Joseph has started completing the square on the equation 3x2 - 7x + 12 = 0. He has worked to the point where he has the expression x2 - x = -4. Use complete sentences describe Joseph’s steps up to this point and whether or not his work is accurate.

Answer:

The correct answer option is B. 2 = 3x + 10x^2

Step-by-step explanation:

We are to determine whether which of the given equations in the answer options can be solved using the following expression:

[tex]x=\frac{-3 \pm\sqrt{(3)^2+4(10)(2)} }{2(10)}[/tex]

Here, [tex]a = 10, b = 3[/tex] and [tex]c=-2[/tex].

These requirements are fulfilled by the equation 4 which is:

[tex]12=3x+10x^2[/tex]

Rearranging it to get:

[tex]10x^2+3x-2=0[/tex]

Substituting these values of [tex]a,b,c[/tex] in the quadratic formula:

[tex]x= \frac{-b \pm \sqrt{b^2-4ac} }{2a}[/tex]

[tex]x= \frac{-3 \pm\sqrt{(3)^2-4(10)(-2)} }{y}[/tex]

Answer:

Step-by-step explanation:

[tex]V_p = b \cdot h \cdot w[/tex], where b, h, w are base, height and width respectively.

[tex]V_{sp} = \frac{1}{3} \cdot {A_b} \cdot h[/tex], where h is the height, and A_b is the area of the base.

First let's calculate the volume of the paralelipiped by the first formula.

[tex]V_p = 7 \cdot 4 \cdot 3 = 84 in^2[/tex]

Then let's find out the height of the pyramid, which is 12 - 4 = 8 in.(See drawing)

Then the area of the base of the square pyramid is simply.

[tex]A_r = 3 \cdot 7 = 21 in^2[/tex]

Now we can find the volume of the pyramid.

[tex]V_{sp} = \frac{1}{3} \cdot 21 \cdot 8 = 56 in^3[/tex]

To find the total volume, let's add both volumes.

[tex]V_T = 84 + 56 = 140 in^3[/tex]

The volume of the composite figure is calculated by adding the volume of the individual shapes that make up the figure. For a figure composed of a rectangular prism and a triangular prism, you would calculate both volumes separately and add them together, giving a total volume of 186 in³ based on the dimensions provided.

In mathematics, the term volume refers to the amount of space occupied by a 3-dimensional object.

The composite figure you've given seems to include a rectangular prism and an extruded triangle (or possibly a rectangular prism attached to a triangular prism, without information about the orientation of the shapes, we cannot be certain).

For example, if it were a rectangular prism attached to a triangular prism, we would find the volume of both separately and then add them together. The volume of the rectangular prism would be found by length*width*height (12 in*4 in*3 in=144 in³). The volume of triangular prism would be 1/2*base*height*length (1/2*4 in*3 in*7 in = 42 in³). Adding them together, the volume of the composite figure would be 186 in³.

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Answer:

$4.75 is how much money is left over

Step-by-step explanation:

3* 1.25= 3.75

2* 2.50= 5.00

1* 6.50+ 6.50

5.00+3.75+6.50= 15.25

20- 15.25= 4.75

Final answer:

Nat spent a total of $15.25 on colas, hot dogs, and a hamburger. After paying with a $20 bill, he has $4.75 left.

Explanation:

To calculate how much money Nat has left after his purchases, we will add up the cost of the items he bought and subtract that total from the $20 bill he used for payment.

Cost of 3 colas: 3 × $1.25 each = $3.75

Cost of 2 hot-dogs: 2 × $2.50 each = $5.00

Cost of 1 hamburger: $6.50

Now we'll sum these amounts to find the total spending:

Total spending = $3.75 + $5.00 + $6.50 = $15.25

Next, we subtract the total spending from the $20 bill:

Change = $20.00 - $15.25 = $4.75

Therefore, Nat has $4.75 left after his purchase.

Answer:

[tex]\large\boxed{A:B:C=\dfrac{1}{10A}}[/tex]

Step-by-step explanation:

[tex]A=\dfrac{B}{2}=\dfrac{C}{5}\\\\A=\dfrac{B}{2}\qquad\text{multiply both sides by 2}\\\\2A=B\to\boxed{B=2A}\\\\A=\dfrac{C}{5}\qquad\text{multiply both sides by 5}\\\\5A=C\to C=5A\\\\A:B:C=A:2A:5A=1:2:5A=\dfrac{1}{2}:5A=\dfrac{1}{2}\cdot\dfrac{1}{5A}=\dfrac{1}{10A}[/tex]

The answer is:

[tex]3x^{2} -14x+7[/tex]

To solve the problem we need to add/subtract like terms. We need to remember that like terms are the terms that share the same variable and the same exponent.

For example, we have:

[tex]x^{2} +2x+3x=x^{2} +(2x+3x)=x^{2}+5x[/tex]

We have that we were able to add just the terms that were sharing the same variable and exponenr (x for this case).

So, we are given the expression:

[tex]5x^2-5x+1-(2x^2+9x-6)=5x^{2}-2x^{2}-5x-9x+1-(-6)\\\\(5x^{2}-2x^{2})+(-5x-9x)+(1-(-6))=3x^{2}-14x+7[/tex]

Hence, the answer is:

[tex]3x^{2} -14x+7[/tex]

Have a nice day!

Answer:64

Step-by-step explanation:

The expression 7 + 3^2(-5 + 1) divided by 2 equals -14.5, after calculating the exponent, solving the parenthesis, multiplying, adding to 7, and then dividing by 2.

To find the value of the expression 7 + 32(-5 + 1) divided by 2, follow these steps:

The value of the expression is -14.5.

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Answer:

x=1, x=7

Step-by-step explanation:

Firstly, you can simplify by dividing both sides by 3, which will make the equation easier to simplify:

(x-4)^2=9

Next, you can take the square root of each side, which will cancel the square, and make the 9 even easier to work with:

(x-4)=(± )3

*Note the (± ) sign, this is because there is a positive and negative answer*

After this, you will basically have two equations:

(x-4)=3

and

(x-4)=-3

You are going to solve both to get both answers, but lets start with the positive first:

x-4=3     Add 4 to both sides to get an x= equation

x=7

And it's the same thing for the negative:

x-4=-3     Add 4 to both sides to get an x= equation

x=1

Hope this helps

Final answer:

To solve the equation 3(x-4)²=27, divide by 3 to get (x-4)²=9, then take the square root to find the solutions x=7 and x=1.

Explanation:

When we're tasked with solving a quadratic equation, such as 3(x-4)²=27, we first need to get it into the standard form of a quadratic equation, which is ax² + bx + c = 0. To do so in this case, we can divide both sides of the equation by 3 to isolate the squared term, yielding (x-4)²= 9. We then take the square root of both sides, giving us x - 4 =3. Solving for x results in two solutions: x = 4 + 3 and x = 4 - 3, which simplifies to x = 7 and x = 1 respectively. These are the solutions to the given equation.

To verify these solutions, you can substitute them back into the original equation to ensure that they satisfy the equation, which in this case, they will. This critical step confirms the accuracy of our solutions.

Answer:

see explanation

Step-by-step explanation:

1

6 = 3x - 9 ( add 9 to both sides )

15 = 3x ( divide both sides by 3 )

5 = x

2

x + 4 < 1 ( subtract 4 from both sides )

x < - 3

3

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

Multiply terms on both sides by 2

x + 6 = - 10 ( subtract 6 from both sides )

x = - 16

Answer:

2) (5,4) and (-1,4)

These have the same y so this line will be horizontal making it parallel to the x-axis.

Step-by-step explanation:

All horizontal lines are parallel to each other.  The x-axis is horizontal. So we are looking for a horizontal line.  On a horizontal line, all the y-coordinates are the same; their equation is in the form y=a number after all.

Anyways we looking for a pair of points with the same y-coordinate.

1) (2,3) and (2,-5)

These have the same x-coordinate so this line is vertical. This would actually be perpendicular to the line given.

2) (5,4) and (-1,4)

These have the same y so this line will be horizontal making it parallel to the x-axis.

3) (2,2) and (-1,-1)

y's aren't the same so not horizontal

x's aren't the same so not vertical

4)  (3,4) and (6,2)

y's aren't the same so not horizontal

x's aren't the same so not vertical

For a pair of points to be parallel to the x-axis, the points have to consist the same x-coordinates.

In this case, only (1) (2,3), (2,-5) have the same x-coordinates.

Therefore, the answer is (1) (2,3), (2,-5).

Hope it helps!

Answer:

Option D. Slope is 1 if the line is vertical

Step-by-step explanation:

we have

Part A)  Slope is a measure of the steepness of the line

The statement is true

Because, the steepness of a line is measured by the absolute value of the slope.

Part B) Slope is the ratio of vertical change to horizontal change

The statement is true

Because the formula to calculate the slope between two points is equal to

[tex]m=\frac{y2-y1}{x2-x1}[/tex]

so

Is the ratio of vertical change to horizontal change

Part C) Slope is zero if the line is horizontal

The statement is true

Because If the line is horizontal, the y-coordinate of two points is the same

therefore

The vertical change is equal to zero

Part D) Slope is 1 if the line is vertical

The statement is Not true

Because, if the line is vertical the slope is undefined

Answer:

Slope is 1

1

if the line is vertical.

Step-by-step explanation:

Answer:

x=5

Step-by-step explanation:

Given

[tex]x=\sqrt{3x+10}[/tex]

Squaring on both sides

[tex]x^2=(\sqrt{3x+10})^2 \\x^2=3x+10\\x^2-3x-10 = 0\\x^2-5x+2x-10 = 0\\x(x-5)+2(x-5) = 0\\(x-5)(x+2)=0\\x=5\\and\\x=-2[/tex]

Since putting -2 gives us:

[tex]-2=\sqrt{3(-2)+10} \\-2=\sqrt{-6+10}\\-2=\sqrt{4}\\-2\neq 2[/tex]

2 is an extraneous solution.

Therefore, the solution to x=√3x+10 is x=5 ..

Answer:

x=7

Step-by-step explanation:

7(x-3) = 28

Distributive property

7x-21 = 28

Addition property of equality

7x = 49

Division property of equality

7x/7 = 49/7

x = 7

The value of x is 7 ..

Answer:

7

Step-by-step explanation:

Answer:

8x^2 + x + 3.

Step-by-step explanation:

(4x^2+1)+(4x^2+x+2)

= 8x^2 + x + 2 + 1

= 8x^2 + x + 3.

Answer:

The polynomial [tex]8x^2+x+3[/tex] represents the sum of given expression.

Step-by-step explanation:

The given expression is

[tex](4x^2+1)+(4x^2+x+2)[/tex]

We need to find the sum of given polynomials.

Open the brackets.

[tex]4x^2+1+4x^2+x+2[/tex]

Combine like terms.

[tex](4x^2+4x^2)+x+(1+2)[/tex]

On further simplification we get

[tex]8x^2+x+3[/tex]

Therefore the polynomial [tex]8x^2+x+3[/tex] represents the sum of given expression.

Answer:

x = 10

Step-by-step explanation:

Given the 2 equations

x - y = 11 → (1)

2x + y = 19 → (2)

Adding the 2 equations term by term eliminates the term in y, that is

(x + 2x) + (- y + y) = (11 + 19), simplifying gives

3x = 30 ( divide both sides by 3 )

x = 10

Answer:

x = 10

Step-by-step explanation:

We know that Rodney is given the following two linear equations and we are to find the value of x at which he would get a solution for this system of linear equations:

[tex] x - y = 1 1 [/tex] - (1)

[tex] 2 x + y = 1 9 [/tex] - (2)

From (1):

[tex]x=11+y[/tex]

Substituting this value of x in (2):

[tex]2(11+y)+y=19[/tex]

[tex]22+2y+y=19[/tex]

[tex]3y=19-22[/tex]

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

[tex]y=-1[/tex]

Substituting this value of y in (1) to find x:

[tex]x=11+(-1)[/tex]

x = 10

The fourth term of the arithmetic series is 38.

Given that, the first number in arithmetic series (a)=8 and common difference (d) =10.

The nth term of the arithmetic series is [tex]a_{n} =a+(n-1) \times d[/tex].

Now, the fourth term= [tex]a_{4} =8+(4-1) \times 10[/tex]

=8+30=38

Therefore, the fourth term of the arithmetic series is 38.

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Answer:

[tex]\large\boxed{b.\ 0,\ \pi,\ \dfrac{\pi}{3},\ \dfrac{5\pi}{3}}[/tex]

Step-by-step explanation:

[tex]\text{Use}\ \sin2x=2\sin x\cos x\\\\\sin2\O=\sin\O\\\\2\sin\O\cos\O=\sin\O\qquad\text{subtract}\ \sin\O\ \text{from both sides}\\\\2\sin\O\cos\O-\sin\O=0\qquad\text{distribute}\\\\\sin\O(2\cos\O-1)=0\iff\sin\O=0\ \vee\ 2\cos\O-1=0[/tex]

[tex]\sin\O=0\iff\O=0\ \vee\ \O=\pi\\\\2\cos\O-1=0\qquad\text{add 1 to both sides}\\\\2\cos\O=1\qquad\text{divide both sides by 2}\\\\\cos\O=\dfrac{1}{2}\iff\O=\dfrac{\pi}{3}\ \vee\ \O=\dfrac{5\pi}{3}[/tex]

Answer:

- 230°

Step-by-step explanation:

A coterminal angle is the given angle ± 360n ( where n is an integer )

Given 130°

Then a coterminal angle = 130° - 360° = - 230°

OR 130° + 360° = 490°

Answer:

y = -1/2 x +5

10 weeks

Step-by-step explanation:

The y intercept is 5  ( That is is the point where x=0)

Our slope is rise over run  or 1 lb /2 weeks

We know this is negative because the cat is losing the weight

slope = -1/2

The equation in slope intercept form is

y= mx + b where m is the slope and b is the y intercept

y = -1/2 x +5

We need to determine when y=0 to find when the cat loses all 5 lbs

0 = -1/2 x +5

Subtract 5 from each side

0-5 = -1/2x +5-5

-5 = -1/2x

Multiply each side by -2 to get x alone

-5 *-2 = -1/2x * -2

10 = x

It will take 10 weeks

Answer:

y = -1/2x + 5

It will take 10 weeks for the cat to lose 5 pounds.

Step-by-step explanation:

Find slope

Take 2 points (2,4) (6,2)

slope = -1/2

y-intercept

(0,5)

y = mx + b form

m represents slope

b represents y-intercept

y = -1/2x + 5

Find the number of weeks it will the cat to lose 5 pounds

Set y = to 0, to find when the cat loses weight

0 = -1/2x + 5

Subtract 5 in both sides

-5 = -1/2x

Isolate the variable by getting rid of the denominator

-1/2x * 2 = -x

2 * - 5 = -10

Simplify

-x = -10

Divide both sides by -1

-x/-1 = x

-10/-1 = 10

Simplify

x = 10

Answer

It will take 10 weeks for the cat to lose 5 pounds

Answer:

A

Step-by-step explanation:

they're proportional

Answer: the answer is A

Step-by-step explanation:

have a good day

Answer:

8

Step-by-step explanation:

So the y-intercept is not given by your table because there is no x that is listed as 0.

But don't fret; we can still find it.

Let's see if the function is linear by seeing if we have the same slope per two points in the table.

For the first pair ( the points (-2,16) and (1,4) ), x increased by 3 and the y decreased by 12 so the slope there is -12/3=-4.

Now looking at the next pair ( the points (1,4) and (2,0) ), x increased by 1 while y decreased by 4 so the slope is -4/1=-4.

So the function appears to be linear.

So the slope-intercept form of a line is y=mx+b where m is slope and b is y-intercept.

We already found the slope from earlier which is m=-4.

So the equation so far is y=-4x+b.

Now to find b, the y-intercept, we need to use a point (x,y) on the line along with y=-4x+b.

Let's see my favorite on the list of points is (2,0).

y=-4x+b with (x,y)=(2,0)

0=-4(2)+b

0=-8+b

8=b

So the y-intercept is 8.

Answer:

Step-by-step explanation:

sin θ= - cos θ

if :    cos θ  ≠  0  

you have :   tan θ = -1

 tan θ = - tan π/4

    tan θ = tan(- π/4 )

θ = - π/4 +kπ       k ∈ Z  

calculate: k   when θ is in quadrant IV :  3π /2 ≤ θ ≤2π

3π /2 ≤  - π/4 +kπ ≤2π

add π/4:         3π /2 + π/4 ≤  - π/4 +kπ+π/4  ≤2π +π/4

                     7π/4 ≤ kπ  ≤9π/4

                   7/4 ≤ k ≤9/4

                  1.75 ≤ k ≤ 2.25

 k ∈ Z  : k =2

so :  θ = - π/4 +2π

     θ = 7π/4

θ = 7(180°)/4 = 315°

Answer:

3√3 cm. = l

√3 cm. = w

Step-by-step explanation:

l = 3w

9 = 3w[w] ↷

9 = 3w² [Divide by 3]

3 = w² [Take the square root]

√3 = w [plug this back into the top equation to get a length of 3√3 cm.]

I am joyous to assist you anytime.

Final answer:

The total amount of money Peter and Angad won and shared in the ratio 6:1, with Peter getting £30 more than Angad, is £72.

Explanation:

Peter and Angad win some money and share it in the ratio of 6:1. If Peter gets £30 more than Angad, to find out how much they got altogether, we can use this ratio to express their winnings in terms of algebra. Let the amount Angad gets be x. Then Peter gets x+£30 since he gets £30 more than Angad. According to the ratio 6:1, Peter's share would be 6 times Angad's share; this gives us the equation 6x = x+ £30.

To solve for x, combine like terms: 6x - x = £30, which simplifies to 5x = £30. Divide both sides by 5 to find x: x = £6. Now that we know Angad receives £6, we can find Peter's share by multiplying Angad's share by 6: 6*£6 = £36. Add the £30 difference to get Peter's total: £36+£30 = £66.

To find the total amount they got altogether, add Angad's share and Peter's share together: £66 + £6 = £72.

Answer:

You should invest $820 in account A and $940 in account B

Step-by-step explanation:

* Lets use the system of linear equations to solve the problem

- Simple Interest Equation I = Prt , Where:

# P = Invested Amount

# I = Interest Amount

# r = Rate of Interest per year in decimal; r = R/100

# t = Time Period involved in months or years

* Lets solve the problem

- The total money invested is $1760

- Account A pays 7​% annual interest

- Account B pays 4​% annual interest

- Let A represent the amount of money invested in the account A

- Let B represent the amount of money invested in the account B

- You would like to earn $ 95 at the end of one year

∴ The interest from both accounts at the end of one year is $95

- Lets write the equations

# Account A :

∵ Account A has $A invested

∴ P = $A

∵ Account A pays 7​% annual interest

∴ r = 7/100 = 0.07

∵ t = 1 year

∵ I = Prt

∴ I = A(0.07)(1) = 0.07A

# Account B :

∵ Account B has $B invested

∴ P = $B

∵ Account A pays 4​% annual interest

∴ r = 4/100 = 0.04

∵ t = 1 year

∵ I = Prt

∴ I = B(0.04)(1) = 0.04B

- The total amount of interest from both accounts at the end of one

  year is $95

∴ I from A + I from B = 95

∴ 0.07A + 0.04B = 95 ⇒ multiply both sides by 100

∴ 7A + 4B = 9500 ⇒ (1)

- The total money to invest in both accounts is $1760

∵ Account A has $A invested

∵ Account B has $B invested

∴ A + B = 1760 ⇒ (2)

* Lets solve the system of equations to find the amount of money

  invested in each account

- Multiply equation (2) by -4 to eliminate B

∵ A + B = 1760 ⇒ × -4

∴ -4A - 4B = -7040 ⇒ (3)

- Add equation (1) and (3)

∵ 7A + 4B = 9500 ⇒ (1)

∵ -4A - 4B = -7040 ⇒ (3)

∴ 7A - 4A = 9500 - 7040

∴ 3A = 2460 ⇒ divide both side by 3

∴ A = 820

- Substitute the value of A in equation (1) or (2)

∵ A + B = 1760 ⇒ (2)

∴ 820 + B = 1760 ⇒ subtract 820 from both sides

∴ B = 940

- From all above

* You should invest $820 in account A and $940 in account B

Answer:

x = 35

Step-by-step explanation:

The sum of the 3 angles in a triangle = 180°

Subtract the sum of the 2 base angles from 180 for angle at vertex

vertex = 180° - (73 + 51)° = 180° - 124° = 56°

The vertex angle is composed of x and 21, so

x + 21 = 56 ( subtract 21 from both sides )

x = 35

Answer:

16 hours

Step-by-step explanation:

Speed=distance/time

Let S be speed, D be distance, and t be time.

Let's solve the speed equation for t since we are looking for time, t.

[tex]S=\frac{D}{t}[/tex]

Multiply both sides by t:

[tex]St=D[/tex]

Divide both sides by S:

[tex]t=\frac{D}{S}[/tex]

So our distance is 239000 and our speed is 250 so plug them in:

[tex]t=\frac{239000}{250}[/tex]

Putting into the calculator now:

t=956 minutes (I knew this was in minutes because the speed was 250 miles per minute)

We need to convert this to hours.  60 minutes=1 hour.

60 minutes=1 hour

956 minutes=x hours

You can setup a proportion if you don't know you just need to take 956 and divide it by 60.  Like so:

[tex]\frac{60}{956}=\frac{1}{x}[/tex]

Cross multiply:

[tex]60x=956(1)[/tex]

[tex]60x=956[/tex]

Divide both sides by 60:

[tex]x=\frac{956}{60}[/tex]

Putting into calculator now:

x=15.9333333333333 which mean we have almost 16 hours

The graph of the inequality x > 34 is graphed

An inequality is an expression that shows the non equal comparison of two or more numbers and variables.

Given the inequality:

3x > 102

Divide the inequality by 3:

x > 34

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