What is the length of the unknown leg in the right triangle?

Answer:

there are no extraneous solutions to the equation

Step-by-step explanation:

Given

[tex]\sqrt{8x+9}[/tex] = x + 2 ( square both sides )

8x + 9 = (x + 2)² ← expand

8x + 9 = x² + 4x + 4 ( subtract 8x + 9 from both sides )

0 = x² - 4x - 5 ← in standard form

0 = (x - 5)(x + 1) ← in factored form

Equate each factor to zero and solve for x

x - 5 = 0 ⇒ x = 5

x + 1 = 0 ⇒ x = - 1

As a check

Substitute these values into the equation and if both sides are equal then they are the solutions.

x = 5 : [tex]\sqrt{8(5)+9}[/tex] = [tex]\sqrt{49}[/tex] = 7

right side = 5 + 2 = 7

left side = right side ⇒ x = 5 is a solution

x = - 1 : [tex]\sqrt{8(-1)+9}[/tex] = [tex]\sqrt{1}[/tex] = 1

right side = - 1 + 2 = 1

left side = right side ⇒ x = - 1 is a solution

There are no extraneous solutions

Answer:

There are no extraneous solutions

Step-by-step explanation:

sqrt(8x+9) = x+2

Square each side

(sqrt(8x+9))^2 = (x+2)^2

8x+9 =  (x+2)^2

FOIL

8x+9 = x^2 +2x+2x+4

Combine like term

8x+9 = x^2 +4x+4

Subtract 8x from each side

8x-8x+9 = x^2 +4x-8x+4

9 = x^2 -4x+4

Subtract 9 from each side

0 = x^2 -4x-5

Factor

0=(x-5) (x+1)

Using the zero product property\

x-5 =0  x+1=0

x=5   x=-1

Check the solutions

x=5

sqrt(8*5+9) = 5+2

sqrt(40+9) = 5+2

sqrt(49) =7

7=7

x=-1

sqrt(8*(-1)+9) = -1+2

sqrt(-8+9) = -1+2

sqrt(1) =1

1=1

Both solutions are true solutions

Answer: True

Step-by-step explanation: that is the correct definition on theoretical probability! :D

. Probability is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes, not unfavorable outcomesThe statement about theoretical probability in the question is false.

The provided statement about theoretical probability is false. According to the definition of theoretical probability, the probability of an event occurring is in fact the number of favorable outcomes divided by the total number of possible outcomes, not the unfavorable outcomes. To calculate the theoretical probability of an event A, one must count the number of outcomes that are favorable for event A (those that result in the event occurring) and divide by the total number of all possible outcomes.

Example:

If a die is rolled, the probability of rolling a three (a favorable outcome) is 1 in 6 (the total number of possible outcomes), as there is one three on a die and six possible outcomes in total.

Therefore, the correct way to express probability is:

Probability = (number of favorable outcomes) / (total number of possible outcomes)

Answer:

9y +9x - 10a-10c -10b = 105

Step-by-step explanation:

Given

a+b+c = -6 and x+y = 5

We have to find 9y +9x - 10a-10c -10b

Rearranging the terms

9x+9y-10a-10b-10c

Simplifying will give us:

= 9 (x+y) - 10(a+b+c)

Putting the values of x+y and a+b+c

= 9(5) -10(-6)

=45+60

=105

Therefore,

9y +9x - 10a-10c -10b = 105 ..

Answer:

9y + 9x - 10a - 10c - 10b  = 105

Step-by-step explanation:

It is given that,  a+ b + c = - 6 and x + y = 5

To find the value of given expression

Let the expression be 9y +9x - 10a-10c -10b

9y +9x - 10a -10c -10b  = 9(y + x) - 10(a + c + b )

 = 9(x + y) - 10(a + b + c)

 = 9 * (5) - 10 * (-6)

 = 45 + 60

 = 105

Therefore 9y + 9x - 10a - 10c - 10b  = 105

Answer:

to find the vertical asymptote, you have to put the rational function in the simplest form, which means to cancel any common factor between the numerator and denominator. here we don't have anything to cancel. then take the denominator and equal it to 0. x-3=0 ,x=3

to find the horizontal asymptote, in this situation, the degree of the numerator and denominator are the same which is 1. therefore, y=the coefficient of the numerator ÷ the coefficient of the denominator. y=6÷1 ,y=6

Answer with explanation:

The given function is:

   [tex]f(x)=\frac{6x-2}{x-3}[/tex]

Horizontal Asymptote

   [tex]y= \lim_{x \to \infty} \frac{6x-2}{x-3}\\\\\text{Dividing Numerator and Denominator by ,x}\\\\\Rightarrow y=\lim_{x \to \infty} \frac{6-\frac{2}{x}}{1-\frac{3}{x}}\\\\\Rightarrow y=\frac{6-0}{1-0}\\\\\Rightarrow y=6[/tex]

Vertical Asymptote

→Substitute Denominator of f(x) =0

⇒x-3=0

⇒x=3

⇒x=3 is Vertical Asymptote , and y=6 is horizontal Asymptote.

Matching with all the options

 Option D

[tex]_nP_k=\dfrac{n!}{(n-k)!}\\\\\\_5P_5=\dfrac{5!}{0!}=120[/tex]

Answer:

A.  120.

Step-by-step explanation:

5P5  =  5! / (5-5)!

= 5! / 0!

= 5!/1

= 5!

= 120.

Answer:

The answer is 7.433

Step-by-step explanation:

Bc:

  25.643  the 5 turned to 15, and the 2 is a 1.

-   18.210

    7.433

Hope my answer has helped you, if not i'm sorry.

Answer:

7.433

Step-by-step explanation:

To solve this problem, you'd need to subtract the numbers from left to right.

25.643-18.21=07.433=7.433

So, the correct answer is 7.433.

I hope this helps!

Thank you for submitting your question!

Answer:

The correct option is Any rational root of f(x) is a factor of 35 divided by a factor of 66....

Step-by-step explanation:

According to the rational root theorem:

if [tex]a_{0}[/tex] and [tex]a_{n}[/tex] are non zero then each rational solution x will be:

x= +/- Factors of [tex]a_{0}[/tex] / Factors of  [tex]a_{n}[/tex]

In the given polynomial we have:

66x4 – 2x3 + 11x2 + 35

[tex]a_{0}[/tex] = 35

[tex]a_{n}[/tex] = 66

Therefore,

x= +/- Factors of 35/ Factors of 66.

Thus the correct option is Any rational root of f(x) is a factor of 35 divided by a factor of 66....

Answer:

A

Step-by-step explanation:

trust bro

Answer:

Option 1 is correct. i.e  2 hours walking; 12 hours running

Step-by-step explanation:

We are given the equations

3w + 6r ≥ 36

3w + 6r ≤ 90

We will check which of the option satisfy the above equations.

1)  2 hours walking; 12 hours running

w = 2 and r =12

3w + 6r ≥ 36

3(2) + 6(12) ≥ 36

6+72 ≥ 36

78 ≥ 36

3w + 6r ≤ 90

3(2) + 6(12) ≤ 90

6+72 ≤ 90

78 ≤ 90

Both equations are satisfied. Option 1 is correct.

2) 4 hours walking; 3 hours running

w = 4 and r =3

3w + 6r ≥ 36

3(4) + 6(3) ≥ 36

12+18 ≥ 36

30 ≥ 36 (this equation doesn't hold as 30 < 36 and not < or equal to 36)

3w + 6r ≤ 90

3(4) + 6(3) ≤ 90

12+18 ≤ 90

30 ≤ 90

So, Option 2 is incorrect.

3) 9 hours running 12 hours walking

w = 9 and r =12

3w + 6r ≥ 36

3(9) + 6(12) ≥ 36

27+72 ≥ 36

99 ≥ 36

3w + 6r ≤ 90

3(9) + 6(12) ≤ 90

27+72 ≤ 90

99 ≤ 90 (this equation doesn't hold because 99 is greater than 90 and not less than 90)

Option 3 is incorrect.

4) 12 hours walking; 10 hours running

w = 12 and r =120

3w + 6r ≥ 36

3(12) + 6(10) ≥ 36

36+60 ≥ 36

96 ≥ 36

3w + 6r ≤ 90

3(12) + 6(10) ≤ 90

36+60 ≤ 90

99 ≤ 90 (this equation doesn't hold because 96 is greater than 90 and not less than 90)

Option 4 is incorrect.

Answer:

Step-by-step explanation:

Answer:

-6/7

Step-by-step explanation:

To find the slope of a line given two points on that line, you can use the slope formula: [tex]\frac{y_2-y_1}{x_2-x_2}[/tex] where you have points [tex](x_1,y_1) \text{ and } (x_2,y_2)[/tex].  

When using that formula, do not put the subscripts when plugging in the answers.

Something that is equivalent to that formula and I think is easier to remember is:

Lining up the points and subtracting vertically and then putting 2nd difference over first difference.

Like this:

 (-2 ,  3)

- ( 5 , -3)

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

-7         6

So the slope is 6/-7 or -6/7.

It doesn't matter what order you do the points. You have done the (5,-3) on top instead.

Like this:

  ( 5, -3)

-  (-2,  3)

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

  7     -6

The slope is still -6/7.

If you don't like that and you do just directly like using that formula, then I will do it that way too.

So applying the formula either gives you: [tex]\frac{3-(-3)}{-2-5)}[/tex] or [tex]\frac{-3-3}{5-(-2)}[/tex].  Both will give us the same number.

[tex]\frac{3-(-3)}{-2-5)}=\frac{6}{-7}=\frac{-6}{7}[/tex]

[tex]\frac{-3-3}{5-(-2)}=\frac{-6}{7}[/tex]

Answer:

(TZ - 2P)/2 = S

Step-by-step explanation:

Multiply both sides by Z

T*Z = Z*2(S + P)/Z

TZ = 2*(S + P)                    Divide by 2

TZ/2 = 2(S + P)/2              Do the division

TZ/2 = S + P                      Subtract P from both sides

TZ/2 - P = S                       That's one answer. You could refine it.

Multiply P by 2 so that you can put TZ - 2P over a common denominator.

(TZ - 2P)/2 = S                   Is another answer.  

I think the second one is probably the best one.

The zeros of f(x) = x^2 - x - 30 are x = 6 or -5

The function is given as:

f(x) = x^2 - x - 30

Expand

f(x) = x^2  + 5x - 6x - 30

Factorize the function

f(x) = x(x  + 5) - 6(x + 5)

Factor out x + 5

f(x) = (x - 6)(x + 5)

Set to 0

(x - 6)(x + 5) = 0

Solve for x

x = 6 or -5

Hence, the zeros of f(x) = x^2 - x - 30 are x = 6 or -5

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Answer:

6

Step-by-step explanation:

In this question it appears that for each burger Tom then has to eat 1 portion of fries. This combination would take 15 + 5 = 20 minutes.

Since 60/20 = 3, he would be able to eat 3 burgers and 3 portions of fries in 60 minutes. Making up a total of 3+3 = 6

Answer:

Tom eats 3 burgers and 3 portions of fries in 1 hour.

Step-by-step explanation:

Tom eats a burger in 15 minutes and a portion of fries in 5 minutes.

Total time he spent = 15 + 5  = 20 minutes in eating both

We have to calculate the number of burgers and fries he is able to eat in one hour.

∵ In 20 minutes Tom eats burger and fries = 1

∴ In 1 minutes he would be able to eat = [tex]\frac{1}{20}[/tex]

∴ In 60 minutes he would be able to eat = [tex]\frac{60}{20}=3[/tex] sets of burgers and fries.

Therefore, Tom eats 3 burgers and 3 portions of fries in 1 hour.

5500% of 4600% is 2530

Order of Operations in Mathematics follow this following rule :

This rule is known as the PEMDAS method.

In working on a mathematical problem, we first calculate operation that is in parentheses, follow by exponentiation, then multiplication or division, and finally addition or subtraction.

Let us tackle the problem.

This problem is about Percentage.

Given:

What percent of 4600% is 2530?​

We can translate above questions into mathematical equation as follows:

[tex]x\% \times 4600 \% = 2530[/tex]

[tex]x\% \times \frac{4600}{100} = 2530[/tex]

[tex]x\% \times 46 = 2530[/tex]

[tex]x\% = 2530 \div 46[/tex]

[tex]x\% = \frac{55}{1}[/tex]

To convert above fraction , we could multiply with 100% :

[tex]x\% = \frac{55}{1} \times 100\%[/tex]

[tex]x\% = \boxed{ 5500 \% }[/tex]

Another Example:

50% of 4600 is 2300

80% of 4600 is 3680

100% of 4600 is 4600

120% of 4600 is 5520

Grade: Middle School

Subject: Mathematics

Chapter: Percentage

Keywords: Linear , Equations , 1 , Variable , Line , Gradient , Point , Multiplication , Division , Exponent , PEMDAS , percentange , percent

The 2530 is 55% of 46. This means that 2530 is 55% of 4600%, as 4600% is equivalent to 46 in decimal form.

To find what percent 2530 is of 4600%, we first need to understand that 4600% is equal to 46. Hence, we need to determine what percent of 46 is 2530.

Let x be the percentage we are trying to find. We can set up the equation:

x% of 46 = 2530

To solve for x, we divide both sides by 46:

x% = 2530 / 46

x% = 55

So, 2530 is 55% of 46.

To summarize, 2530 is 55% of 46. This means that 2530 is 55% of 4600%, as 4600% is equivalent to 46 in decimal form.

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Answer:

According to the graph of f(x), the function has three real solutions to x= -2, x= -1 and x=1.

Then, given that the function is a 5 degree polynomial, we can conclude that the function has 2 complex zeros.

Answer:

I assume your goal is to rationalize the denominator -

So you need to multiply the denominator by its conjugate.

So we have:

[tex]\frac{\sqrt{x} }{\sqrt{x}-\sqrt{7} } *\frac{\sqrt{x} +\sqrt{7} }{\sqrt{x} +\sqrt{7}}[/tex]

->

[tex]\frac{x-\sqrt{7}\sqrt{x} }{x-7}[/tex]

gg

Answer:

90°

Step-by-step explanation:

we know that

The side TD is perpendicular to the plane formed by the square base ABCD of the cube

therefore

The measure of angle ∠TDB is 90 degrees

The correct answer is d) 90

Step-by-step explanation:

Answer:

g is a dependent variable

u is an independent variable

Step-by-step explanation:

Given relationship is:

g=748u

Here two terms need to be defined

Independent Variable : The variable which can take any value. The independent variable is used to calculate the value of dependent variable

Dependent Variable: The dependent variable is determined by the independent variable.

Therefore,

The correct answer is:

g is a dependent variable

u is an independent variable ..

Answer:

(Choice A)

Harry stains his shirt during the dinner.

Step-by-step explanation:

If Harry and Larry are having a barbecue dinner and the probability of Harry staining his shirt during the dinner is 1/5 and the probability of Larry staining his shirt during the dinner is 15%, Harry stains his shirt during the dinner is more likely.

Harry - 1/5 as a percent: 20%

Larry - 15% as a decimal: 0.15

Therefore 20% is more than 15 percent.

This makes the answer, Harry stains his shirt during the dinner.

Answer:

(Choice A)

Harry stains his shirt during the dinner.

Step-by-step explanation:

If Harry and Larry are having a barbecue dinner and the probability of Harry staining his shirt during the dinner is 1/5 and the probability of Larry staining his shirt during the dinner is 15%, Harry stains his shirt during the dinner is more likely.

Harry - 1/5 as a percent: 20%

Larry - 15% as a decimal: 0.15

Therefore 20% is more than 15 percent.

This makes the answer, Harry stains his shirt during the dinner.

Answer:

The distance formula d = rt, 'd' stands for distance, 'r' stands for rate and 't' stands for time.

It's a well known formula used in physics to solve problems that involves questions like "how fast", "how far", or "for how long".

In the formula d = rt, the variable t represents time. It's a consistent representation across situations whenever you are dealing with equations for motion, speed, or other physics contexts.

The variable t in the formula d = rt stands for time. This formula is used to calculate distance based on the rate of speed (r) and the time (t) traveled. To solve for time, you would rearrange the formula to t = d/r, which means time equals distance divided by rate.

In various physics and mathematical contexts, as shown in the reference information, the letter t consistently represents time. It could be the time an object takes to cover a certain distance at a given rate, or more complex scenarios involving non-Euclidean worlds or quantum mechanics. Nevertheless, whenever you see t in such equations, it is typically referring to time, unless otherwise specified.

Answer:

(x + 4)(x + 2)

Step-by-step explanation:

Consider the factors of the constant term ( + 8) which sum to give the coefficient of the x- term ( + 6)

The factors are + 4 and + 2, since

+ 4 × + 2 = + 8 and + 4 + 2 = + 6, hence

x² + 6x + 8 = (x + 4)(x + 2) ← in factored form

The two binomials that are factors of this trinomial x^2+6x+8 would be (x + 4)(x + 2).

Factorization is expressing a mathematical quantity in terms of multiples of smaller units of similar quantities.

For an integer, factorization is the decomposition of that integer in terms of smaller integers which when multiplied with each other give back that considered number.

Those composing integers are called factors of that considered integers.

Consider the factors of the constant term ( + 8) which sum to give the coefficient of the x- term ( + 6)

x² + 6x + 8

The factors are - 4 and + 2, since

+ 4 × + 2 = + 8 and + 4 + 2 = + 6,

hence , in the factored form

x² + 6x + 8 = (x + 4)(x + 2

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Answer:

11 cartons

Step-by-step explanation:

Number of large boxes that can be contained in a carton = 8

Number of small boxes that can be contained in a carton = 10

Total number of boxes sent in one shipment = 96

This shipment contains both large and small boxes. Let there be x cartons of large boxes and y cartons of small boxes in one shipment. So, we can say,

Total number of large boxes in one shipment = Number of boxes in one carton * Total number of cartons = 8x

Similarly,

Total number of small boxes in one shipment = 10y

Since, total number of boxes in one shipment is 96, we can set up the equation as:

8x + 10y = 96

This equation can have following possible solutions:

We are given in the statement that are more large boxes than the small boxes, the valid solutions are:

Assuming that he sent both large and small boxes, the only valid solution will be:

This means, the merchant sent 7 cartons of large boxes i.e. 56 large boxes and 4 cartons of small boxes i.e. 40 small boxes. So in total he sent 11 cartons.

Answer:

11 boxes total   4 small    7 large  

Step-by-step explanation:

1 small box *10 = 10             96-10 = 86 (NO MULTIPLE OF 8)

2 small boxes*10=20          96-29=76 ( no multiple of 8)

3 small boxes  *10=30        96-30= 66(no multiple of 8)

4 small boxes*10 = 40        96-40 = 56 (YES multiple of 8)      56/8 = 7   (7 large boxes)

4 large boxes and 7 large boxes

Answer:

2

Step-by-step explanation:

There are four 11-year olds and x amount of 14-years old.

We are given the average of this group of 11 and 14 year olds is 12.

This gives us the following equation to solve for x:

[tex]\frac{4(11)+x(14)}{4+x}=12[/tex]

Doing some simplifying on the left and writing 12 as 12/1 on the right because I'm going to cross-multiply in the step to follow:

[tex]\frac{44+14x}{4+x}=\frac{12}{1}[/tex]

Cross multiplying:

[tex]1(44+14x)=(4+x)(12)[/tex]

Distribute:

[tex]44+14x=48+12x[/tex]

Subtract 12x on both sides:

[tex]44+2x=48[/tex]

Subtract 44 on both sides:

[tex]2x=4[/tex]

Divide both sides by 2:

[tex]x=2[/tex]

There are two 14-year olds.

Let's confirm.

You have four 11 year olds and two 14 years old.

So you have:

11,11,11,11,14,14

We want to find the average of this set of numbers.

You add the numbers and divide by the number of numbers.

[tex]\frac{11+11+11+11+14+14}{6}=\frac{72}{6}=12[/tex]

This is the average the problem wanted us to obtain so the work we did above is correct.

-- Hamburgers got 33 total votes.

-- Tacos got 84 total votes.

-- Pizza got 82 total votes.

Tacos and pizza are almost equally preferred.

-- Tacos are out in front of pizza, but only by 2 votes.

-- Hamburgers are way behind, with less than half the votes of either tacos OR pizza.

It is found that Spaghetti and Pizza are preferred almost equally.

The unitary method is a method for solving a problem by the first value of a single unit and then finding the value by multiplying the single value.

Given that Ethel surveyed a random sample of 100 students from his middle school to learn more about what type of school lunches are preferred based on his observation the table of the following is :

                       Hamburgers             Spaghetti         Pizza

Sample 1:                 13                              43                     44

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

Sample 2:                20                              41                     39

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

Total:                       33                               84                   83

Here, we see that the total number of spaghetti and Pizza chosen are almost equal as:

Total number of Spaghetti preferred=84.

Total number of Pizza preferred=83.

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Answer:

q = 5

Step-by-step explanation:

Given

- 12 = q - 17 ( add 17 to both sides )

5 = q ⇒ q = 5

Answer:

q=5

Step-by-step explanation:

The equation is  -12=q-17​

In this equation we have to find the value of q.

So move the constant to the L.H.S

-17 will become +17 when it moves to the L.H.S

17-12=q

5=q

Thus the value of q is 5....

Answer:

11.2 miles

Step-by-step explanation:

What has been describe here is a right triangle.

He went 10 miles east, and he turned left and went 5 miles.  He started at home ended at work.  We want to make a road from his home directly to his workplace (no turns; straight there).

So we have by Pythagorean Theorem:

[tex]5^2+10^2=c^2[/tex]

[tex]25+100=c^2[/tex]

[tex]125=c^2[/tex]

Square root both sides:

[tex]\sqrt{125}=c[/tex]

[tex]11.2 \approx c[/tex]

11.2 miles would be the length of that direct path.

To find the number or rows, divide the total number of students by the number in each row.

63 / 7 = 9 rows.

Hello There!

If there are 63 students marching in the band and there is 7 rows, we can find out how many students are in each row by dividing 63 "number of students" by 7 "number of rows" and we will get an answer of 9.