What is the solution to the equation below?
log 20х3 - 2logx = 4
x=25
x=50
x=250
x=500​

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:

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:

tan O = √3/ 3.

Step-by-step explanation:

The tangent is  positive in quadrant 3.

The adjacent side has length  √((2^2 - (-1)^2)

= -√3.

So tan O  = -1 / -√3

= 1 / √3

= √3/ 3.

To determine tan(θ) for an angle θ where sin(θ) = -1/2 and θ is in quadrant III, we perform the following steps:
1. **Understand the given information:** When sin(θ) = -1/2, it means the opposite side to angle θ in a right triangle is -1 and the hypotenuse is 2. Since θ lies in the third quadrant, both sine and cosine values are negative, implying that the adjacent side length must also be negative.
2. **Use the Pythagorean theorem:** For a right triangle:
  \[ \text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2 \]
  Substituting hypotenuse and opposite side values:
  \[ 2^2 = (-1)^2 + \text{adjacent}^2 \]
  Simplify and solve for the adjacent side:
  \[ 4 = 1 + \text{adjacent}^2 \]
  \[ 3 = \text{adjacent}^2 \]
  \[ \text{adjacent} = \sqrt{3} \] or \[ \text{adjacent} = -\sqrt{3} \]
  Since the angle is in the third quadrant where cosine is also negative, we must take the negative square root:
  \[ \text{adjacent} = -\sqrt{3} \]
3. **Calculate tan(θ):** By definition,
  \[ \tan(θ) = \frac{\text{opposite}}{\text{adjacent}} \]
  Substitute the known side lengths:
  \[ \tan(θ) = \frac{-1}{-\sqrt{3}} \]
4. **Simplify the expression:**
  \[ \tan(θ) = \frac{1}{\sqrt{3}} \]
  To rationalize the denominator:
  \[ \tan(θ) = \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} \]
  \[ \tan(θ) = \frac{\sqrt{3}}{3} \]
Thus, if sin(θ) = -1/2 and θ is in quadrant III, then tan(θ) = √3/3. Note that in the context of trigonometry, it is often assumed that the angle and the ratios are dimensionless, so a negative 'opposite' simply reflects the direction along the coordinate axis rather than a literal negative length.

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.

Least to greatest:

When ordering decimals from least to greatest, it's crucial to understand place value. When ordering, I usually like to start with whole numbers and put them first and then I think about the numbers after the decimal.

Answer:

A large pizza costs $5 and a small pizza costs $3.

Let 'x' be the number of large pizzas and 'y' the number of small pizzas. We have that:

x + y = 150

5x + 3y = 570

Solving the system of equations:

x + y = 150 → 3x + 3y = 450 → 3y = 450 - 3x

5x + 3y = 570 → 5x + 450 - 3x = 570

2x = 120

x = 60

y = 150 - 60 = 90

Therefore. Carmen y Carlos will make 60 large pizzas and 90 small pizzas.

The equation of the circle is [tex](x+8)^{2} +(y -15)^{2} = 289[/tex].

The general equation of the circle is [tex](x-h)^{2} +(y-k)^{2} = r^{2}[/tex].

Where, (h, k) is the center and r is the radius of the circle.

Let the equation of the circle be

[tex](x-h)^{2} +(y-k)^{2} =r^{2}..(i)[/tex]

According to the given question.

The center of the circle is (-8, 15).

⇒ (h, k) = (-8, 15)

And, the circle is passing through origin i.e. (0, 0).

Since, the center of the circle is (-8, 15).

So, the equation (i) can be written as

[tex](x-(-8))^{2} +(y - 15)^{2} = r^{2}[/tex]

Also, the circle is passing through (0, 0).

Therefore,

[tex](0+8)^{2} +(0-15)^{2} = r^{2}[/tex]

[tex]\implies 64+ 225 = r^{2} \\\implies 289 = r^{2} \\\implies r=\sqrt{289} \\\implies r = 17[/tex]

So, the radius of the circle is 17 unit and its center is (-8, 15).

Therefore, the equation of the circle is

[tex](x+8)^{2} +(y-15)^{2} = 289[/tex]

Hence, the equation of the circle is [tex](x+8)^{2} +(y -15)^{2} = 289[/tex].

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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.

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:

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:

20 ≤ f(x) ≤ 28.14

Step-by-step explanation:

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:

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

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:

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:

b=2

Step-by-step explanation:

9x + 12y = 21

6x + 8y = 7b

To have an infinite number of solutions, the two equations must be equal

Take the first equation and divide by 3, since it is a factor of all the terms

9x/3 + 12y/3 = 21/3

3x+4y = 7

Then multiply by 2 to get the x term equal

2*3x + 2*4y = 2*7

6x +8y = 7*2

Comparing this to the second equation

6x + 8y = 7b

b must equal 2

[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: 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)

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.

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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-- 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:

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:

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:

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:

The first line is a dashed line which joins ordered pairs negative 1, 5 and negative 1, negative 4. The second line is a dashed line and joins ordered pairs negative 2, negative 2 and 4, 4. The portion common to the right of the first line and above the second line is shaded

Step-by-step explanation:

we have

First inequality

x+1 > 0

x > -1

The solution of the first inequality is the shaded area at right of the dashed line x=-1 (vertical line)

Second inequality

y-x > 0  

y> x

The solution of the second inequality is the shaded area above the dashed line y=x

therefore

The first line is a dashed line which joins ordered pairs negative 1, 5 and negative 1, negative 4. The second line is a dashed line and joins ordered pairs negative 2, negative 2 and 4, 4. The portion common to the right of the first line and above the second line is shaded

see the attached figure to better understand tye problem

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.

Answer:

c = sqrt( 29x^2 + 44x + 52)

Step-by-step explanation:

This is an application of the Pythagorean theorem

c = sqrt(a^2 + b^2)

a = 2x - 4

b = 5x + 6

c = sqrt( (2x - 4)^2 + (5x + 6)^2 )                           Substitute and expand

c = sqrt( 4x^2 -  16x + 16 + 25x^2 + 60x + 36)     Collect like terms

c = sqrt( 4x^2 + 25x^2 - 16x + 60x + 16+36)        Combine the like term pairs.

c = sqrt( 29x^2 + 44x + 52)

This does not factor into anything very nice.

Answer:

Madeline is standing 2x - 4 feet from the base of a tree, and the height of the tree is 5x + 6

Now we want to know the distance from Madeline's feet to the top of the tree.

You could picture it as a triangle rectangle, where the cathetus is the distance between Madeline and the tree and the distance between the floor and the top of the tree, in this case the distance between Madeline's feet and the top of the tree is the hypotenuse of such triangle rectangle, and can be obtained using the Pythagorean theorem: "the square of the hypotenuse is equal to the sum of the square cathetus"

then:

[tex]H^2 = (2x - 4)^2 + (5x + 6)^2[/tex]

[tex]H^2 = (4x^2 -16x + 16) + (25x^2  + 60x +36)[/tex]

[tex]H^2 = (29x^2 + 44x + 52)[/tex]

[tex]H = \sqrt{ (29x^2 + 44x + 52)}[/tex]

this is the distance from Madeline's feet to the top of the tree in terms of x.