u · v = (2)(-6) + (5)(1) = -12 + 5 = -7
Since the dot product is not equal to zero, u and v are not orthogonal.
Step:2. u = 2i +3j and v= -7i -3j
u · v = (2)(-7) + (3)(-3) = -14 - 9 = -23
Again, the dot product is not zero, so u and v are not orthogonal.
Step:4. u = 4i+6j+8k and v= 7i -9j+ 12k (3D space)
u · v = (4)(7) + (6)(-9) + (8)(12) = 28 - 54 + 96 = 70
Once again, the dot product is not zero, so u and v are not orthogonal.
Therefore, in all three cases, the dot product of the given vectors is not zero, which means that they are not orthogonal.

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The variance of the given distribution is 0.9.

Calculating the anticipated value or distribution mean is the first step in determining a probability distribution's variance. The anticipated value is a weighted average of all potential outcomes, with each outcome's probability serving as the weight. The expected value can be expressed mathematically as follows:

[tex]E(X) =[/tex] Σ[tex][xi[/tex] × [tex]P(xi)][/tex]

where μ is the mean of the data.

Then, calculate μ:

μ [tex]= (1+2+3+4+5)/5[/tex]

[tex]=(15/5)[/tex]

[tex]= 3[/tex]

and replace this value and the values of xn and P(xn) into the formula for the variance, just as follow:

Hence, the variance of the given distribution is 0.9.

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Answer:

[tex]\large \boxed{\boxed{\textsf{$(x+2)(x-4)$}}}[/tex]

[tex]\boxed{\boxed{\large \textsf{$x=-2, x=4$}}}[/tex]

This is a quadratic expression, in the form:

[tex]\boxed{\large \textsf{$ ax^2+bx+c$, where$\ a \neq 0$}}[/tex]

To factorise this expression, we will need to have 4 terms. Currently there are only 3. To do this, we need to find 2 integers, that add together to form the middle term, -2, and multiply together to form the constant term, -8.

[tex]\large \textsf{the 2 integers $\Rightarrow$ 2 and -4}\\ \textsf{$-4+2 = -2$\ (coefficient of middle term)}\\ \textsf{$-4 \times 2=-8$\ (constant term)}[/tex]

Now we can split the middle term into 2 terms, using the integers we just found:

[tex]\large \textsf{$x^2+2x-4x-8$}[/tex]

Now we can factorise this.

[tex]\large \textsf{Group each pair of terms together, and take out a common factor.}\\ \\ \large \textsf{$x(x+2)-4(x+2)$}\\ \\ \large \textsf{Now take out the common factor from the expression: (x+2)} \\ \\ \large \textsf{$(x+2)(x-4)$}[/tex]

This leaves us with our fully factorised expression:

[tex]\large \boxed{\boxed{\textsf{$(x+2)(x-4)$}}}[/tex]

To solve the quadratic expression, we can make it equal to zero:

[tex]\large \textsf{$x^2-2x-8=0$}[/tex]

Primarily, to solve this, we can used the factorised form from above, and apply the zero-product property.

The zero-product property states that:[tex]\large \textsf{If $a\times b=0$, then $a=0$ or $b=0$ (or both $a=0$ AND b=0)}[/tex]

[tex]\large \textsf{$\therefore$ if $(x+2)(x-4)=0$, then $(x+2)=0$, and/or $(x-4)=0$ }[/tex]

[tex]\large \textsf{$\implies \boxed{\boxed{x=-2, x=4}}$ }[/tex]

Similarly, we can also use the quadratic formula to solve this equation:

[tex]\boxed{\Large \textsf{$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$} \large \textsf{, for $ax^2+bx+c=0$}}[/tex]

[tex]\large \textsf{$\Rightarrow a=1, b=-2, c=-8$}\\ \\ \Large \textsf{$x=\frac{-(-2)\pm \sqrt{(-2)^2-4(1)(-8)}}{2(1)}$}\\ \\ \boxed{\boxed{\large \textsf{$\therefore x=4, x=-2$}}}[/tex]

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