A circle with radius r is inscribed into a right triangle. Find the perimeter of the triangle if: The point of tangency divides the hypotenuse into 5 cm and 12 cm segments.

Answer:

15 million

Step-by-step explanation:

This is a one-tailed test with a significance level of 0.025. The test statistic is -1.83, and we reject the null hypothesis. The p-value is approximately 0.0344.

This is a one-tailed test because the alternative hypothesis (H1) is specifying a less than condition (<) for the population mean.

The decision rule for a one-tailed test with a significance level of 0.025 is to reject the null hypothesis (H0) if the test statistic is less than the critical value.

The test statistic is calculated by subtracting the hypothesized population mean from the sample mean and dividing by the standard deviation divided by the square root of the sample size. In this case, the test statistic is [(215 - 220) / (15 / sqrt(64))] = -1.83 (rounded to 3 decimal places).

In order to make a decision regarding H0, we compare the test statistic with the critical value. If the test statistic is less than the critical value, we reject H0. Otherwise, if the test statistic is greater than or equal to the critical value, we fail to reject H0. In this case, -1.83 is less than the critical value, so we reject H0.

The p-value is the probability of obtaining a test statistic as extreme as the one observed, assuming that the null hypothesis is true. To find the p-value, we look up the test statistic in the standard normal distribution table. In this case, the p-value is approximately 0.0344 (rounded to 4 decimal places).

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Answer:

  24 cm

Step-by-step explanation:

The product of height and width will remain the same, so the new width (w) will be given by ...

  w·(new height) = (old width)·(old height)

  w·(1.25·40 cm) = (30 cm)(40 cm)

  w = (30 cm)/1.25 = 24 cm . . . . . . . divide by 1.25·40 cm and simplify

_____

This derives from the fact that volume and length are unchanged. The formula for the volume in terms of length, width, and height is ...

  V = LWH

Then the product of W and H is the constant ...

  V/L = WH . . . . . divide by L

For our purpose, we only need to know that V and L are unchanged, so the product WH is unchanged. We don't need to know their values.

Of course, increasing the height by 25% is equivalent to multiplying it by 1.25:

  H + 25/100·H = H·(1 + 0.25) = 1.25H

Answer:

Option A is correct.

Step-by-step explanation:

64 < x < 68

This inequality represent that the height x should be greater than 64 and less than 68.i.e.

x>64 and x<68

So, Option A is correct.

Answer:

x>64 and x<68

Step-by-step explanation:

The answer for your question is A,D.

Answer:

D. supply curve shifts to the left

C. price increases and quantity decreases

Step-by-step explanation:

Since the number of sellers decreases, the quantity available at the same price decreases. This shifts the supply curve to the left.

When the supply curve shifts to the left, the equilibrium point shifts to the left (and up the demand curve). Hence the price increases and the quantity decreases.

Answer:

y= 2x+8

Step-by-step explanation:

The standard form of an equation of line in slope-intercept form is:

y= mx+b

Where m is the slope of the line and b is the y-intercept of the line.

To obtain the equation from the given information we need to put the values of m and b into the standard form of equation.

We are given

m = 2

and

b = 8

So putting the values of m and b

y = (2)x +8

y= 2x+8 ..

THE ANSWER IS :y=2x +8

just know its right.

Answer:

  (n+54)/9 = 21

Step-by-step explanation:

A number: n

Is increased by 54: n+54

The sum is divided by 9: (n +54)/9

and the result is 21:

  (n +54)/9 = 21

Answer:

  see attachment

Step-by-step explanation:

It is convenient to use a spreadsheet for this purpose.

The first row of numbers is constant at $45, as there is no daily charge associated with that payment method.

The second row of numbers uses the formula ...

  cost = $12 + $4×(number of days)

The third row of numbers uses the formula ...

  cost = $6×(number of days)

___

Of course, you can use these formulas to fill in the numbers by hand. For example, for 15 days, the charges are ...

Answer:

2/4

when simplified = 1/2

Your answer is 1/2

Step-by-step explanation:

Amt. of Baseball caps = 4

Blue caps = 2

which can be written as 2/4

when simplified 2/4 ÷ 2/2 = 1/2

Willie has 4 baseball caps, of which 2 are blue. To find the fraction of caps that are blue, divide the number of blue caps (2) by the total number of caps (4), resulting in a fraction of 1/2.

Willie has 4 baseball caps in total, and 2 of those caps are blue. To determine what fraction of the caps is blue, you divide the number of blue caps by the total number of caps.

So, the calculation would be:

Therefore, the fraction of Willie's caps that are blue is 1/2, which means half of the caps are blue.

x= 8 is the correct answer

Answer:

x = 8

Step-by-step explanation:

Given equation is,

[tex]log_5(10x-1)=log_5(9x+7)[/tex]

We know that,

[tex]log_a(b)=log_a(c)\implies b = c[/tex]

[tex]\implies 10x -1 = 9x + 7[/tex]

Subtracting 9x on both sides,

[tex]x - 1 = 7[/tex]

Adding 1 on both sides,

[tex]x = 8[/tex]

Hence, the solution would be x = 8

Dang that’s tuff I feel bad

Answer:

Step-by-step explanation:

Formula

abs(a + b) + abs(c)

Givens

a = - 3

b = - 7

c = - 15

Solution

abs(-3 - 7) + abs(-15)

abs(-10) + abs(-15)

10 + 15

25

- The solution/answer is 10.

The two angles are Supplementary angles and need to eaual 1980 degrees.

x-2 + 5x +2 = 180

Simplify:

6x = 180

Divide both sides by 6:

x = 180 /6

x = 30

Answer:

Step-by-step explanation:

The volume of a rectangular box is width times length times height:

V = wlh

After the cardboard is folded, the width is 12 - 2x, the length is 16 - 2x, and the height is x.

So the volume is:

V = (12 - 2x) (16 - 2x) x

If we graph this, we get a wave: desmos.com/calculator/rsjosgzuxz

The wave is the highest at around x = 2.3 in.

If we set x = 2.3:

w = 12 - 2x = 7.4

l = 16 - 2x = 11.4

h = x = 2.3

[tex]\bf slope = m = \cfrac{rise}{run} \implies \cfrac{ f(x_2) - f(x_1)}{ x_2 - x_1}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array}\\\\[-0.35em] \rule{34em}{0.25pt}\\\\ f(x)= 2x^2-1\qquad \begin{cases} x_1=0\\ x_2=5 \end{cases}\implies \cfrac{f(5)-f(0)}{5-0} \\\\\\ \cfrac{[2(5)^2-1]~~-~~[2(0)^2-1]}{5}\implies \cfrac{50-(-1)}{5}\implies \cfrac{50+1}{5}\implies \cfrac{51}{5}\implies 10\frac{1}{5}[/tex]

Surface area of [tex]\(r_2(u,v)\)[/tex]  is [tex]\(25\)[/tex] times the area of parameter domain[tex]\(D\),[/tex] yielding [tex]\(1600\)[/tex] if [tex]\(D\)[/tex] is [tex]\(8 \times 8\).[/tex]

Let's break it down step by step:

step:-1. **Define the parametric surfaces**: We have two parametric surfaces: [tex]\( r_1(u,v) = f(u,v)i + g(u,v)j + h(u,v)k \)[/tex] and [tex]\( r_2(u,v) = 5r_1(u,v) \).[/tex]

step:-2. **Calculate the partial derivatives**: Compute the partial derivatives of [tex]\( r_1 \)[/tex]  with respect to [tex]\( u \)[/tex]  and [tex]\( v \)[/tex] denoted by [tex]\( r_{1u} \)[/tex] and [tex]\( r_{1v} \).[/tex]

step:-3. **Multiply by 5**: Since[tex]\( r_2(u,v) = 5r_1(u,v) \),[/tex]  the partial derivatives of [tex]\( r_2 \)[/tex] with respect to [tex]\( u \)[/tex] and [tex]\( v \)[/tex] will be 5 times the corresponding partial derivatives of [tex]\( r_1 \)[/tex], denoted by [tex]\( r_{2u} \)[/tex] and [tex]\( r_{2v} \).[/tex]

step:-4. **Calculate the cross product**: Compute the cross product of [tex]\( r_{2u} \)[/tex] and[tex]\( r_{2v} \),[/tex] denoted by [tex]\( \| r_{2u} \times r_{2v} \| \)[/tex]. This will be 25 times the magnitude of the cross product of [tex]\( r_{1u} \)[/tex] and [tex]\( r_{1v} \),[/tex] as the cross product is linear with respect to the vectors involved.

step:-5. **Surface area integral**: Use the formula for the surface area integral: [tex]\( A = \iint_D \| r_u \times r_v \| \, dA \),[/tex] where [tex]\( \| r_{2u} \times r_{2v} \| \)[/tex] replaces [tex]\( \| r_{u} \times r_{v} \| \).[/tex]

step:-6. **Calculate the integral**: Integrate[tex]\( \| r_{2u} \times r_{2v} \| \)[/tex] over the parameter domain [tex]\( D \)[/tex]. Since [tex]\( \| r_{2u} \times r_{2v} \| \)[/tex] is constant and equal to 25 times the magnitude of the cross product of [tex]\( r_{1u} \)[/tex] and [tex]\( r_{1v} \)[/tex], the integral becomes [tex]\( 25 \times \text{Area of } D \).[/tex]

step:-7. **Determine the area of the parameter domain**: If the parameter domain [tex]\( D \)[/tex] is a rectangle with sides of length 8 in both directions, its area is [tex]\( 8 \times 8 = 64 \).[/tex]

step:-8. **Final calculation**: Multiply the area of [tex]\( D \)[/tex] by 25 to get the surface area of[tex]\( r_2(u,v) \)[/tex], which is[tex]\( 25 \times 64 = 1600 \).[/tex]

So, the surface area of the parametric surface given by [tex]\( r_2(u,v) = 5r_1(u,v) \)[/tex] is 1600.

The surface area of the parametric surface [tex]\( \mathf{r}_2(u, v) \)[/tex] is 25.

To find the surface area of the parametric surface given by [tex]\( \mathf{r}_2(u, v) = 5 \mathf{r}_1(u, v) \)[/tex] where [tex]\( -4 \leq u \leq 4 \)[/tex] and [tex]\( -4 \leq v \leq 4 \)[/tex], given that the surface area of [tex]\( \mathf{r}_1(u, v) \)[/tex] over the same parameter range is 1, follow these steps:

Surface Area of [tex]\( \mathf{r}_1(u, v) \)[/tex]

  The surface area of [tex]\( \mathf{r}_1(u, v) \)[/tex] is given to be 1.

Relationship Between [tex]\( \mathf{r}_1 \) and \( \mathf{r}_2 \)[/tex]

 [tex]\[ \mathf{r}_2(u, v) = 5 \mathf{r}_1(u, v) \][/tex]

Effect of Scaling on Surface Area

  When a surface is scaled by a factor k, the surface area is scaled by a factor of k². This is because surface area is a two-dimensional measure, and scaling each dimension by k multiplies the area by k².

Calculation for [tex]\( \mathf{r}_2(u, v) \)[/tex]

  In this problem, the scaling factor k is 5. Therefore, the surface area of [tex]\( \mathf{r}_2(u, v) \)[/tex] will be [tex]\( 5^2 \)[/tex] times the surface area of [tex]\( \mathf{r}_1(u, v) \)[/tex].

[tex]\[\text{Surface area of } \mathf{r}_2(u, v) = 5^2 \times \text{Surface area of } \mathf{r}_1(u, v)\][/tex]

Substitute the Given Surface Area

  The surface area of [tex]\( \mathf{r}_1(u, v) \)[/tex] is 1.

[tex]\[\text{Surface area of } \mathf{r}_2(u, v) = 5^2 \times 1 = 25\][/tex]

Answer:

  C. The statement is true because the 95th percentile for men is greater than the 5th percentile for women.

Step-by-step explanation:

We suspect relevant data is missing, including exactly what clearance the the statement is referring to. It would be nice to know where the various men's and women's percentiles lie.

We give the above answer on the basis of the assumption that the largest of men are always larger than the smallest of women.

Answer:

Range = {2,1,4,5}

Step-by-step explanation:

When a function is given in the form if a relation. i.e. ordered pairs.

When the function is given in the form of ordered pairs then the set of first elements i.e. x-coordinates of all ordered pairs forms domain while the set of second elements i.e. y-coordinate of all ordered pair is called range.

So in the given function:

Range = {2,1,4,5}

The repeating values are only written once..

Answer:

d. 4x^2 +24x +23

Step-by-step explanation:

Evaluating polynomials is sometimes easier when they are written in Horner form:

f(x) = (4x +8)x -9

Substituting (x+2) for x, we have ...

f(x+2) = (4(x+2)+8)(x+2) -9

= (4x +16)(x +2) -9

= 4x^2 +24x +32 -9

= 4x^2 +24x +23 . . . . . matches choice D

_____

Alternative method

You can observe that the answer choices differ in the coefficient of the x-term. So, to make the correct selection, you only need to find the coefficient of the x-term. That will be the sum of coefficients of the x-terms in 4(x+2)^2 and 8(x+2). Those x-terms are 4·4x and 8x, so have a sum of (16+8)x = 24x. This matches choice D.

Use the midpoint formula:

(-7,-4) (-1,8)

(-7 + -1) /2 ,  (-4 +8) /2

-8/2 , 4/2

Midpoint = (-4,2)

8 students know Spanish since 12 students know French and 7 students know Italian

Answer:

8 students learn Spanish.

Step-by-step explanation:

In a class total number of students are 30.

Students who know Italian = 15

Students who know French = 10

Students who know both = 3

Rest all students know Spanish.

By Venn diagram attached number of students who know Spanish

= [ 30 - (12 + 3 + 7 )]

= 30 - (22)

= 8 students.

8 students learn Spanish.

Answer:

(-5,2)

Step-by-step explanation:

The given system is

1st equation: y = 4x + 22

2nd equation:  4x – 6y = –32

We plug in the first equation into the second equation to obtain:

4x – 6(4x + 22) = –32

We expand the parenthesis to obtain:

4x – 24x -132= –32

Group similar terms;

4x – 24x = –32+132

Combine similar terms

-20x =100

Divide both sides by -20

x =-5

Put x=-5 into the 1st equation

y = 4(-5) + 22

y=-20+22

y=2

The solution is:

(-5,2)

Answer:

* The largest area can be enclosed is 6050 feet²

Step-by-step explanation:

* Lets explain the situation to solve the problem

- There is a rectangular parking

- The parking will surrounded by fencing from three sides only

- The length of fencing is 220 feet

- Lets consider the width of the rectangle is x and the length of it is y

- The side along the street will not fence

* Lets put all of these data in equation

∵ The width of the parking is x

∵ The length of the parking is y

- He will not fence the side along the street

∴ The perimeter of the parking = x + y + x

∴ The perimeter of the parking = 2x + y

- The length of the fencing = the perimeter of the park

∵ The length of the fencing = 220 feet

∵ The perimeter of the parking = 2x + y

∴ 2x + y = 220 ⇒ (1)

- Lets find the area of the parking

∵ The area of any rectangle is length × width

∵ The width of the rectangle is x

∵ The length of the rectangle is y

∴ The area of the parking (A) = x × y

∴ The area of the parking = xy ⇒ (2)

- Lets find the value of y from equation (1) and substitute this value

 in equation (2)

∵ 2x + y = 220 ⇒ subtract 2x from both sides

∴ y = 220 - 2x

- Substitute this value in equation (2)

∵ A = xy

∴ A = x(220 - 2x) ⇒ open the bracket

∴ A = 220x - 2x²

- To find the largest area differentiate the area with respect to x

  and equate the result by 0 to find x which gives the largest area

∵ A = 220x - 2x²

- Lets remember the differentiation rules

# If y = a x^n, where a is the coefficient of x then dy/dx = (an) x^(n-1)

# If y = ax, then dy/dx = a

# If y = a, where a is constant then dy/dx = 0

∴ dA/dx = 220 - 2(2) x^(2-1)

∴ dA/dx = 220 - 4x

- Put dA/dx = 0 ⇒ for largest area

∵ dA/dx = 0

∴ 220 - 4x = 0 ⇒ add 4x to both sides

∴ 220 = 4x ⇒ divide both sides by 4

∴ 55 = x

* The width of the parking is 55 feet

- Substitute this value of x in the equation of the area to find the

 largest area

∵ A = 220x - 2x²

∵ x = 55

∴ A = 220(55) - 2(55)² = 12100 - 6050 = 6050 feet²

* The largest area can be enclosed is 6050 feet²

To find the largest area that can be enclosed by the given amount of fencing, we can use the concept of optimization and solve for a rectangular lot's dimensions that maximize the area. The largest area that can be enclosed is 6050 square feet.

To find the largest area that can be enclosed, we can use the concept of optimization. Let's assume the length of the rectangular lot is x feet. The remaining length, which is not fenced, will be 220 - 2x feet. The width of the lot will be y feet. So we have 2x + y = 220. To find the largest area, we need to express the area in terms of a single variable. A = xy, substitute y = 220 - 2x. So A = x(220 - 2x) = 220x - 2x^2. To find the maximum value of A, we can find the vertex of the parabola, which corresponds to the maximum. The x-coordinate of the vertex is -b/2a. In this case, a = -2, b = 220, so x = -220/(-2*2) = 55.Plug this value of x into the equation y = 220 - 2x. y = 220 - 2(55) = 110. So the dimensions of the lot that will enclose the largest area are 55 feet by 110 feet. Substituting these values into the area formula A = xy, we get A = 55 * 110 = 6050 square feet.

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Answer:

B. -4, 6

Step-by-step explanation:

Given the quadratic equation;

x^2 - 2x - 24 = 0

we can determine the solution by first factoring the expression on the left hand side. We determine two numbers whose product is -24 and sum -2. By trial and error the two numbers are found to be;

-6 and 4

We replace the middle term, -2x, with these two values;

x^2 + 4x -6x -24 = 0

x(x+4) -6(x+4) = 0

(x-6)(x+4) = 0

x-6 = 0 or

x + 4 = 0

x = 6 or x = -4

Answer: the answer is B

Step-by-step explanation: -4 & 6 are both zeros of the function .

For this case we have that by definition, the perimeter of the quadrilateral shown is given by the sum of its sides:

Let "p" be the perimeter of the quadrilateral, then:

[tex]p = 7 + y + 7 + x\\p = 7 + 7 + x + y\\p = 14 + x + y[/tex]

So, the perimeter of the figure is: [tex]14 + x + y[/tex]

Answer:

[tex]p = 14 + x + y[/tex]

Answer:

  perimeter = x + y + 14

Step-by-step explanation:

The perimeter is the sum of the side lengths:

  perimeter = x + 7 + y + 7

The constants can be combined to give ...

  perimeter = x + y + 14

___

An "equation" will have an equal sign somewhere. Since you want to find the perimeter, it makes sense for the equation to show you how to find the perimeter. (I have used "perimeter" to represent the perimeter. It is often represented using the letter P. Of course, you can choose any variable or other representation you like.)

Answer: D) 60x-45y = - 1 where A = 60 , B = -45 , and c = - 1

Step-by-step explanation: Clear it : 45y=60x+1

Step 2: Isolate the constant on one side: ( -1 = -45y +60x)

step 3: A= 60 , B= - 45 , C= - 1

Answer:

60x-45y = - 1 where A = 60 , B = -45 , and c = - 1

Step-by-step explanation:

Answer:

Part 1) The height of the triangle when θ = 30° is equal to [tex]8.66\ ft[/tex]

Part 2) The height of the triangle when θ = 40° is equal to [tex]12.59\ ft[/tex]

Part 3) The area of triangle with θ = 30° is less than the area of triangle with θ = 40°

Step-by-step explanation:

Part 1) What is the height of the triangle when θ = 30 ° ?

we have

[tex]y=15tan(\theta)[/tex]

substitute the value of theta in the equation and find the height

[tex]y=15tan(30\°)=8.66\ ft[/tex]

Part 2) What is the height of the triangle when θ = 40 ° ?

we have

[tex]y=15tan(\theta)[/tex]

substitute the value of theta in the equation and find the height

[tex]y=15tan(40\°)=12.59\ ft[/tex]

Part 2) Vance is considering using either θ = 30 ° or θ = 40 ° for his garden

Compare the areas of the two possible gardens

step 1

Find the area when θ = 30 °

The height is [tex]8.66\ ft[/tex]

Remember that the area of a triangle  is equal to the base multiplied by the height and divided by two

so

[tex]A=(1/2)(30)(8.66)=129.9\ ft^{2}[/tex]

step 2

Find the area when θ = 40°

The height is [tex]12.59\ ft[/tex]

Remember that the area of a triangle  is equal to the base multiplied by the height and divided by two

so

[tex]A=(1/2)(30)(12.59)=188.85\ ft^{2}[/tex]

Compare the areas of the two possible gardens

The area of triangle with θ = 30° is less than the area of triangle with θ = 40°

Answer:

∠JML = 104°

Step-by-step explanation:

JL bisects angle KLM, so ...

∠KLM = 2·∠KLJ = 2·38° = 76°

Adjacent angles in any parallelogram are supplementary, so ...

∠JML = 180° -∠KLM = 180° -76°

∠JML = 104°

To find the multiplicative inverse of x modulo n, we need to find an integer s such that (x * s) mod n = 1. We can calculate the multiplicative inverses for the given values: (a) x = 52, n = 77: s = 65. (b) x = 77, n = 52: s = 33. (c) x = 53, n = 71: s = 51. (d) x = 71, n = 53: s = 9.

To find the multiplicative inverse modulo n of x, we need to find an integer s such that (x * s) mod n = 1. Let's calculate the multiplicative inverses for the given values:

a) For x = 52 and n = 77:

x * s ≡ 1 (mod n)

52 * s ≡ 1 (mod 77)

s ≡ 65 (mod 77)

So, the multiplicative inverse of 52 modulo 77 is 65.

b) For x = 77 and n = 52:

77 * s ≡ 1 (mod 52)

s ≡ 33 (mod 52)

The multiplicative inverse of 77 modulo 52 is 33.

c) For x = 53 and n = 71:

53 * s ≡ 1 (mod 71)

s ≡ 51 (mod 71)

The multiplicative inverse of 53 modulo 71 is 51.

d) For x = 71 and n = 53:

71 * s ≡ 1 (mod 53)

s ≡ 9 (mod 53)

The multiplicative inverse of 71 modulo 53 is 9.

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Answer with Step-by-step explanation:

On spinning the spinner twice,we have 16 different outcomes:.

We write the outcomes with their product:

                      Product

2    2                   4

2    3                   6

2    4                  8

2    9                  18

3    2                   6

3    3                   9

3    4                   12

3    9                   27

4    2                   8

4    3                    12

4    4                   16

4    9                   36

9    2                    18

9    3                    27

9    4                    36

9    9                    81

outcomes have a product less than 20 and contain at least one even number are in bold letters.

Hence, outcomes have a product less than 20 and contain at least one even number are:

10

There are 8 outcomes of spinning the spinner twice that result in a product less than 20 and contain at least one even number.

To determine how many outcomes of spinning a spinner twice result in a product less than 20 and contain at least one even number, we start by listing all possible outcomes.

The spinner is divided into four equal sections: 2, 3, 4, and 9. When spun twice, there are 16 outcomes in total:

We now filter these outcomes to find those with a product less than 20 and at least one even number:

The total number of outcomes that meet the criteria is 8.

Answer:

  (2 +4i)(5 -6i) . . . or . . . (4 -2i)(6 +5i)

Step-by-step explanation:

The product of two complex numbers is ...

  (a +bi)(c +di) = (ac -bd) +(bc +ad)i

So, we're looking for pairs of numbers that can be combined in different ways to give 34 and 8. The numbers we found (by trial and error) are ...

  2, 4, 5, 6

where 4*6 +2*5 = 34 and 4*5 -2*6 = 8. Because of the effect if i^2 on the sign, we need to have the imaginary parts have opposite signs.

Each of the solutions shown above is representative of 4 solutions. For example, for the first one, you could have ...

  (2 +4i)(5 -6i) = (2·5 +4·6) + (4·5 +2(-6))i = 34 +8i

  (5 -6i)(2 +4i) = (5·2 +6·4) + (-6·2 +5·4) = 34 +8i . . . . . order of factors swapped

  (-2 -4i)(-5 +6i) = ((-2)(-5) -(-4)(6)) + ((-4)(-5) +(-2)(6))i = 34 +8i . . . . both factors in the first solution negated

  (-5 +6i)(-2 -4i) = ((-5)(-2) -(6)(-4)) +(6(-2) +(-5)(-4))i = 34 +8i . . . . factors swapped and negated

___

Likewise, the second shown solution above is representative of 4 solutions.

Possible solutions are ...

with sign and order variations.

_____

Comment on trial and error

Actually, we did an exhaustive search of the 441 products of single-digit numbers [-9, 9] to see which pairs of them differed by 34. Then, among those, we looked for product pairs that added to 8. In the end, we found the 8 solutions described above.