how do you simplify this rational expression? please show your work.

Answer:

c) [tex]x=\frac{y^2\pm\sqrt{y^4-4a}}{2}[/tex]

d) [tex]x= \frac{-3}{p-q^2u}[/tex]

Step-by-step explanation:

c) [tex]y= \frac{\sqrt{x^2 + a} }{x}[/tex]

Solving the question:

[tex]y= \frac{\sqrt{x^2+a}}{x}\\Taking\,\,square\,\,on\,\,both\,\,sides\\(y)^2= (\frac{\sqrt{x^2+a}}{x})^2\\y^2= \frac{x^2+a}{x}\\y^2.x = x^2+a\\x^2 +a - y^2x =0\\Rearranging\\x^2 -y^2x +a =0\\Solving \,\,using\,\,quadratic\,\,equation\,\,\\x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\\where\,\, a= 1, b= -y^2 and c= a\\x=\frac{-(-y^2)\pm\sqrt{(-y^2)^2-4(1)(a)}}{2(1)}\\x=\frac{y^2\pm\sqrt{y^4-4a}}{2}[/tex]

d) [tex]\sqrt{\frac{px+3}{ux}}=q[/tex]

Solving to find value of x

[tex]\sqrt{\frac{px+3}{ux}}=q\\ Taking\,\, square\,\, on\,\, both\,\, sides\,\,\\(\sqrt{\frac{px+3}{ux}})^2=q^2\\\frac{px+3}{ux} = q^2\\px+3 = q^2.ux\\px = q^2.ux -3\\px - q^2.ux = -3\\x(p-q^2u) = -3\\x= \frac{-3}{p-q^2u}[/tex]

Answer: $8

Step-by-step explanation:

If 2.00 was 25%, then 25% * 4 = 100

2 * 4 = 8

1/2 *30*12=180

The area of the triangle is 180 meters squared

Hope this helped!

Answer:

180 meters

Step-by-step explanation:

Answer:

y=1/2(x-6)^2+3/2

Step-by-step explanation:

General equation of parabola centered at (h,k) and axis of symmetry is parallel to y-axis is given as

(x-h)^2=4p(y-k)^2

where

vertex is at (h,k)

p shows the distance of focus from vertex, f = (h, k + p)

and directrix is given at y = k - p

if value of p is >0 then the focus is above vertex

if value of p<0 then focus is below vertex

Given:

focus= (6,2)

directrix y=1

Comparing above with standard formula of parabola, we get

Also distance p is half the distance from the (6,2) to the directix y=1 which is  

1/2=p

As p>0, that means given parabola focus lies above vertex

hence parabola opens upwards

Finding vertex of parabola:

As focus is is at (6,2) and also 1/2 above the vertex so we get vertex at

V=(6,2-1/2)

V=(6,1.5)

Now by comparing above with the standard formula of parabola we have

h=6, k=1.5

Putting values in the formula we get

(x-6)^2=4(1/2)(y-1.5)

(x-6)^2=2(y-1.5)

(x-6)^2=2y-3

Or by re-arranging the terms equation can also be written as

f(x)=1/2(x-6)^2+3/2 !

The equation for a parabola with a focus at (6,2) and a directrix at y=1 is y = (x²-12x+37)/2. It is derived using the definition of a parabola.

The equation of a parabola can be derived using the definition of a parabola: the set of all points that are equidistant from a given point (the focus) and a given line (the directrix). The distance between a point (x,y) on the parabola and the focus (6,2) is equal to the absolute value of the difference between y-coordinate of the point and y-coordinate of the directrix line.

Mathematically, the distance to the focus is (x-6)² + (y-2)² and the distance to the directrix is |y-1|. For any point on the parabola, these distances are equal, so we have (x-6)² + (y-2)² = (y-1)². Solving for y, we obtain the quadratic equation y = (x²-12x+37)/2, which represents the equation of a parabola with a focus at (6,2) and a directrix at y=1.

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For this case we have to subtract the two distances (subtract the polynomials) and the difference will be given by the shorter miles of the second route.

[tex]6x ^ 2 + 2x-2- (5x ^ 2-8x-6) =[/tex]

Taking into account that:

[tex]- * + = -\\- * - = +\\6x ^ 2 + 2x-2-5x ^ 2 + 8x + 6 =[/tex]

Adding similar terms:[tex]6x ^ 2-5x ^ 2 + 2x + 8x-2 + 6 =\\x ^ 2 + 10x + 4[/tex]

So, the correct option is C

ANswer:

Option C

Answer:

The answer is B.

Step-by-step explanation:

Let us go through each of the points one by one:

The label A represents the radius of the base of the cone.

The label B represents the height of the cone.

The label C represents the origin of the base of the cone.

The label D represents the vertex of the cone (where the cone ends).  

So it is choice B that represents the height of the cone.

P.S: it is tempting to pick label D to represent the height, but since label A already points to a line that is the height of the cone, we don't pick Label D.

Answer:

the probability would be 8/18

when you simplify (divide by 2) the answer is 4/9

Step-by-step explanation:

Final answer:

The probability of selecting a chicken tortilla soup can from a total of 18 cans, 8 of which are chicken tortilla, is 4/9.

Explanation:

The probability that a randomly selected can will be chicken tortilla soup is calculated by dividing the number of chicken tortilla soup cans by the total number of cans. In this case, there are 8 chicken tortilla soup cans out of 18 total cans. To find the probability, you would perform the following calculation:

Probability = Number of chicken tortilla soup cans / Total number of cans
            = 8 / 18
            = 4 / 9

Therefore, the simplified fractional probability of selecting a chicken tortilla soup can is 4/9.

Answer:

The x-intercepts are (4,0) and (-9,0)

Step-by-step explanation:

We want to find the x-intercepts of the function: [tex]f(x)=x^2+5x-36[/tex]

At x-intercept, [tex]f(x)=0[/tex]

[tex]\implies x^2+5x-36=0[/tex]

We split the middle term to obtain;

[tex]x^2+9x-4x-36=0[/tex]

Factor by grouping:

[tex]x(x+9)-4(x+9)=0[/tex]

[tex](x-4)(x+9)=0[/tex]

Apply the zero product principle.

[tex](x-4)=0,(x+9)=0[/tex]

[tex]x=4,x=-9[/tex]

Hence the x-intercepts are (4,0) and (-9,0)

Answer:

Option A.

Step-by-step explanation:

Let A represents have soup and B represents having salad for lunch.

If two events are not dependent on each other, then they are known as independent events.

Probability of having soup is not dependent on Probability of having salad.

[tex]P(A\cap B)=P(A)P(B)[/tex]

Using the formula of conditional probability, we get

[tex]P(A|B)=\frac{P(A\cap B)}{P(B)}\Rightarrow \frac{P(A)P(B)}{P(B)}=P(A)[/tex]

[tex]P(B|A)=\frac{P(A\cap B)}{P(A)}\Rightarrow \frac{P(A)P(B)}{P(A)}=P(B)[/tex]

Having soup or salad for the lunch are two independent event because [tex]P(A|B)=P(A)[/tex] and [tex]P(B|A)=P(B)[/tex].

Therefore, the correct option is A.

The correct answer is:

c) Having soup and salad are not independent events because P(A|B) ≠ P(A) and P(B|A) ≠ P(B).

Given that two events A and B, A represent having soup and B represent having salad for lunch.

We need to determine if the events are independent.

To determine if having soup (A) and salad (B) for lunch are independent events, we need to compare the conditional probabilities with the individual probabilities.

The events A and B are independent if and only if:

P(A|B) = P(A) and P(B|A) = P(B)

where P(A|B) is the probability of having soup given that salad is already chosen, P(A) is the probability of having soup in general, P(B|A) is the probability of having salad given that soup is already chosen, and P(B) is the probability of having salad in general.

Now, let's evaluate each statement:

a) Having soup and salad for lunch are independent events because P(A|B) = P(A) and P(B|A) = P(B).

This statement suggests that both conditional probabilities are equal to the individual probabilities, which would mean the events are independent. However, this contradicts the definition of independent events, where both conditional probabilities should be equal for independence.

b) Having soup and salad for lunch are not independent events because P(A|B) = P(A) and P(B|A) = P(B).

This statement is also incorrect because it states that the conditional probabilities are equal to the individual probabilities, which would indicate independence, but that is not the case.

c) Having soup and salad are not independent events because P(A|B) ≠ P(A) and P(B|A) ≠ P(B).

This statement correctly states that both conditional probabilities are not equal to the individual probabilities, indicating that the events are not independent. This is the correct answer.

d) Having soup and salad for lunch are independent events because P(A|B) ≠ P(A) and P(B|A) ≠ P(B).

This statement is incorrect because it suggests that the events are independent based on the fact that the conditional probabilities are not equal to the individual probabilities. However, this reasoning is flawed, and the correct interpretation for independence is when the conditional probabilities are equal to the individual probabilities.

Therefore, the correct answer is:

c) Having soup and salad are not independent events because P(A|B) ≠ P(A) and P(B|A) ≠ P(B).

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Answer:

The third side must be smaller than 20 inches and greater than 4 inches

[tex]4<c<20[/tex]

Step-by-step explanation:

Let a, b and c be the lengths of triangle's sides. Then

[tex]a+b>c\\ \\a+c>b\\ \\b+c>a[/tex]

Use this rule in your case. So, if a=12 and b=8, then

[tex]12+8>c\\ \\12+c>8\\ \\8+c>12[/tex]

Hence, you get

[tex]c<20\\ \\c>-4\\ \\c>4[/tex]

From these inequalities, you can state that

[tex]4<c<20[/tex]

So, c must be smaller than 20 inches and greater than 4 inches.

Answer:

The possible lengths are all the real numbers greater than 4 inches and less than 20 inches

Step-by-step explanation:

we know that

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Let

c----> the length of the third side of triangle

Applying the triangle inequality theorem

1) 12+8 > c

20 > c

Rewrite

c < 20 in

2) 8+c > 12

c > 12-8

c < 4 in

therefore

4 in < c < 20 in

The possible lengths are all the real numbers greater than 4 inches and less than 20 inches

Answer:

Option A. 30 days

Step-by-step explanation:

Find the area of the brownie pan

The area of the trapezoid is equal to

step 2

To calculate the number of days, divide the total area of the brownie pan by 30 square centimeters of brownie

so

Answer:

C. 12.5,114,125

Step-by-step explanation:

:3:3:3:3::33

Answer:

D

Step-by-step explanation:

We have to find the list of equivalent numbers

A.1.25,[tex]1\frac{1}{4}[/tex],12.5%

[tex]1\frac{1}{4}=\frac{5}{4}=1.25[/tex]

12.5%=[tex]\frac{125}{1000}=0.125[/tex]

Given numbers are not equivalent

Hence, option A is false.

B.0.125,[tex]\frac{1}{4}[/tex],12.5%

[tex]\frac{1}{4}=0.25[/tex]

12.5%=[tex]\frac{125}{1000}=0.125[/tex]

Given numbers are not equivalent.

Hence, option B is false.

C.12.5,[tex]1\frac{2}{12}[/tex],125%

[tex]1\frac{2}{12}=\frac{14}{12}=1.167[/tex]

125%=[tex]\frac{125}{100}=1.25[/tex]

Given numbers are not equivalent.

Hence, option C is false.

D.1.25,[tex]1\frac{1}{4}[/tex],125%

[tex]1\frac{1}{4}=\frac{5}{4}=1.25[/tex]

125%=[tex]\frac{125}{100}=1.25[/tex]

Given numbers are equal.

Hence, option D is true.

Answer:

1 + 1 = 2

Other people say 1 +1 = window.

She gets 10000$hope dat helped

Answer:

the answer is $841.53

explanation:

this probably won't help now but it will help others :D

Answer:

(x - 4)(x + 1)

Step-by-step explanation:

Given

x² - 3x - 4

To factor the trinomial

Consider the factors of the constant term (- 4) which sum to give the coefficient of the x- term (- 3)

The factors are - 4 and + 1, since

- 4 × 1 = - 4 and - 4 + 1 = - 3, hence

x² - 3x - 4 = (x - 4)(x + 1)

#3 When you set the proportion make sure you pick sides that oppose same angle and take them in same order (for example big triangle first and small second)

( 2x+2)/10= (x-2+3)/(x-2);

(2x+2)(x-2)=10(x+1);

2x^2 -4x+2x-4=10x+10;

2x^2 -2x-4-10x-10=0;

2x^2 -12x-14=0; divide by 2

X^2 -6x-7=0; factor

(X-7)(x+1)=0 so x=7 and x=-1

Reject x=-1 because we can’t have negative dimensions so x=7 is the solution

Answer: 3 units

Step-by-step explanation:

the volume of a cone C = πr²h/3

C = 18π, h =2x, r = diameter/2 = 2x/2 =x

18π = π* x²* 2x/3

divide through by π

18 = x² * 2x/3

multiply through by 3

54 = 2x³

divide through by 2

27 = x³

x =∛27 = 3

x = 3 units = radius

Answer:

a. [tex]d=2.0t[/tex]

b. 26 meters

c. picture attached

Step-by-step explanation:

- Since the turtle travels 2.0 meters per minute, it will travel a total distance of [tex]2.0t[/tex] in [tex]t[/tex] minutes. We know that the total distance is [tex]d[/tex], so we can equate both total distances to create our rule:

[tex]d=2.0t[/tex]

- To find the distance of the turtle after 13 minutes we just need to replace [tex]t[/tex] with 13 in our rule:

[tex]d=2.0t[/tex]

[tex]d=2.0(13)[/tex]

[tex]d=26[/tex]

The turtle cover a distance of 26 meters after 13 minutes.

- To graph the function we are going to evaluate our function at two points, and then we are joining those two points with a line (check the attached picture)

We already know that the turtle cover a distance of 26 meters after 13 minutes, since the distance depends on the time our point will have coordinates (time, distance) (13 minutes, 26 meters) (13, 26)

Notice that when time is zero, distance is zero as well, so our second point is (0, 0)

Now we can join the points using a line:

Answer: second one

Step-by-step explanation:

Answer:

P = 2l + 2w

x = "a number"

54 = 2(l+w)

27 = l + w

27 = (x+2) + (2x-5)

27 = 3x - 3

27 = 3 (x-1)

9 = x-1

10 = x

Step-by-step explanation:

To solve for the unknown number given the rectangle's dimensions and perimeter, we set up an equation in terms of the unknown number, which represents the width, and solve for it, finding that the number is approximately 8.33.

The question involves finding a specific number given the perimeter of a rectangle and the relationships between its dimensions. First, let's represent the unknown number as x. Given, the length (L) is 2 cm more than x, so L = x + 2. The width (W) is twice x, so W = 2x. The perimeter (P) of a rectangle is given by 2(L + W), and we're told it is 54 cm.

Substituting the expressions for L and W into the perimeter formula gives: 2((x + 2) + 2x) = 54. Simplifying, we get 6x + 4 = 54, and further simplifying results in 6x = 50. Dividing both sides by 6 gives x = 8.33. Therefore, the unknown number is approximately 8.33.

Answer:

the third from the top option

Step-by-step explanation:

Answer:

a. x² + x - 30 = (x + 6)(x - 5)

b. -3x² + 23x - 14 = -[(3x - 2)(x - 7)]

c. 2x² - 5x + 4 can not factorize by this way

d. 6x² + 10x - 24 = 2[(3x - 4)(x + 3)]

Step-by-step explanation:

* To factor a trinomial in the form ax² ± bx ± c:

- Look at the c term

# If the c term is positive

∵ c = r × s ⇒ r and s are the factors of c

∴ r and s will have the same sign (sign of b)

∵ a = h × k ⇒ h , k are the factors of a

∴ rk + hs = b

∴ (hx + r)(kx + s) ⇒ if b +ve  OR  (hx - r)(kx - s) ⇒ if b -ve

# If the c term is negative

∵ c = r × s ⇒ r and s are the factors of c

∴ r and s will not have the same sign

∵ a = h × k ⇒ h and k are the factors of a

∴ rk - hs = b OR hs - rk = b

(hx + r)(kx - s) OR (hx - r)(kx + s)

* Now lets solve the problem

a. x² + x - 30

∵ ax² + bx + c

∴ a = 1 , b = 1 , c = -30

∵ c is negative

∴ r and s have different signs

∵ a = h × k

∵ 1 = 1 × 1

∴ h = 1 , k = 1

∵ c = r × s

∵ c = -30

∴ r × s = -30

∵ 6 × -5 = -30

∴ r = 6 , s = -5

∴ hs = 6

∴ rk = -5

∵ hs - rk = 6 - 5 = 1 ⇒ same value of b

∴ (x + 6)(x - 5)

* x² + x - 30 = (x + 6)(x - 5)

b. -3x² + 23x - 14 ⇒ take -1 as a common factor

∴ -(3x² - 23x + 14)

∵ ax² + bx + c

∴ a = 3 , b = -23 , c = 14

∵ c is positive

∴ r and s have same sign (-ve) because b is negative

∵ a = h × k

∵ 3 = 3 × 1

∴ h = 3 , k = 1

∵ c = r × s

∵ 14 = 2 × 7

∴ r = 2 , s = 7

∴ hs = 3 × 7 = 21

∴ rk = 2 × 1 = 2

∵ hs + rk = 21 + 2 = 23 ⇒ same value of b

∴ (3x - 2)(x - 7)

* -3x² + 23x - 14 = -[(3x - 2)(x - 7)]

c. 2x² - 5x + 4

∵ ax² + bx + c

∴ a = 2 , b = -5 , c = 4

∵ c is positive

∴ r and s have same sign (-ve) because b is negative

∵ a = h × k

∵ 2 = 2 × 1

∴ h = 2 , k = 1

∵ c = r × s

∵ 4 = 2 × 2

∴ r = 2 , s = 2

∴ hs = 2 × 2 = 4

∴ rk = 2 × 1 = 2

∵ hs + rk = 4 + 2 = 6 ⇒ not same value of b

∴ We can not factorize it

* 2x² - 5x + 4 can not factorize by this way

d. 6x² + 10x - 24 ⇒ take 2 as a common factor

∴ 2(3x² + 5x - 12)

∵ ax² + bx + c

∴ a = 3 , b = 5 , c = -12

∵ c is negative

∴ r and s have different signs

∵ a = h × k

∵ 3 = 3 × 1

∴ h = 3 , k = 1

∵ c = r × s

∵ -12 = -4 × 3

∴ r = -4 , s = 3

∴ hs = 3 × 3 = 9

∴ rk = -4 × 1 = -4

∵ hs - rk = 9 - 4 = 5 ⇒ same value of b

∴ (3x - 4)(x + 3)

* 6x² + 10x - 24 = 2[(3x - 4)(x + 3)]

the square root is 2

Answer:

2

Step-by-step explanation:

2 times 2 equals 4

Answer:

Total area equation = tex]w(w+6)=91[/tex]

b/2 = 3

Dimensions of the patio: width = 7 feet, length = 13 feet

Step-by-step explanation:

The area of a rectangle is given the formula:

[tex]A=wl[/tex]

where

[tex]w[/tex] is the width

[tex]l[/tex] is the length

We know from our problem that the area of the patio is 91 square feet, so [tex]A=91[/tex]. We also know that the length is 6 feet longer then the width, so [tex]l=w+6[/tex].

Replacing values in our area equation

[tex]A=wl[/tex]

[tex]91=w(w+6)[/tex]

[tex]w(w+6)=91[/tex]

Expanding the left side:

[tex]w*w+6w=91[/tex]

[tex]w^2+6w=91[/tex]

Remember that to complete the square we need to add half the coefficient of the linear term squared. The lineal term is [tex]w[/tex], so its coefficient is 6. Now, half its coefficient or [tex]\frac{b}{2} =\frac{6}{2} =3[/tex]. Finally, [tex]3^2=9[/tex].

To complete the square we need to add 9 to both sides of the equation:

[tex]w^2+6w+9=91+9[/tex]

[tex]w^2+6w+9=100[/tex]

Notice that the left side is a perfect square trinomial (both [tex]w^2[/tex] and 9 are perfect squares), so we can express it as:

[tex](w+3)^2=100[/tex]

Now that we completed the square, we can solve our equation

- Take square root to both sides

[tex]\sqrt{(w+3)^2} =\pm\sqrt{100}[/tex]

[tex]w+3=\pm10[/tex]

- Subtract 3 from both results

[tex]w=10-3,w=-10-3[/tex]

[tex]w=7,w=13[/tex]

Since length cannot be negative, [tex]w=7[/tex] is the solution of our equation.

We now know that the width of our rectangular patio is 7 feet, so we can find its length:

[tex]l=w+6[/tex]

[tex]l=7+6[/tex]

[tex]l=13[/tex]

We can conclude that half the coefficient of the width is [tex]\frac{b}{2}=3[/tex], the width of the patio is 7 feet, and its length is 13 feet.

Answer:

r = 16.9 cm

Step-by-step explanation:

We are given the circumference of a circle to be 106.81 cm and with the help of this, we are to find the radius of the circle.

We know that the formula for the circumference is given by:

[tex] 2\pi r [/tex]

so equating the given value to get:

[tex] 2 \pi r = 106.81 [/tex]

[tex] r = \frac { 106.81 } { 2\times \pi } [/tex]

r = 16.9 cm

For this case we have that by definition, the circumference of a circle is given by:

[tex]C = \pi * d[/tex]

Where:

d: It is the diameter of the circle.

[tex]d=2r[/tex]

Clearing we have:

[tex]C=2\pi*r\\r=\frac{C}{2\pi}\\r=\frac{106.81}{2\pi}\\r=17,0079[/tex]

Rounding the amount and [tex]\pi=3.14[/tex]

Then, the radius of the circle is 17 centimeters

ANswer:

[tex]r = 17 \ cm[/tex]

Answer:

No congruency statement can be made because the side lengths are unknown.

Step-by-step explanation:

For two triangles to be considered congruents they have to similar and of the same size.

In this case, we can say both triangles are similar (same shape) since their internal angles are the same (90°, 30° and 60°), but we cannot say if they are congruent (same size in addition to being similar) because we don't know anything about the length of their sides, they could be the same, or not.

Hot air rises and cold air sinks

Cold air is heavier and more dense then hot air and therefore sinks. Hot air rises. An interesting reason that hot air rises is because cold air sinks. When the cold air sinks it pushes up the hotter air.

Hope this helped!

[tex]

3\sqrt{8}\cdot4\sqrt{3} \\

12\sqrt{24} \\

12\cdot2\sqrt{6} \\

\boxed{24\sqrt{6}}

[/tex]

Final answer:

The product of (3 the square root of 8)(4 the square root of 3) is simplified to 24√6, which is the final simplified form as the radical cannot be simplified further.

Explanation:

The product of (3 the square root of 8) and (4 the square root of 3) involves using properties of radicals and multiplication. First, simplify the square root of 8 to 2√2 because 8 is 4 times 2, and 4 is a perfect square whose square root is 2. Now the expression becomes (3∙2√2)∙(4√3). We then multiply the coefficients (3∙2) and (4), which is 24, and the radicals √2 and √3 which is √6. The final product is 24√6, which cannot be simplified further as 6 is not a perfect square.

Answer:

11.5 (i think)

Step-by-step explanation:

23 is the diameter and half of the diameter is the radius so 23 divided by 2 is 11.5.

Answer:

Step-by-step explanation:

In the picture you have a diameter of d = 23 cm.

The diameter is twice the radius: d = 2r.

Therefore, r = d : 2 → r = 23 cm : 2 = 11.5 cm

Answer:

14x-6y=4 and 14x-28y=1

Step-by-step explanation:

we have

7x-3y=4 ----> equation A

2x-4y=1 ----> equation B

Multiply the equation A by 2 both sides

2*(7x-3y)=4*2

14x-6y=8

Multiply the equation B by 7 both sides

7*(2x-4y)=1*7

14x-28y=7

therefore

The system of equations that is not equal to the system of equations above is

 14x-6y=4 and 14x-28y=1