What are the zeros of g(x)= x³ + 6x² - 9x-54?

Answer:

see explanation

Step-by-step explanation:

the equation of parabola in vertex form is

y = a(x - h)² + k

where (h, k ) are the coordinates of the vertex and a is a multiplier.

here (h, k ) = (3, 1 ) , then

y = a(x - 3)² + 1

to find a substitute any other point on the graph into the equation.

using (0, 7 )

7 = a(0 - 3)² + 1 ( subtract 1 from both sides )

6 = a(- 3)² = 9a ( divide both sides by 9 )

[tex]\frac{6}{9}[/tex] = [tex]\frac{2}{3}[/tex] = a

y = [tex]\frac{2}{3}[/tex] (x - 3)² + 1 ← in vertex form

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

the equation of a parabola in factored form is

y = a(x - a)(x - b)

where a, b are the zeros and a is a multiplier

here zeros are - 1 and 3 , the factors are

(x - (- 1) ) and (x - 3), that is (x + 1) and (x - 3)

y = a(x + 1)(x - 3)

to find a substitute any other point that lies on the graph into the equation.

using (0, - 3 )

- 3 = a(0 + 1)(0 - 3) = a(1)(- 3) = - 3a ( divide both sides by - 3 )

1 = a

y = (x + 1)(x - 3) ← in factored form

The required sample size is 694.

Given a student wants to make a guess within 0.03 of the true ratio with a 95% confidence level.

Margin of Error, E = 0.03

significance level α=0.05  {95% confidence}

A given estimate of the percentage of population p is p = 0.79.

The critical value for the significance level α = 0.05 is Zc = 1.96. This can be determined using either Excel or a normal probability table.

Use the following formula to calculate the minimum sample size required to estimate the percentage of population p within the required margin of error.

n≥p(1-p)(Zc÷E)²

n=0.796×(1-0.796)(1.96÷0.03)²

n=693.1016

Therefore, the  resample sizequired to meet the condition is n ≥ 693.1016 and must be an integer. From this, we conclude that the minimum sample size required is n = 694.

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Answer:

-ab^2

Step-by-step explanation:

7b^2a - 5b^2a - 3ab^2

= 2b^2a - 3ab^2

= (2 × b^2 × a) - (3 × a × b^2)

= -ab^2

b^2a and ab^2 is the same thing. hope it helps!

Answer:

B. Linear increasing

Step-by-step explanation:

decay means decreasing

exponential graph would have a curve

alison, beatrice and chole each of them have at first 56, 8, 32 books.

alison, beatrice and chole each had some books. alison gave beatrice and chole some books that doubled the number of books they had, lastly chole gave alison and beatrice some books that doubled the number of books they had. each of them had 32 books at the end.

Working backward is process in which calculation done in reverse direction.

let alison, beatrice and chole has x,y and z books initially,

alison gave beatrice and chole some books that doubled the number of books they had
alison, beatrice and chole has (96-2y-2z), (2y) , (2z)
chole gave alison and beatrice some books that doubled the number of books they had. each of them had 32 books at the end.

alison, beatrice and chole has 2(96-2y-2z), 2(2y) , (96-2(96-2y-2z)-4y)

at the end,
beatrice = 32

4y=32

y = 8

chole =32

96-2(96-2y-2z)-4y)=32
z=32

alison = 96-y-z

x=96-8-32
x=56

Thus the required alison, beatrice and chole each of them have at first 56, 8, 32 books.

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Answer:  x = 7

Step-by-step explanation:

[tex]\frac{x}{8} =\frac{28}{32}[/tex]

We cross multiply, and we get:

32x = 8(28)

32x = 224

    x = 7

Answer: a. -52 1/2

Step-by-step explanation:

    Given:

3/4(2/3 + 4/3) - 9 / 1/2 x 1/3

[tex]\displaystyle \frac{3}{4}(\frac{2}{3} +\frac{4}{3} )-\frac{9}{\frac{1}{2} *\frac{1}{3} }[/tex]

    Add and multiply:

[tex]\displaystyle \frac{3}{4}(\frac{6}{3} )-\frac{9}{\frac{1}{6} }[/tex]

    Simplify and divide:

[tex]\displaystyle \frac{3}{4}(2 )-54[/tex]

    Multiply:

[tex]\displaystyle 1.5-54[/tex]

    Subtract:

         a. -52 1/2

By apply the distributive property of multiplication, the value of k which makes the statement true is 1.

Mathematically, the distributive property of multiplication is given by tis expression:

x(y + z) = xy + xz.

Next, we would apply the distributive property of multiplication to the given equation by distributing to the left-hand side using xy⁴:

k(2x⁴y⁴ + 7x³y⁸) = 2x⁴y⁴ + 7x³y⁸

Dividing both sides by the common factor, we have:

k = (2x⁴y⁴ + 7x³y⁸)/(2x⁴y⁴ + 7x³y⁸)

k = 1.

In conclusion, the value of k which makes the statement true is 1.

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Complete Question:

What value of k makes the statement true?

[tex]x^k[/tex]y⁴(2x³ + 7x²y⁴) = 2x⁴y⁴ + 7x³y⁸

Answer: (1, -3)

Step-by-step explanation:

[tex]8x+4y=-4\\-4x-5y=11[/tex]

[tex]2(-4x-5y)=11(2)\\-8x-10y=22[/tex]

new system:

[tex]8x+4y=-4\\-8x-10y=22[/tex]

add.

[tex]-6y=18\\y=-3[/tex]

plug in -3 for y

[tex]8x+4(-3)=-4\\8x-12=-4\\8x=8\\x=1[/tex]

system:

[tex](1,-3)[/tex]

Answer:

A midpoint of a line segment implies a point on the line that is equidistant to its two ends. So that the required proof is shown below:

Given: line segment MN

           x, the midpoint of MN

           MX = RX

Then;

MX = XN (midpoint property of a line)

So that,

MN = MX + XN

      = MX + MX (since XN = MX)

MN = 2MX

Thus,

MX = XN = RX (given that MX = RX)

Therefore, it can be concluded that;

XN = RX

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Answer:

a.) The pizzas will cost the same at 4 toppings

Step-by-step explanation:

Medium Pizza:

$5.99 + $0.75x

Large Pizza:

$6.99 + $0.50x

Find x

$5.99 + $0.75x =  $6.99 + $0.50x

0.25x = 1

x = 4

Jane goes to the beach if and only if it is a sunny day.

========================================================

Reason:

A regular conditional statement is in the form "If P, then Q" where P and Q are placeholders for other statements.

For example, we can replace "P" with "it rains" and replace "Q" with "the grass gets wet". This means "If P, then Q" becomes "If it rains, then the grass gets wet".

It's hopefully clear that the example above is a one way street. It points in only one direction. The act of raining leads directly to the grass being wet. However, we cannot go in reverse. If we see the grass is wet, it doesn't mean it rained. Perhaps someone turned on a hose or sprinkler system. P leads to Q, but Q does not lead to P.

In short, all of what is mentioned so far is considered a regular conditional statement. So far we haven't addressed bi-conditional statements.

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

Bi-conditional statements are conditionals that work in reverse.

An example would be "If a figure is a square, then it has 4 congruent sides and 4 right angles". That conditional statement works in reverse. Therefore "If a figure has 4 congruent sides and 4 right angles, then the figure is a square" is also true simply by what it means to be a square.

So the format "If P, then Q" can be reversed to "If Q, then P" to have an equivalently true statement. The common practice is to use "if and only if" to help shorten things.

We would have the template "P if and only if  Q" which is the same as "Q if and only if P". The order doesn't matter since we can reverse things just fine.

Often you'll find bi-conditional statements when it comes to definitions. The term "weekend" literally means the day is either Saturday or Sunday, which is why choice C is another bi-conditional. A very similar situation applies to choice D as well.

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

We've found that choices A, C and D are bi-conditional statements. They can be ruled out. We're left with choice B.

This is not a bi-conditional. Why not? It's assumed that Jane would go to the beach on a sunny day, but perhaps she enjoys the beach just fine on cloudy days. There's also the fact she could be doing other things on sunny days. The presence of the sun doesn't automatically mean the beach. If you said "It gets warm if and only if it's a sunny day", then I think this is more in line with a bi-conditional.

This is why choice B is the final answer.

The simplified form of the expression ( (2⁻²)(3⁴) )⁻³ × ( (2⁻³)(3²) )²  is  [tex]\frac{1}{3^8}[/tex].

Hence, option B is the correct answer.

Given the expression;

( (2⁻²)(3⁴) )⁻³ × ( (2⁻³)(3²) )²

First, we use the properties of exponents.

( 2⁻²ˣ⁻³ × 3⁴ˣ⁻³ ) × ( 2⁻³ˣ² × 3²ˣ² )

( 2⁶ × 3⁻¹² ) × ( 2⁻⁶ × 3⁴ )

We take out the parentheses, such that;

2⁶ × 2⁻⁶ × 3⁻¹² × 3⁴

Note that: 2⁶ × 2⁻⁶ = 1, as [tex]2^6*2^{-6}=2^{6+(-6)}=2^0=1[/tex]

Hence, we have

1 × 3⁻¹² × 3⁴

3⁻¹²⁺4

3⁻⁸

Next, we express with positive exponent

[tex]\frac{1}{3^8}[/tex]

The simplified form of the expression ( (2⁻²)(3⁴) )⁻³ × ( (2⁻³)(3²) )²  is  [tex]\frac{1}{3^8}[/tex].

Hence, option B is the correct answer.

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Answer:

Step-by-step explanation:

By applying the triangle midpoint theorem to the two triangles (see attachment), the correct statements are:

A. One-halfQP = UT

D. SU ∥ RP

Triangle midpoint theorem states that the line segment which connects the midpoints of two (2) sides of a triangle must be parallel to the third side, and it's congruent to one-half of the third side.

By applying the triangle midpoint theorem to the two triangles (see attachment), we can infer and logically deduce that one-half of side QP is equal to side UT and side SU is parallel to side RP (SU ∥ RP).

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Complete Question:

Points S, U, and T are the midpoints of the sides of ΔPQR. ΔSUT is inside of ΔPQR. Points S, U, and T are the midpoints of ΔPQR. Which statements are correct? Check all that apply.

A. One-half QP = UT

B. One-half TS = RQ

C. SU = PR

D. SU ∥ RP

E. UT ⊥ RP

Answer: a and d

Step-by-step explanation:

[tex]y=2(x+1)^2+6[/tex]

The vertex (−1,6)

Equation of axis of symmetry is x = -1

minimum at (-1,6)

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

The vertex (1,-4)

Equation of axis of symmetry is x = 1

maximum at (1,-4)

A graph can be defined as a pictorial representation or a diagram that represents data or values.

[tex]y=2(x+1)^2+6[/tex]

The vertex (−1,6)

Equation of axis of symmetry is x = -1

minimum at (-1,6)

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

The vertex (1,-4)

Equation of axis of symmetry is x = 1

maximum at (1,-4)

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Answer: The values are a = 4, b = 2, c = 16, d = 9

simplified further

c = 16

d = 9

Using translation concepts, the equation of the graph is:

y = |0.5x - 2.5| + 3.

A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction in it's definition.

The function is:

y = |0.5x - 2| + 3.

When it is shifted one unit to the right, we have that x -> x - 1, hence:

y = |0.5(x - 1) - 2| + 3

y = |0.5x - 2.5| + 3.

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The inverse function of f(x) is [tex]f^{-1}(x) = \sqrt{x - 6}[/tex] and the domain is x > 6

The function is given as:

f(x) = x^2 + 6

Rewrite as:

y = x^2 + 6

Swap x and y

x = y^2 + 6

Subtract 6 from both sides

y^2 = x - 6

Take the square root

[tex]y = \sqrt{x - 6}[/tex]

Rewrite as:

[tex]f^{-1}(x) = \sqrt{x - 6}[/tex]

Hence, the inverse function of f(x) is [tex]f^{-1}(x) = \sqrt{x - 6}[/tex] and the domain is x > 6

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The value on the table above will be $30.6. Then the correct option is D.

The quantity of anything is stated as though it were a fraction of a hundred.

Raj was friends with his waitress, so he left her a 90 percent tip.

If his tip was $27.54.

Then the value in the table above will be

⇒ $27.54 / 0.90

⇒ $30.6

Then the correct option is D.

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The true statement about the function which is represented by this graph is: D. the function has an inverse because it passes the horizontal line test.

The horizontal line test states that if the graph of function (f) is intersected by any horizontal line more than once, then, the function (f) doesn't have an inverse.

Conversely, if the graph of function (f) isn't intersected by any horizontal line more than once, then, the function (f) has an inverse.

By critically observing the graph of the given function (f) shown in the image attached below, we can logically deduce that it has an inverse because it passed the horizontal line test.

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Answer: D. The function has an inverse because it passes the horizontal line test.

Step-by-step explanation:

Got it right on test.

Answer:

The building is around 47 meters tall

Step-by-step explanation:

Pole:

tall/long

2.8/1.79

Building:

tall/long

x/42.25

Solve for x:

2.8/1.79 = x/42.25

1.79x = 84.5

x ≈ 47

Answer:

LCM(24,30,40)=120.The Least common multiple for three numbers 24,30 and 40 is 120

Step-by-step explanation:

The rates of each cyclist are 17mi/h and 20mi/h

What is rate?

Rate is a ratio that compares two different quantities which have different units.

In this comparison, one unit is taken to be a standard unit in which the other is compared with.

Analysis:

Let the speeds of both cyclists be x,  x+3

distance travelled by x cyclist = 2x

distance travelled by (x+3) cyclist = 2(x+3)

Total distance = 2x + 2(x+3) = 74 miles

2x +2x +6 = 74

4x + 6 = 74

4x = 74 - 6

4x = 68

x = 68/4 = 17mi/h

The other cyclist speed is x +3 = 17 +3 = 20mi/h

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Answer:

Step-by-step explanation:

Comment and answer

ABC is similar to XBY           The triangles are similar because

BX = 2                                             Given

XA = 3                                             Given

BY = 3                                              Given

YC = x                                              Object of question

BX/(BX + XA) = BY / (BY + YC)       Parts of two similar triangles

2/(2 + 3) = 3/(3+ x)                           Combine the left

2/5 = 3/(3 + x)                                  Cross Multiply

2*(3 + x) = 3*5                                  Remove the brackets

6 + 2x = 15                                       Subtract 6 from both sides

6-6+2x = 15 - 6                                Combine

2x = 9                                               Divide by 2

2x/2 = 9/2

x = 4.5

Answer

x = 4.5 or D

Answer:

3x² + 6x + 4

Step-by-step explanation:

6x + 7 + x² + 2x² - 3

→ Collect like terms

x² + 2x² + 6x + 7 - 3

→ Simplify

3x² + 6x + 4

Answer:

3x^2+6x+4

Step-by-step explanation:

We can find the sum of the polynomials by combining like terms.

(6x+7+x²)+(2x²-3)

x^2+2x^2+6x+7-3

3x^2+6x+4

Answer:  The base should be 4.

Step-by-step explanation:

1. We draw a triangle shape like the image.

(The steps was showed in the image).

Answer:

x = 1

Explanation:

[tex]\sf \rightarrow log(3x) + log(x + 4) = log(15)[/tex]

Rule: log(a) + log(b) = log(ab)

[tex]\sf \rightarrow log(3x(x + 4)) = log(15)[/tex]

cancel out log on both sides

[tex]\sf \rightarrow 3x(x + 4) = 15[/tex]

relocate constant variable

[tex]\sf \rightarrow 3x^2 + 12x -15 = 0[/tex]

take 3 as a common factor

[tex]\sf \rightarrow 3(x^2 + 4x -5) = 0[/tex]

divide both sides by 3

[tex]\sf \rightarrow x^2 + 4x -5 = 0[/tex]

middle term split

[tex]\sf \rightarrow x^2 + 5x -x-5 = 0[/tex]

factor common terms

[tex]\sf \rightarrow x(x + 5) -1(x+5)= 0[/tex]

collect into groups

[tex]\sf \rightarrow (x-1)(x+5)= 0[/tex]

set to zero

[tex]\sf \rightarrow x-1 = 0 , \ x+5= 0[/tex]

relocate variables

[tex]\sf \rightarrow x = 1, \ x = -5[/tex]

There must be a positive solution for log, so the solution is only x = 1

Answer:

[tex]x=1[/tex]

Step-by-step explanation:

Given:

[tex]\log (3x)+\log(x+4)=\log (15)[/tex]

As the logs have no base, assume that the base is 10.

[tex]\textsf{Apply log Product law}: \quad \log_ax + \log_ay=\log_axy[/tex]

[tex]\implies \log_{10} (3x)+\log_{10}(x+4)=\log_{10} (15)[/tex]

[tex]\implies \log_{10} \left(3x(x+4)\right)=\log_{10} (15)[/tex]

Expand the brackets:

[tex]\implies \log_{10} \left(3x^2+12x\right)=\log_{10} (15)[/tex]

[tex]\textsf{Apply the log Equality law}: \quad \textsf{if }\: \log_ax=\log_ay\:\textsf{ then }\:x=y[/tex]

[tex]\implies 3x^2+12x=15[/tex]

Subtract 15 from both sides:

[tex]\implies 3x^2+12x-15=0[/tex]

Factor out the common term 3:

[tex]\implies 3(x^2+4x-5)=0[/tex]

Divide both sides by 3:

[tex]\implies x^2+4x-5=0[/tex]

Split the middle term:

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

Factorize the first two terms and the last two terms separately:

[tex]\implies x(x+5)-1(x+5)=0[/tex]

Factor out the common term (x + 5):

[tex]\implies (x-1)(x+5)=0[/tex]

Therefore:

[tex]x-1=0 \implies x=1[/tex]

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

As logs cannot be taken of negative numbers, [tex]x=-5[/tex] is an extraneous solution.  Therefore, the only valid solution is:  [tex]x=1[/tex]

A total of 800 tickets for the covered pavilion and 700 tickets for lawn seats were sold, solved using a system of equations.

We assume the number of covered pavilion tickets sold to be x and the number of lawn seat tickets sold to be y.

As a total of 1500 tickets were sold, we can form an equation:

x + y = 1500 ... (i),

as we had only those two types of tickets.

The cost of each covered pavilion ticket = $20.

Hence, the total revenue from covered pavilion tickets = $20x.

The cost of each lawn seat ticket = $10.

Hence, the total revenue from lawn seat tickets = $10y.

The total receipts were said to be $23,000.

Thus, we can represent this as an equation:

20x + 10y = 23000 ...(ii).

Combining (i) and (ii), we get a system of equations:

x + y = 1500 ... (i).

20x + 10y = 23000 ... (ii).

Dividing equation (ii) by 10, we get:

2x + y = 2300 ... (iii).

Subtracting (i) from (iii), we get:

2x + y = 2300.

 x + y = 1500.

(-)  (-)     (-)

____________

x = 800.

Substituting x = 800 in (i), we get:

x + y = 1500,

or, 800 + y = 1500,

or, y = 1500 - 800,

or, y = 700.

Thus, a total of 800 tickets for the covered pavilion and 700 tickets for lawn seats were sold, solved using a system of equations.

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