Find the zeros of y = x2 - 6x - 4 by completing the square.
A. x=-34/13
B. X- 3+ 118
CX13
D. x 316

Answer:

See explanation below for further details.

Step-by-step explanation:

A rational consist of two real numbers such that:

[tex]\frac{a}{b} =c[/tex]

If c is a polynomial with a certain grade, then, both the numerator and the denominator must be also polynomials and the grade of the numerator must be greater than denominator.

If c is linear function, that is, a first order polynomial, then a must be a (n+1)-th polynomial and b must be a n-th polynomial.

Example:

If [tex]a = x^{2}[/tex] and [tex]b = x[/tex], then:

[tex]c = \frac{x^{2}}{x}[/tex]

[tex]c = x[/tex]

If c is a quadratic function, that is, a second order polynomial, then a must be a (n+1)-th polynomial and b must be a n-th polynomial.

Example

If [tex]a = 3\cdot x^{3}[/tex] and [tex]b = x[/tex], then:

[tex]c = \frac{3\cdot x^{3}}{x}[/tex]

[tex]c = 3\cdot x^{2}[/tex]

But if c is an exponential, both the numerator and the denominator must be therefore exponential function and grade of each exponential function must different to the other.

Example

If [tex]a = 10^{2x}[/tex] and [tex]b = 3\cdot 10^x[/tex], then:

[tex]c = \frac{10^{2\cdot x}}{3\cdot 10^{x}}[/tex]

[tex]c = \frac{1}{3}\cdot 10^{2\cdot x -x}[/tex]

[tex]c = \frac{1}{3}\cdot 10^x[/tex]

Otherwise, c would be equal to a constant function, that is, a polynomial with a grade 0.

If [tex]a = 5\cdot e^{x}[/tex] and [tex]b = -3\cdot e^{x}[/tex], then:

Example

[tex]c = \frac{5\cdot e^{x}}{-3\cdot e^{x}}[/tex]

[tex]c = -\frac{5}{3}[/tex]

It is worth to add that exponential functions can be a linear combination of single exponential function, similar to polynomials.

Example

[tex]5\cdot a^2\cdot x -9[/tex]

Answer:

[tex]1.\ \ p^2q = (\frac{p^5}{p^{-3}q^{-4}})^{\frac{1}{4}}[/tex]

[tex]2.\ \ pq^{\frac{3}{2}}} = (\frac{p^2q^7}{q^{4}})^{\frac{1}{2}}[/tex]

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

[tex]4.\ \ p^2q^{\frac{1}{2}} = (p^6q^{\frac{3}{2}})^{\frac{1}{3}}[/tex]

Step-by-step explanation:

Required

Match each expression to their simplified form

1.

[tex](\frac{p^5}{p^{-3}q^{-4}})^{\frac{1}{4}}[/tex]

Simplify the expression in bracket by using the following law of indices;

[tex]\frac{a^m}{a^n} = a^{m-n}[/tex]

The expression becomes

[tex](\frac{p^{5-(-3)}}{q^{-4}})^{\frac{1}{4}}[/tex]

[tex](\frac{p^{5+3}}{q^{-4}})^{\frac{1}{4}}[/tex]

[tex](\frac{p^8}{q^{-4}})^{\frac{1}{4}}[/tex]

Split the fraction in the bracket

[tex](p^8*\frac{1}{q^{-4}})^{\frac{1}{4}}[/tex]

Simplify the fraction by using the following law of indices;

[tex]\frac{1}{a^{-m}} = a^m[/tex]

The expression becomes

[tex](p^8*q^4)^{\frac{1}{4}}[/tex]

Further simplify the expression in bracket by using the following law of indices;

[tex](ab)^m = a^m * b^m[/tex]

The expression becomes

[tex](p^{8*\frac{1}{4}}\ *\ q^4*^{\frac{1}{4}})[/tex]

[tex](p^{\frac{8}{4}}\ *\ q^{\frac{4}{4}})[/tex]

[tex]p^2q[/tex]

Hence,

[tex](\frac{p^5}{p^{-3}q^{-4}})^{\frac{1}{4}} = p^2q[/tex]

2.

[tex](\frac{p^2q^7}{q^{4}})^{\frac{1}{2}}[/tex]

Simplify the expression in bracket by using the following law of indices;

[tex]\frac{a^m}{a^n} = a^{m-n}[/tex]

The expression becomes

[tex]({p^2q^{7-4}}})^{\frac{1}{2}}[/tex]

[tex]({p^2q^3}})^{\frac{1}{2}}[/tex]

Further simplify the expression in bracket by using the following law of indices;

[tex](ab)^m = a^m * b^m[/tex]

The expression becomes

[tex]{p^{2*\frac{1}{2}}q^{3*\frac{1}{2}}}}[/tex]

[tex]pq^{\frac{3}{2}}}[/tex]

Hence,

[tex]pq^{\frac{3}{2}}} = (\frac{p^2q^7}{q^{4}})^{\frac{1}{2}}[/tex]

3.

[tex]\frac{(pq^3)^{\frac{1}{2}}}{(pq)^{\frac{-1}{2}}}[/tex]

Simplify the numerator as thus:

[tex]\frac{p^{\frac{1}{2}} * q^3*^{\frac{1}{2}}}{(pq)^{\frac{-1}{2}}}[/tex]

[tex]\frac{p^{\frac{1}{2}} * q^{\frac{3}{2}}}{(pq)^{\frac{-1}{2}}}[/tex]

Simplify the denominator as thus:

[tex]\frac{p^{\frac{1}{2}} * q^{\frac{3}{2}}}{p^{\frac{-1}{2}}q^{\frac{-1}{2}}}[/tex]

Simplify the expression in bracket by using the following law of indices;

[tex]\frac{a^m}{a^n} = a^{m-n}[/tex]

The expression becomes

[tex]p^{\frac{1}{2} - (\frac{-1}{2} )} * q^{\frac{3}{2} - (\frac{-1}{2}) }[/tex]

[tex]p^{\frac{1}{2} +\frac{1}{2} } * q^{\frac{3}{2} + \frac{1}{2} }[/tex]

[tex]p^{\frac{1+1}{2}} * q^{\frac{3+1}{2}}[/tex]

[tex]p^{\frac{2}{2}} * q^{\frac{4}{2}}[/tex]

[tex]pq^2[/tex]

Hence,

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

4.

[tex](p^6q^{\frac{3}{2}})^{\frac{1}{3}}[/tex]

Simplify the expression in bracket by using the following law of indices;

[tex](ab)^m = a^m * b^m[/tex]

The expression becomes

[tex]p^6*^{\frac{1}{3}}\ *\ q^{\frac{3}{2}}*^{\frac{1}{3}}[/tex]

[tex]p^{\frac{6}{3}}\ *\ q^{\frac{3*1}{2*3}}[/tex]

[tex]p^2 *\ q^{\frac{3}{6}}[/tex]

[tex]p^2 *\ q^{\frac{1}{2}[/tex]

[tex]p^2q^{\frac{1}{2}[/tex]

Hence

[tex]p^2q^{\frac{1}{2}} = (p^6q^{\frac{3}{2}})^{\frac{1}{3}}[/tex]

Answer:

No

Step-by-step explanation:

We have to check if 3/5 = 6/25. If we convert 3/5 into a fraction with denominator of 25 so that we can compare the two we get 15/25 and this is clearly not equal to 6/25. Therefore, the answer is no.

Answer:  1. [tex]\dfrac{9}{8}[/tex] .

Step-by-step explanation:

The given expression  [tex]2x^2-3x+1[/tex]

Using completing the square method.

First, we divide the entire expression by 2 and write in the form of [tex]x^2+bx=c[/tex] , we get

[tex]x^2-\dfrac{3}{2}x+\dfrac{1}{2}[/tex]

Such that [tex]b=\dfrac{3}{2}\ \ \ , c=\dfrac{1}{2}[/tex]

Now add and subtract [tex](\dfrac{b}{2})^2[/tex] on both sides.

[tex](\dfrac{b}{2})^2=(\dfrac{\dfrac{3}{2}}{2})^2\\\\=(\dfrac{3}{2\times2})^2\\\\=(\dfrac{3^2}{4^2})\\\\=\dfrac{9}{8}[/tex]

Here, we add and subtract [tex]\dfrac{9}{8}[/tex] .

Hence, the correct answer is  1. [tex]\dfrac{9}{8}[/tex] .

There are 55 two-bedroom units  in the apartment.

Let x represent the number of one bedroom units and y represent the number of two bedroom units.

Since the apartment building contains 100 units, hence:

x + y = 100      (1)

The total monthly rent paid by the tenants is $55,275, hence:

495x + 600y = 55275   (2)

Solving equation 1 and 2 simultaneously gives:

x = 45, y = 55

Hence, there are 55 two-bedroom units  in the apartment.

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Answer: 2 1/10, 3.122, 3.19, 61/10

Step-by-step explanation:

Let's first turn each number into decimal form: (1/10=.1, 2/10=.2,etc.)

2.1, 6.1, 3.122, 3.19.  Obviously, the smallest number is 2.1, because it has a 2 in the ones place.  Next, we see a tie in the ones place between 3.122 and 3.19.  Then we look at the tenths place and see another tie.  Then, at the hundredths place, we see that 3.19 is greater that 3.122.  Thus, the order of the numbers is 2.1,3.122,3.19,6.1.

Hope it helps <3

Answer:

117,600 ways

Step-by-step explanation:

This problem can be solved using fundamental principle of counting , according to which , if there are m different things and n different things,

then there m*n ways to combine.

Example: if there 5 shirts and 3 ties,

then number of ways in which different combination shirt and tie can be worn is

5*3 = 15 ways.

____________________________________

In the problem,\

number of states is 50,

let those states be

A1, A2, A3, A4 ......A50

in each state there are 2 senate

Condition

3-Senator committee in which no two members are from the same state.

then

first member can be chosen from any of the 50 state,

hence first member can be chosen in 50 ways

for simplicity lets take state A1, you can choose any one you wish

second member cannot be from state A1, then number of state left = 49

second member can be chosen from  any of 49 state left

hence second member can be chosen in 49 ways

for simplicity lets take state A2,

Third member cannot be from state A1 and A2, then number of state left = 48

Third member can be chosen from  any of 48 state left

hence Third member can be chosen in 48 ways

for simplicity lets take state A3,

Hence no. of ways of choosing  3-Senator committee be formed such that no two Senators are from the same state = 50*49*48 = 117,600 ways

Answer:

Step-by-step explanation:

Let width = w units

Length = 4w - 3

Cost for lining corners of wall = $ 18.50

Perimeter of the wall = 18.50/0.25 =  74 ft

2*(length  +width ) = 74

2*(4w - 3 + w) =74

2*(5w - 3) =74

10w - 6 = 74   {Add 6 to both sides}

10w - 6 + 6 = 74 + 6

10w = 80

Divide both sides by 10

10w/10 = 80/10

w =  8 ft

Length = 4w - 3 = 4*8 - 3 = 32 - 3

Length = 29 ft

Area of the wall = length * width

                           = 29 * 8

                           =  232 square ft

Cost of painting the wall per square feet = 20 cents = 0.20

Cost of painting 232 square feet = 232 * 0.20

                                                             = $ 46.40

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

Explanation:

Label the friends A,B,C,D,E

There are 5 choices for the first ticket, then 4 for the second, and 3 for the third. We have 5*4*3 = 20*3 = 60 permutations. One permutation is ABC and another permutation is CBA if order mattered. However, order does not matter. All that counts is the group overall rather than the individual. This is because there is no ranking and all the friends are effectively treated equally.

With a set like {A,B,C} there are 3! = 3*2*1 = 6 ways to arrange this set of three items. This means we have overcounted by a factor of 6. Divide the previous result (60) by 6 to get 60/6 = 10

So there are exactly 10 different groups possible.

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

You can use the combination formula

[tex]_n C _r = \frac{n!}{r!(n-r)!}[/tex]

with n = 5 and r = 3 to get the same result.

Or you could use Pascal's triangle to find "10" located in the row that has 1,5,10,10,5,1. Notice how 10 is located in slot 3. You start counting at slot 0.

Answer:

15

Step-by-step explanation:

you need to get to 35. if one bag is 5 pounds what else plus 5 is 35. so each bag is 15 pounds

Answer: 15

Step-by-step explanation:

35 - 5 = 30

30 ÷ 2 = 15

Hope this helped.

(15 each)

Answer:

Your correct answer is option C. Equilateral triangles

Step-by-step explanation:

In similar polygons, it is all the corresponding angles that are congruent.

Answer:

45

Step-by-step explanation:

g(4) = 3 * 4 - 7 = 5

f[g(4)] = f(5) = 2 * 5² - 5 = 45

Answer:

45

Step-by-step explanation:

Answer:

A is the correct answer

Step-by-step explanation:

Graph (A) represents the system of equations 4x + y = 3 and 2x - 3y = 3 option (A) is correct.

It is defined as the relation between two variables, if we plot the graph of the linear equation we will get a straight line.

If in the linear equation, one variable is present, then the equation is known as the linear equation in one variable.

We have:

The system of equations

4x + y = 3 and

2x - 3y = 3

In the first line, the slope is -4, and the y-intercept at (0,3)

In  the second line, the slope is 2/3, and the y-intercept at (0,-1)

Thus, graph (A) represents the system of equations 4x + y = 3 and 2x - 3y = 3 option (A) is correct.

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Answer:

answer is (0,1)

Step-by-step explanation:

check

by substituting x=0 and y=1 in the equation above

- 2(0)+1> -4

1> -4

Thereby (0,1) is the required solution to the inequality

Answer:

24 Combinations Total

Step-by-step explanation:

METHOD 1 (no advanced math skills needed)

i. Number the friends: a, b, c, and d

ii. List all possible combinations...

To speed up the process instead of sitting here and listing 24 combinations, after two sections you can see that with friend "a" sitting all the way at the left there are six combinations and the same goes for friend "b". You can then multiply 6 (number of combinations with that friend first) and 4 (friends) to get 24.

METHOD 2

n! means the total number for permutations calculation (1 x 2 x ... until you reach n ) = 24

The function that models the inverse variation is y=96/x. Thus, the correct option is A.

Let there are two variables p and q

Then, p and q are said to be directly proportional to each other if

p = kq

where k is some constant number called the constant of proportionality.

This directly proportional relationship between p and q is written as

p∝q where that middle sign is the sign of proportionality.

In a directly proportional relationship, increasing one variable will increase another.

Now let m and n be two variables.

Then m and n are said to be inversely proportional to each other if

[tex]m = \dfrac{c}{n} \\\\ \text{or} \\\\ n = \dfrac{c}{m}[/tex]

(both are equal)

where c is a constant number called the constant of proportionality.

This inversely proportional relationship is denoted by

[tex]m \propto \dfrac{1}{n} \\\\ \text{or} \\\\n \propto \dfrac{1}{m}[/tex]

As visible, increasing one variable will decrease the other variable if both are inversely proportional.

Given that x and y vary inversely. Therefore, the relation can be written as,

y  ∝ 1/x

y = k × 1/x

y = k/x

Also, when y =8, the value of x is 12, therefore, if we substitute the points in the above-formed equation,

y = k/x

8 = k /12

k = 96

Hence, the function that models the inverse variation is y=96/x. Thus, the correct option is A.

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Step-by-step explanation:

What was Given was that Point G is rotated 90°. The coordinate of the pre-image point G was (7, –5), and its image G’ is at the coordinate (5, 7).

What you need to find was : What is the direction of the rotation.

So coming to conclusions is we have given that Point G is rotated 90°.

The Pre-image of G = ( 7, -5).

Translated image G' = ( 5 ,7).

By the Rotation rule of 90° is : (-y, x) 90 degree rotation counterclockwise around the origin.

(y, -x) 90 degree rotation clockwise about the origin.

We can from G (7,-5) and G'( 5 ,7) it is rotation counterclockwise around the origin.

Therefore, Direction of rotation is counterclockwise around the origin.

Answer:

The answer is COUNTERCLOCKWISE

Step-by-step explanation:

edg

Answer:

-2

Step-by-step explanation:

The product of the slope of perpendicular lines= -1.

Slope of given line= ½,

since the equation is in the form of y=mx +c, where m is the slope of the line.

½(slope of line)= -1

slope of line

= -1 ÷½

= -1 ×2

= -2

Answer:

Step-by-step explanation:

Slope of perpendicular line = -1

y = (1/2)x + 5

Slope [tex]m_{1}\\[/tex] = 1/2

[tex]m_{1}*m_{2}=-1\\\\\frac{1}{2}*m_{2}=-1\\\\m_{2} = -1*\frac{2}{1}\\\\ m_{2}= -2[/tex]

y = -2x + 7

Hey there! :)

Answer:

C. (0, 5)

Step-by-step explanation:

Where the equation crosses the y axis is known as the y-intercept.

In the equation y = mx + b, the b  value represents the y-intercept where x = 0. Therefore:

Rearrange the equation pictured:

y = 5 - 3x --> y = -3x + 5

The b value in this equation is 5, so

The y-intercept of this equation is (0, 5).

Answer:

Option C is the correct option.

Here,

When X=0

y= 5-3*0

= 5-0

=5

x=0

y=5

Hope this helps...

Good luck on your assignment..

Answer:

40/46+x=30/46-x

Step-by-step explanation:

Speed=distance/time

s=d/t

x=speed of the boat

Downstream speed=x+46

Upstream speed=x-46

Downstream

s=d/t

x+46=40/t

t=40/(x+46)

Upstream

s=d/t

x-46=30/t

t=30/(x-46)

Equate upstream and downstream

30/x-46=40/x+46

30(x+46)=40(x-46)

30x+1380=40x-1840

30x+1380-40x+1840=0

-10x+3220=0

-10x=-3220

Divide both sides by -10

x=322

x=322 miles/hour

Answer:

19 : 25

Step-by-step explanation:

Answer:

4

Step-by-step explanation:

I'm guessing that each number represents the number of rectangles that overlap. Where you see 1, there is only one rectangle, 2 shows where two rectangles overlap, and so on.

At the question mark, it appears as though four rectangles overlap. So a 4 would go there.

Answer:

y = (x - 2)² + 1

Step-by-step explanation:

Gnerela equation of a parabola in vertex form:

y = a(x - h)² + k

where (h, k) is the vertex

From the description of the graph:

Replacing into the general equation:

y = a(x - 2)² + 1

5 = a(0 - 2)² + 1

5 - 1 = a*4

4/4 = a

1 = a

Then, the equation is: y = (x - 2)² + 1

Answer:

The correct answer is A, "y = (x - 2)² + 1"

Step-by-step explanation:

I just completed the test and got full credit using this option.

Answer:

Table 2

Step-by-step explanation:

The question is to identify the table where the relative frequency of orders for blue pants is half the relative frequency of their orders for black shorts.

We are interested in the number of blue pants and black shorts.

If we name them as p and b respectively, then we are looking for a table where s/p=2

Let's verify relevant data in all tables:

Table 1:

Table 2:

Table 3:

Table 4:

So only one of tables- the second one from top represents the required data.

Answer:

The second table

Step-by-step explanation:

I did the test.

Answer:

Step-by-step explanation:

Well since AB/DE = 3/4 and AB/DE = BC/EF, then BC/EF must also be 3/4.

We now only need to find a fraction that simplifies to 3/4.

8/12 = 2/3 [tex]\neq[/tex] 3/4

12/16 = 3/4 ✅

4/6 = 2/3 [tex]\neq[/tex] 3/4

6/9 = 2/3 [tex]\neq[/tex] 3/4

This means that the correct answer is B. 12 and 16.

Answer:

B. 12 and 16

Step-by-step explanation:

Answer:

A. 10/(3x^2+3x)

Step-by-step explanation:

2x5=10

3x(x+1) = 3x^2+3x

Answer: 3.5 pi.

Step-by-step explanation: Since we know the circumference formula of a circle is 2*pi*radius, we just have to substitute 7 in there, and divide by 4. So here ya go, 3.5 pi.

Answer:

27. 4x + 44  (D)

28. 17.5 in (A)

29. 189 cm² (C)

Step-by-step explanation:

27. 4 (2x + 11 - x) → 4 (x + 11) → 4x + 44

28. 369.25 ÷ 42.2 = 8.75 → 8.75 × 2 = 17.5 in

29. 21 × 9 = 189 cm²

Hope this helps! :)

Answer:

27.D

28.A

29.C

Step-by-step explanation:

27.   8x+44-4x= 4x+44         D

28. (369.25*2)/42.2=17.5      A

29.     9*21=189 cm^2           C