A teacher needs to choose seven students to hand out papers. The total number of ways he may choose the students can be found using a combination

Answer:

y = [tex]\frac{30-3x}{5}[/tex]

Step-by-step explanation:

Given

3x + 5y = 30

Isolate the term in y by subtracting 3x from both sides

5y = 30 - 3x ( divide both sides by 5 )

y = [tex]\frac{30-3x}{5}[/tex]

Answer:

[tex]y=\frac{-3}{5}x+6[/tex]

Step-by-step explanation:

3x+5y=30 is the given equation.

We are asked to isolate y.

The term that contains y is 5y. I'm going to first isolate 5y.

To isolate 5y we see that we have +3x with it.  To undo addition of 3x we will need to subtract 3x on both sides.

3x+5y=30

-3x          -3x

     5y=-3x+30

We now have 5y by itself.  There is actually one step left to get the y by itself.

The last step is undo this multiplication between 5 and y.

To undo multiplication you divide.

So we will be dividing 5 on both sides. This gives us:

3x+5y=30

-3x        -3x

     5y=-3x+30

     --     ---------

     5          5

So the whole reason we did that is because 5y/5 is just y.  I'm going to separate that fraction on the right hand side.  This gives me:

     y=-3/5  x   +  30/5

I'm going to simplify the 30/5 to 6.

    y=-3/5   x          +6

Answer:

The measure of arc x is 70°

Step-by-step explanation:

we know that

The measurement of the outer angle is the semi-difference of the arcs it encompasses.

and

The inscribed angle is half that of the arc it comprises.

so

The arc that comprises the inscribed angle of 15 degrees is equal to

15(2)=30 degrees

The outer angle of 20 degrees is equal to

20°=(1/2)[x-30°]

40°=[x-30°]

x=40°+30°=70°

Answer:

f(-7)=0.

f(2)=-9/5.

f(3) doesn't exist because 3 isn't in the domain of the function.

Step-by-step explanation:

[tex]f(x)=\frac{x+7}{x^2-9}[/tex] is the given function.

We are asked to find:

[tex]f(-7)[/tex]

[tex]f(2)[/tex]

[tex]f(3)[/tex].

f(-7) means to replace x in the expression called f with -7:

Evaluate [tex]\frac{x+7}{x^2-9}[/tex] at [tex]x=-7[/tex]

[tex]\frac{(-7)+7}{(-7)^2-9}[/tex]

[tex]\frac{0}{49-9}[/tex]

[tex]\frac{0}{40}[/tex]

[tex]0[/tex]

So f(-7)=0.

f(2) means to replace x in the expression called f with 2:

Evaluate [tex]\frac{x+7}{x^2-9}[/tex] at [tex]x=2[/tex]

[tex]\frac{2+7}{2^2-9}[/tex]

[tex]\frac{9}{4-9}[/tex]

[tex]\frac{9}{-5}[/tex]

[tex]\frac{-9}{5}[/tex]

So f(2)=-9/5

f(3) means to replace x in the expression called f with 3:

Evaluate [tex]\frac{x+7}{x^2-9}[/tex] at [tex]x=3[/tex]

[tex]\frac{3+7}{3^2-9}[/tex]

[tex]\frac{10}{9-9}[/tex]

[tex]\frac{10}{0}[/tex]

Division by 0 is not allowed so 3 is not in the domain of our function.

Answer:

Yes.

Step-by-step explanation:

It can be noticed that 1/e can be written as e^(-1). Whenever a number or a variable is shifted from the numerator to the denominator or vice versa, the power of that number or that variable becomes negative. This means that 1/e = e^(-1). Since f(x) = (1/e)^x, f(x) can be written as:

f(x) = (e^(-1))^x.

Since there are two powers and the base is same, so both the powers will be multiplied. Therefore:

f(x) = e^(-x).

It can be seen that e is involved in the function and it has a negative power of x, so this means that f(x) is a decreasing exponential function!!!

Answer:

Measure of arc PQR is 190°

Step-by-step explanation:

First have a sketch of the quadrilateral PQRS inside a circle

You should notice that the intercepted angle is ∠PSR

You should remember that the intercepted arc PQR is twice the intercepted angle ∠PSR

Find the intercepted angle ∠PSR

Remember that in a quadrilateral opposite angles add up to 180°

Hence;

∠PQR+∠PSR=180°

85 + ∠PSR=180°

∠PSR=180°-85°=95°

Find arc PQR

Arc PQR =2×∠PSR

Arc PQR=2×95°

=190°

Answer:

-4

Step-by-step explanation:

So we are given f(t)=-2t+5.

We are asked to find the value a such that f(a)=13.

If f(t)=-2t+5, then f(a)=-2a+5.  I just replaced old input,x, with new input, a.

So we want to solve f(a)=13 for a.

f(a)=13

Replace f(a) with -2a+5:

-2a+5=13  

Subtract 5 on both sides:

-2a    =8

Divide both sides by -2:

 a       =-4

Check:

Is -2t+5=13 for t=-4?

-2(-4)+5=8+5=13, so yep.

The value of t is -4 for which the function will give 13 as output.

A mathematical relationship from a set of inputs to a set of outputs is called a function.

The given function is F(t) = -2t+5

So,   -2t + 5 = 13

⇒  -2t = 8

⇒ t = -4

So, for the value of t = -4 the given function will give the output 13

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Answer:

So the solution is -2<x<4 with it's graph is

       O~~~~~~O

------(-2)--------(4)-----------

Step-by-step explanation:

The intersection symbol means it has to be included in both sets.

We have that x<4 and x>-2.

If we graph this, where do we see both shadings:

~~~~~~~~~~~~~~~0                   x<4

         O~~~~~~~~~~~~~~~~~    x>-2

-------(-2)-------------(4)------------

Both shadings happen between -2 and 4 (and not outside that area).

So the solution is -2<x<4

It's graph is

       O~~~~~~O

------(-2)--------(4)-----------

Disclaimer:

I didn't see any equal signs in your problem.

You know the [tex]\ge[/tex] or the [tex]\le[/tex] so we didn't have any closed dots.

The graph is, °←--------------------→°

                   -2____0_________4

The intersection of two sets is the set of all those elements which are common to both of the sets.

Here, given that, {x | x < 4} ∩ {x | x > -2}

It indicates that, x is less than 4, i.e. x= 3,2,1,0,-1,-2,-3,......

Again, x is greater than -2, i.e. x = -1,0,1,2,3,4,....................

Now, as it is intersection of this two sets

So, we have, x= -1,0,1,2,3.

Hence, the required graph is,

       °←------------------------------------→°

_(-2)__________0____________4_____

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Supplementary angles are 2 angles that have the sum of 180 degrees. Therefore, the requirement of supplementary angles is to have the 2 angles to equal 180 degrees.

Hope this helps!

Answer:

the sum of angles must be 180 degrees

Step-by-step explanation:

The definition of supplementary angles is

"Two angles are supplementary angles if their sum is 180 degrees".

For example, if x and y are supplementary angles then

x + y = 180

Therefore, we can conclude that the requirement of supplementary angles is " the sum of angles must be 180 degrees".

Answer:

The correct option is A.

Step-by-step explanation:

The given expression is:

(a^-8b/a^-5b^3)^-3

We have to keep one thing in our mind that in a division, same base, the exponents are subtracted.

We will change the division into multiplication: The denominator will become numerator and the signs of the exponent become opposite.

(a^-8 *a^+5 * b *b^-3)^-3

=(a^-8+5 * b^1-3) ^-3

=(a^-3*b^-2)^-3

=(a)^-3*-3 (b)^-2*-3

= a^9 b^6

Thus the correct option is A....

Answer:

a^9b^6

Step-by-step explanation:

Answer:

Option B is correct.

Step-by-step explanation:

we are given [tex]f(x) = \frac{x}{2}-3[/tex]

and [tex]g(x) = 3x^2+x-6[/tex]

We need to find (f+g)(x)

We just need to add f(x) and g(x)

(f+g)x = f(x) + g(x)

[tex](f+g)(x)=(\frac{x}{2}-3)+(3x^2+x-6)\\(f+g)(x)=\frac{x}{2}-3+3x^2+x-6\\(f+g)(x)=3x^2+\frac{x}{2}+x-3-6\\(f+g)(x)=3x^2+\frac{3x}{2}-9\\[/tex]

So, Option B is correct.

For this case we have the following functions:

[tex]f (x) = \frac {x} {2} -3\\g (x) = 3x ^ 2 + x-6[/tex]

We must find [tex](f + g) (x).[/tex] By definition, we have to:

[tex](f + g) (x) = f (x) + g (x)[/tex]

So:

[tex](f + g) (x) = \frac {x} {2} -3+ (3x ^ 2 + x-6)\\(f + g) (x) = \frac {x} {2} -3 + 3x ^ 2 + x-6\\(f + g) (x) = + 3x ^ 2 + x + \frac {x} {2} -3-6\\(f + g) (x) = + 3x ^ 2 + \frac {2x + x} {2} -9\\(f + g) (x) = + 3x ^ 2 + \frac {3x} {2} -9[/tex]

Answer:

Option B

Answer:

3 packages

Step-by-step explanation:

To find how many packages you can buy for $9, you have to divide 9 by 3.

Since each package costs $3.

So, 9/3 = 3

So you can buy 3 packages of blueberries.

Answer:

3 packages

Step-by-step explanation:

Since you can buy a package for $3, and you only have $9, how many $3 can you fit? Only 3. In conclusion, you can only buy 3 packages.

Answer:

Choice A. [tex]F(x) = -(x - 4)^{2} -3[/tex].

Step-by-step explanation:

Both F(x) and G(x) are quadratic equations. The graphs of the two functions are known as parabolas. All four choices for the equation of F(x) are written in their vertex form. That is:

[tex]y = a(x - h)^{2} + k[/tex], [tex]a \ne 0[/tex]

where

For the parabola G(x),

In other words,

Hence choice A. [tex]F(x) = -(x - 4)^{2} -3[/tex].

The only equation that correctly represents the graph of F(x) as a vertically stretched and reflected version of the graph of G(x), shifted to the right by 4 units.

Option A. [tex]F(x)=-(x-4)^{2}-3[/tex] is correct.  

To answer this question, we can consider the following transformations of the graph of [tex]G(x)=x^2$:[/tex]

Vertical stretch: [tex]$F(x)=kx^2$[/tex] for some positive constant k.

This will stretch the graph of G(x) vertically by a factor of k.

Vertical shift: [tex]$F(x)=x^2+h$[/tex] for some constant $h$.

This will shift the graph of G(x) up or down by $h$ units.

* **Horizontal shift: [tex]F(x)=(x+h)^2[/tex] for some constant $h$. This will shift the graph of G(x) to the left or right by h units.

* **Reflection: [tex]F(x)=-x^2[/tex]. This will reflect the graph of G(x) across the x-axis.

The graph of F(x) in the image appears to be a vertically stretched and reflected version of the graph of G(x), shifted to the right by 4 units. This suggests that the equation of F(x) could be of the form:

F(x)=-k(x-h)^2 for some positive constant k and some constant h.

To find the values of k and h, we can use the fact that the graph of F(x) passes through the points (-2,3) and (4,-3).

Substituting x=-2 and y=3 into the equation above, we get:

[tex]3=-k(-2-h)^2[/tex]

and substituting x=4 and y=-3 into the equation above, we get:

[tex]-3=-k(4-h)^2[/tex]

Dividing the second equation by the first equation, we get:

[tex]\frac{1}{3}=\frac{(-2-h)^2}{(4-h)^2}[/tex]

Taking the square root of both sides, we get:

[tex]\pm \frac{1}{\sqrt{3}}=\frac{-2-h}{4-h}[/tex]

Solving for $h$, we get:

[tex]h=4\pm \frac{4}{\sqrt{3}}[/tex]

Substituting this value of h into either of the original equations, we can solve for k.

For example, substituting [tex]$h=4+\frac{4}{\sqrt{3}}$[/tex] into the first equation, we get:

[tex]3=-k\left(-2-4+\frac{4}{\sqrt{3}}\right)^2[/tex]

Solving for $k$, we get:

[tex]k=3\cdot \left(-6+\frac{4}{\sqrt{3}}\right)^2[/tex]

Therefore, one possible equation for $F(x)$ is:

[tex]F(x)=-3\left(x-4+\frac{4}{\sqrt{3}}\right)^2[/tex]

Another possible equation for $F(x)$ is:

[tex]F(x)=-3\left(x-4-\frac{4}{\sqrt{3}}\right)^2[/tex]

These two equations are equivalent, since they both represent the same reflected and vertically stretched version of the graph of G(x),  shifted to the right by 4 units.

Therefore, the answer to the question is:

[tex]A. F(x)=-(x-4)^{2}-3[/tex]

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Answer:

I measures 60° ⇒ 3rd answer

Step-by-step explanation:

* Look to the attached figure

- To prove that the two triangles are similar by SAS we must to

  find two proportional pairs of corresponding side and the measure

  of the including angles between them are equal

- The given is:

  m∠ F = 60°

  EF = 40 , FG = 20

  HI = 20 , IJ = 10

- To prove that the Δ EFG is similar to Δ HIJ we must to prove

# EF/ HI = FG/IJ ⇒ two pairs of sides proportion

# m∠ F = m∠ I ⇒ including angles equal

∵ EF = 40 and HI = 20

∴ EF/HI = 40/20 = 2

∵ FG = 20 and IJ = 10

∴ FG/IJ = 20/10 = 2

∵ EF/HI = FG/IJ = 2

∴ The two pairs of sides are proportion

∵ ∠ F is the including angle between EF and FG

∵ ∠ I is the including angle between HI and IJ

∴ m∠ F must equal m∠ I

∵ m∠ F = 60° ⇒ given

∴ m∠ I = 60°

* I measures 60°

To prove triangles are similar by the SAS similarity theorem when dealing with equilateral triangles, the angle between the proportional sides needs to be congruent. In this context, showing that the angle measures 30° would be necessary if one angle is already known to be 60°. Thus, either angle I or J should be proven to measure 30°, depending on which is between the proportional sides.

To prove that triangles are similar by the SAS similarity theorem, two conditions must be met: corresponding sides are in proportion, and the angles between those sides are congruent. In the context of proving triangle similarity with equilateral triangles, by using a bisector, we can form a smaller triangle whose hypotenuse is twice as long as one of its sides. Consequently, as the sides preserve this ratio, the angles approach 30° and 60°. Therefore, if we must prove that a triangle in a given figure is similar through the SAS similarity criterion, and we have a side-angle-side situation where one angle is already known to be 60°, the angle we would need to prove congruent would reasonably be 30°.

This conclusion is supported by the nature of equilateral triangles, where bisecting the angle will give us angles of 30° and 60°. Additionally, the similarity of triangles is further backed by the congruency of angles and the proportionality of sides, as mentioned in other contexts within the provided information. Therefore, based on the given context and the properties of equilateral triangles, we would aim to prove that angle I or angle J (whichever is between the proportional sides) measures 30°.

Answer:

7

Step-by-step explanation:

The equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

y = 8x + 7 ← is in slope- intercept form

with y- intercept c = 7

Answer:

The domain is {x : x ∈ R} or (-∞ , ∞)

Step-by-step explanation:

* Lets explain how to find the domain

- The domain of the function is the values of x which make the

  function defined

- The quantity under the square root must be ≥ 0 because there is

  no square root for negative value

* Lets solve the problem

∵ f(x) = √(x² - x + 6)

∴ The value of (x² - x + 6) must be greater than or equal zero because  

   there is no square root for negative value

- Graph the function to know which values of x make the quantity

  under the root is negative that means the values of x which make

  the graph under the x-axis

∵ The graph doesn't intersect the x-axis at any point

∵ All the graph is above the x-axis

∴ There is no value of x make f(x) < 0

∴ x can be any real number

∴ The domain of f(x) is all real numbers

∴ The domain is {x : x ∈ R} or (-∞ , ∞)

Answer:

(27,000 - 22,900)/27,000 = 0.1519 or 15.19% decrease in price.

Step-by-step explanation:

For this case, we propose a rule for three:

27,000 -----------> 100%

22,900 -----------> x

Where "x" represents the percentage associated with 22,900.

[tex]x = \frac {22,900 * 100} {27,000}\\x = 84.8148148148[/tex]

Now we look for the percentage of decrease:

100% -84.8148148148% = 15.1851851852%

Rounding off we have:

15.19%

Answer:

15.19%

Answer: 1.6 or 8/5

Step-by-step explanation: If you want to simplify it, the answer would be 8/5 because each number can be divided by 3. If you need the decimal, it would be 1.6.

Step-by-step explanation:

[tex]\dfrac{24}{15}=\dfrac{24:3}{15:3}=\dfrac{8}{5}\\\\\\\dfrac{8}{5}=\dfrac{5+3}{5}=1\dfrac{3}{5}\\\\\\1\dfrac{3}{5}=1\dfrac{3\cdot2}{5\cdot2}=1\dfrac{6}{10}=1.6[/tex]

Answer:

Step-by-step explanation:

A cube has 6 faces and faces are square in shape. Thus to find the surface area of a cube, first we will find the area of one face. To find the area of a face we will simply multiply the side length twice.

20*20=400

Thus the area of one face = 400 units square.

Now to find the surface area  of a cube we will simply multiply the area of one face by the number of total faces of a cube.

400*6 = 2400

Therefore the net folded to form a cube has a surface area of 2400 units square....

Answer:

L ≅ P

Step-by-step explanation:

To prove congruence through ASA, we need the measures of two angles that are adjacent to a given side. If we are given that two angles are congruent to other angles in another triangle respectively and both of those angles are adjacent to one side (in which each side of each triangle is congruent) then the triangles are congruent through ASA.

To prove congruence using the ASA congruence theorem, we need to show that the angle measures of the triangles are equal. The additional information needed is that angle NM is congruent.

To prove that the triangles are congruent using the ASA congruence theorem, we need to show that the angle measures of the triangles are equal.

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Answer:

Please see attached image.

Step-by-step explanation:

When you multiply a function by -1 you are basically reflecting the values of the function over the x-axis.

The values that were negative become positive and the positive values become negative.

Please, take a look at the attached graph below, where there are examples with two functions.

Answer:

[tex]y=\frac{3}{5} x+1[/tex] and [tex]5y=3x-2[/tex] are parallel.

[tex]10x-6y=-4[/tex] is neither parallel nor perpendicular.

Step-by-step explanation:

First, you have to simplify each equation in terms of y.

[tex]y=\frac{3}{5} x+1\\5y=3x-2\\10x-6y=-4[/tex]

Your first equation is already in terms of x, so simplify your second equation.

[tex]5y=3x-2\\y=\frac{3}{5} x-\frac{2}{5}[/tex]

Now you can simplify your third equation.

[tex]10x-6y=-4\\-6y=-10x-4\\y=\frac{5}{3} x+\frac{2}{3}[/tex]

These are your three equations in terms of y:

[tex]y=\frac{3}{5} x+1\\\\y=\frac{3}{5} x-\frac{2}{5} \\\\y=\frac{5}{3} x+\frac{2}{3}[/tex]

Now, all you have to know is how to tell using your slope if a line is parallel or perpendicular to another.

Two parallel lines will have the exact same slope.

Two perpendicular lines will have slopes which are opposite reciprocals. For example, a line with a slope of 2 is perpendicular to a line with a slope of [tex]-\frac{1}{2}[/tex], as they have opposite signs and are reciprocal (2/1 versus 1/2) to each other.

Your first two equations have the same slope and are therefore parallel.

Your third equation is a reciprocal, but it is not opposite, and is therefore not parallel nor perpendicular.

The first and the second lines are parallel as they both have a slope of 3/5. None of the lines is perpendicular to the others.

In geometry and algebra, lines can be either parallel, perpendicular, or neither. To determine this, we need to look at the slope of each line. The slope of a line is the value of 'm' in the equation of the line, y = mx + c.

The equations you have provided are:

When two lines have the same slope, they are parallel. When two lines have slopes that are negative reciprocals of each other (meaning their product is -1), they are perpendicular. In this case, the first and the second line are parallel (both have a slope of 3/5), and none of these lines is perpendicular to the others.

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Answer:

Her oven will cool to 75°F in 100 minutes, or 1 hour 40 minutes.

Step-by-step explanation:

Let T(t) represent the oven temperature, and t the time in minutes.

Then:

T(t) = 400°F - (3.25°F/min)t

Find the time, t, at which T(t) will = 75°F:

400°F - (3.25°F/min)t = 75°F

Subtracting 75°F from both sides, we get:

(3.25°F/min)t = 325°F.

Dividing both sides by (3.25°F/min) yields:

t = 100 minutes.

Her oven will cool to 75°F in 100 minutes, or 1 hour 40 minutes.

It will take 100 minutes for Laura's oven to cool down from 400°F to 75°F at a cooling rate of 3.25°F per minute.

The temperature must drop from 400°F to 75°F, so the temperature difference: 400°F - 75°F = 325°F.

325°F / 3.25°F per minute = 100 minutes.

Answer:

So, Option A is correct.

Step-by-step explanation:

If the triangles are similar, then the sides are proportional

If triangle ABC is similar to triangle PQR

then sides

AB/PQ = BC/QR = AC/PR

We need to find PQ

We are given AB = 6, BC =8 and QR=24

AB/PQ = BC/QR

Putting values:

6/PQ = 8/24

Cross multiplying:

6*24 = 8*PQ

144/8 = PQ

=> PQ = 18

So, Option A is correct.

Answer: Option A

[tex]PQ=18[/tex]

Step-by-step explanation:

Two triangles are similar if the length of their sides is proportional.

In this case we have the triangle ∆ABC and ∆PQR so for the sides of the triangles they are proportional it must be fulfilled that:

[tex]\frac{AB}{PQ}=\frac{BC}{QR}=\frac{AC}{PR}[/tex]

In this case we know that:

[tex]BC=8[/tex]

[tex]QR=24[/tex]

[tex]AB=6[/tex]

Therefore

[tex]\frac{BC}{QR}=\frac{AB}{PQ}[/tex]

[tex]\frac{8}{24}=\frac{6}{PQ}[/tex]

[tex]PQ=6*\frac{24}{8}[/tex]

[tex]PQ=18[/tex]

Answer:

y ≤ 3x - 2  and  4x + y ≥2

Step-by-step explanation:

We can easily solve this problem by using a plotting tool or any graphing calculator.

We test each case, until we arrive to the correct option

Case 1

y ≤ 3x - 2  and  4x + y ≥2

Correct Option.

Please see attached image for graph

Answer:

x = 40

Step-by-step explanation:

The three angles in a triangle add to 180 degrees

x + 30 + (3x-10) = 180

Combine like terms

4x +20 = 180

Subtract 20 from each side

4x+20-20 = 180-20

4x= 160

Divide each side by 4

4x/4 =160/4

x = 40

Answer:

* sin Ф = -15/17     * cos Ф = 8/17      * tan Ф = -15/8

* csc Ф = -17/15    * sec Ф = 17/8      * cot Ф = -8/15

Step-by-step explanation:

Lets revise the trigonometric function of angle Ф

- Angle θ is in standard position

- Point  (8, -15) is on the terminal ray of angle θ

- That means the terminal is the hypotenuse of a right triangle x and y

  are its legs

∵ x-coordinate is positive and y-coordinate is negative

∴ angle Ф lies in the 4th quadrant

- The opposite side of angle Ф is the y-coordinate of the point on the

  terminal ray of angle Ф and the adjacent side to angle Ф is the

  x-coordinate of that point

∵ The length of the hypotenuse (h) = √(x² + y²)

∴ h = √[(8)² + (-15)²] = √[64 + 225] = √[289] = 17

∴ The length of the hypotenuse is 17

- Lets find sin Ф

∵ sin Ф = opposite/hypotenuse

∵ The opposite is y = -15

∵ The hypotenuse = 17

∴ sin Ф = -15/17

- Lets find cos Ф

∵ cos Ф = adjacent/hypotenuse

∵ The adjacent is x = 8

∵ The hypotenuse = 17

∴ cos Ф = 8/17

- Lets find tan Ф

∵ tan Ф = opposite/adjacent

∵ The opposite is y = -15

∵ The adjacent = 8

∴ tan Ф = -15/8

- Remember csc Ф is the reciprocal of sin Ф

∵ csc Ф = 1/sin Ф

∵ sin Ф = -15/17

∴ csc Ф = -17/15

- Remember sec Ф is the reciprocal of cos Ф

∵ sec Ф = 1/ cos Ф

∵ cos Ф = 8/17

∴ sec Ф = 17/8

- Remember cot Ф is the reciprocal of tan Ф

∵ cot Ф = 1/tan Ф

∵ tan Ф = -15/8

∴ cot Ф = -8/15

Answer:

x = - 5, x = 4

Step-by-step explanation:

Given

f(x) = x² + x - 20

To find the zeros equate f(x) to zero, that is

x² + x - 20 = 0

Consider the factors of the constant term ( - 20) which sum to give the coefficient of the x- term ( + 1)

The factors are + 5 and - 4, since

5 × - 4 = - 20 and + 5 - 4 = + 1, hence

(x + 5)x - 4) = 0 ← in factored form

Equate each factor to zero and solve for x

x + 5 = 0 ⇒ x = - 5

x - 4 = 0 ⇒ x = 4

Answer:

The zeroes are {-5, 4}.

Step-by-step explanation:

f(x)  =  x^2 + x - 20 = 0

To factor this function we need 2 numbers whose sum is + 1 ( x = 1x) and whose product is -20. They are 5 and - 4 so we have:

(x + 5)(x - 4) = 0

x + 5 = 0 gives x = -5

and x - 4 = 0 gives x =  4.

Answer:

Step-by-step explanation:

√(6-a)^2+(-2-a)^2+(0-a)^2 = 10

36-12a+a^2+4+4a+a^2+a^2 =100

3a^2 -8a -60=0

(3a+10)(a-6)=0

a= -3/10 or 6

Answer:

A

Step-by-step explanation:

The definition of a parabola.

Any point on the parabola (x, y) is equidistant from a point ( the focus ) and a line ( the directrix ).

Parabola is best describe by:

"Parabola is defined as the set of all the points in a plane that are equidistant from a fixed point known as focus and a fixed line called directrix."

According to the question,

A. The set of all points in a plane that are equidistant from a single

point and a single line.

It represents the parabola.

Option(A) is the correct answer.

B. The set of all points in a plane that are equidistant from two points.

It represents the line.

Option (B) is not the correct answer.

C. The set of all points in a plane at a given distance from a given

point.

It represents the circle.

Option (C) is not the correct answer.

D. The set of all points in a plane that are equidistant from two points

     and a single line.

It represents the perpendicular bisector.

Option (D) is not the correct answer.

Hence, Option(A) is the correct answer.

Learn more about parabola here

brainly.com/question/10572747

#SPJ2

Answer:

2nd choice.

Step-by-step explanation:

Let's compare the following:

[tex]f(x)=a\sin(b(x-c))+d[/tex] to

[tex]f(x)=-3\sin(4x-\pi))-5[/tex].

They are almost in the same form.

The amplitude is |a|, so it isn't going to be negative.

The period is [tex]\frac{2\pi}{|b|}[/tex].

The phase shift is [tex]c[/tex].

If c is positive it has been shifted right c units.

If c is negative it has been shifted left c units.

d is the vertical shift.

If d is negative, it has been moved down d units.

If d is positive, it has been moved up d units.

So we already know two things:

The amplitude is |a|=|-3|=3.

The vertical shift is d=-5 which means it was moved down 5 units from the parent function.

Now let's find the others.

I'm going to factor out 4 from [tex]4x-\pi[/tex].

Like this:

[tex]4(x-\frac{\pi}{4})[/tex]

Now if you compare this to [tex]b(x-c)[/tex]

then b=4 so the period is [tex]\frac{2\pi}{4}=\frac{\pi}{2}[/tex].

Also in place of c you see [tex]\frac{\pi}{4}[/tex] which means the phase shift is [tex]\frac{\pi}{4}[/tex].

The second choice is what we are looking for.

Answer: Second Option

Amplitude = 3; period = pi over two; phase shift: x equals pi over four

Step-by-step explanation:

By definition the sinusoidal function has the following form:

[tex]f(x) = asin(bx - c) +k[/tex]

Where

[tex]| a |[/tex] is the Amplitude of the function

[tex]\frac{2\pi}{b}[/tex] is the period of the function

[tex]-\frac{c}{b}[/tex] is the phase shift

In this case the function is:

[tex]f(x) = -3 sin(4x - \pi) - 5[/tex]

Therefore

[tex]Amplitude=|a|=3[/tex]

[tex]Period =\frac{2\pi}{b} = \frac{2\pi}{4}=\frac{\pi}{2}[/tex]

[tex]phase\ shift = -\frac{(-\pi)}{4}=\frac{\pi}{4}[/tex]