What are the exact solutions of x2 = 4 − 7x?

Answer:

18.

Step-by-step explanation:

y^2 = 2x + 6

2x = y^2 - 6

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

The points of intersection of the  curves are  calculated:

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

2y + 2 = y^2 - 6

y^2 - 2y - 8 = 0

( y - 4)(y + 2) = 0

y = 4, -2

when y = 4, x = 5 and when y = -2 x = -1.

Integrating between  y = 4 and y = -2

      4

      INT  [ ( y + 1 + 1/2 (6 - y^2) ]dy

      -2

=    4

     {   y^2 / 2 + y + 3y - y^3/6 }

  -2

= 13 1/3 - (-4 2/3)

13 1/3 + 4 2/3

= 18  answer.

The area bounded by the two given curves which are y² = 2x + 6 and x = y + 1 is; 18

We want to find the area bounded by the curves;

y² = 2x + 6 and x = y + 1

Put y + 1 for x in first equation to get;

y² = 2(y + 1) + 6

y² - 2y - 8 = 0

Using quadratic equation calculator gives;

y = -2, 4

Now, we can rewrite the first two equations like this;

x = (y²/2) - 3 and x = y + 1

Thus, the area is;

A = [tex]\int\limits^4_{-2}[/tex]  [y + 1 - (¹/₂y² - 3)]dy

Integrating this gives us;

A = [4y + ¹/₂y² - ¹/₆y³]⁴₋₂

Solving by plugging in the boundary values gives;

A = 18

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Answer:

y=-12/5x+62

Step-by-step explanation:

to solve this i turned two point form, 5,50 and 17 1/2,20, into slope intercept form by using the formula y-y1=(y2-y1/x2-x1)(x-x1), which when input with the data becomes y-50=(20-50/17 1/2-5)(x-5) which then becomes y-50=-12/5(x-5), then y-50=-12/5x+12, and finally y=-12/5x+62

B. The graph that best represents the equation y =  |x| - 1 is the option B.

To solve this problem we have to try with some values, the symbol |x| is the absolute value which means any number either positive or negative always is positive |-5| = 5 and |5| = 5.

Let's take x = -3, -2, -1, 0, 1, 2, 3.

For x = -3

y = |-3| - 1 = 3 - 1

y = 2

For x = -2

y =  |−2| - 1 = 2 - 1

y = 1

For x = -1

y =  |−1| - 1 = 1 - 1

y = 0

For x = 0

y =  |0| - 1 = 0 - 1

y = -1

For x = 1

y =  |1| - 1 = 1 - 1

y = 0

For x = 2

y =  |2| - 1 = 2 - 1

y = 1

For x = 3

y =  |3| - 1 = 3 - 1

y = 2

y  ║  x

2      -3

1       -2

0       -1

-1       0

0       1

1        2

2       3

If we graph the points obtain in the table above, the result is a graph with the characteristics of the option B.

Answer:

A

Step-by-step explanation:

Group the 4 terms into groups of 2 without changing their order.  That looks like this:

[tex](2x^3+6x^2)+(5x+15)[/tex]

Now within each group pull out what's common:

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

Now we have (x + 3) common between the 2 terms, so let's pull that out.  With grouping, you know that it "works" if what's in the parenthesis are both exactly the same term.  Ours are both (x + 3).  What's "left over" are the things we originally pulled out, so group those together and you're done!

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

Ted's debt can be expressed as 2w - 28

Data;

This is a mathematical expression written to represent word problems.

Given that Ted's debt is 14 less than half of Mike's debt and Mike's debt is represented as w.

Let x represent Ted's debt

[tex]x = \frac{w}{2} - 14\\ x = 2w -28[/tex]

The expression can be written as 2w - 28

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Answer:

2264 students

Step-by-step explanation:

3330/100=33.3

100-32=68

33.3 x 68=2264.4

Hope This Helps! :D

Answer:

2264 students get 8 or more hours of sleep.

Step-by-step explanation:

We are given that 32% of students say they get less than the recommended eight hours of sleep per night.

We are to find the number of students who get 8 or more hours of sleep.

Percentage of students who get 8+ hours of sleep = 100 - 32 = 68%

Number of students who get 8 or more hours = 68/100 - 3330 = 2264

Answer:

40 mph and 68 mph

Step-by-step explanation:

Let x mph be the speed of the slower car, then 2x-12 mph is the speed of the faster car.

In 6 hours the faster car got to San Diego, so it travels the distance

[tex]6\cdot (2x-12)=12x-72\ miles.[/tex]

By that time the slower car was 168 miles away from the destination, so the slower car travels the distance

[tex]12x-72-168=2x-240\ miles[/tex]

It takes the slower car 6 hours to travel 12x-240 miles, so

[tex]6x=12x-240\\ \\6x-12x=-240\\ \\-6x=-240\\ \\6x=240\\ \\x=40\ mph.[/tex]

Now,

[tex]2x-12=2\cdot 40-12=80-12=68\ mph[/tex]

Answer:

$18.750

Step-by-step explanation:

Answer:  B) $11,093

Step-by-step explanation:

[tex]\text{The formula for depreciation is: }\\EV=P(1-r)^t\\\\\bullet EV=expected\ value\\\bullet P=principal\ \text{(original value)}\\\bullet r=rate\ of\ depreciation\\\bullet t=time\ \text{(in years)}\\\\\\\text{The given values are: }\\\bullet P=\$25,000\\\bullet r=15\%\quad (0.15)\\\bullet t=5\\\\\\EV=25,5000(1-0.15)^5\\.\quad =25,000(0.85)^5\\.\quad =25,000(0.443705)\\.\quad =\large\boxed{11,092.63}[/tex]

Answer:

Linear

Step-by-step explanation:

It is not quadratic or exponential since the term to term sequence is +2.

- 5 ⇔ -3

( Adding 2 )

- 3 ⇔ -1

( Adding 2 )

- 1 ⇔ 1

( Adding 2 )

if your teacher allows the pi symbol then your answer is 2513. if your teacher requires you to use 3.14 for pi then your answer is 2512. basically you use the Pythagorean theorem to find the height, and then plug the height (24) and the radius which is half the diameter (10) into the equation to find the volume of the cone. if you have any other questions about it please ask! hope this helps:)

Answer:

  -12

Step-by-step explanation:

To put the equation into vertex form, you can factor out the leading coefficient, then add and subtract a value equal to the square of half the x-coefficient (inside parentheses).

  = 4(x^2 +2x) -8

  = 4(x^2 +2x +1) -8 -4

  = 4(x +1)^2 -12

The y-value of the vertex is -12.

The answer would be -12

Answer:

  f(-1) = 7

Step-by-step explanation:

Put -1 where x is and do the arithmetic.

  f(-1) = 2(-1) +9 = 7

Answer:

zero

Step-by-step explanation:

Given a rational function then the denominator cannot be zero as this would make the function undefined.

Answer:

zero

Step-by-step explanation:

We have to fill the correct word in the blank space.

Suppose

[tex]f(x)=\frac{1}{x}[/tex]

Substitute x=0 then we get

[tex]f(x)=\frac{1}{0}=\infty[/tex]

The function is not defined at x=0

Because the denominator of the function at x=0 is zero which makes the function not define.

When the definition of a function involves  a fraction , the function is undefined at any value that would make the denominator of the function zero.

It intersects both nappes of the double napped cone, but does not go through the vertex. This weird intersection will create a conic curve that is called the...

The cross-section that is obtained on slicing the double napped cone such that it does not pass through the vertex of the cone is:

                                Hyperbola.

We know when a plane intersects only one nape of a double napped cone then the cross section that is obtained is either a parabola, ellipse or a circle.

But when it passes through both the napes of a double napped cone then the cross-section obtained is a hyperbola.

               Hence, the answer is Hyperbola.

Answer:

See below in Bold.

Step-by-step explanation:

If he scored 76 points he must have answered 76/4 = 19 questions correctly.

If the total number of questions is q then our equation is q = 10 + 76/4

= 10 + 19 = 29 questions.

Step-by-step explanation:

The population is all possible apartments that Greg could rent.  The sample is the ones he finds listed in newspapers and online.

So the answer is the third one.

The sample represents a part of the population from which data is actually taken. In this scenario, the apartments listed in the newspaper and Internet site are the sample, and all the apartments near the college form the population. Therefore, the correct answer is C.

In this question, Greg is trying to estimate the cost of living in an apartment. The population refers to the entire group of entities from which he might theoretically collect data. In this case, the population would be all available apartments near the college. That's because these apartments represent the full scope of possible information Greg would need for his research.

On the other hand, the sample is a subset of this population that Greg is actually reviewing for data. In this case, it would be the listings in a local newspaper and on an Internet site since these listings are the specific data points that Greg is examining.

So, the correct answer is C: 'Sample: listings in a local newspaper and on an Internet site; Population: all available apartments near the college'

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Answer: $186.25

Step-by-step explanation:

To find the original price of the Digital Media Player before the 20% reduction, you can follow these steps:
1. Understand the discount:
A 20% reduction means the item is sold for 80% of its original price (100% - 20% = 80%).
2. Set up the equation to represent this situation:
Let the original price be \( P \). If the player is marked down by 20%, then it is now being sold for 80% of its original price. So we can write the following equation:
\[ 0.80 \cdot P = 149 \]
3. Solve for \( P \) (the original price):
To find the original price, divide the reduced price by 0.80 (which represents the 80% it is now being sold for).
\[ P = \frac{149}{0.80} \]
4. Calculate the result:
\[ P = \frac{149}{0.80} \]
\[ P = 186.25 \]
So the original price of the Digital Media Player was $186.25.

Answer:

[tex]\frac{\sqrt{17} }{7}[/tex]

Step-by-step explanation:

Using the Pythagorean identity

sin²x + cos²x = 1, then

sinx = [tex]\sqrt{1-cos^2x}[/tex]

Note that ([tex]\frac{4\sqrt{2} }{7}[/tex])² = [tex]\frac{32}{49}[/tex]

sinΘ = [tex]\sqrt{1-(32/49)}[/tex]

        = [tex]\sqrt{\frac{17}{49} }[/tex] = [tex]\frac{\sqrt{17} }{7}[/tex]

Answer:

The sine function is y = 4 sin (4 x) - 3

Step-by-step explanation:

The explanation in the attached file

Answer:

4.4 in

Step-by-step explanation:

If a radius is perpendicular to a chord, it bisects that chord. You can use Pythagorean theorem here

[tex] {3.7}^{2} + {2.4}^{2} = {x}^{2} [/tex]

Once solved you'll find x to be roughly 4.4 in

Answer:

x = 4.4 in

Step-by-step explanation:

The segment from the centre of the circle to the chord is a perpendicular bisector, hence

7.4 ÷ 2 = 3.7

Consider the right triangle with legs 3.7 and 2.4 and hypotenuse x

Using Pythagoras' identity in the right triangle, then

x² = 2.4² + 3.7² = 5.76 + 13.69 = 19.45

Take the square root of both sides

x = [tex]\sqrt{19.45}[/tex] ≈ 4.4 in

Answer:

30

Step-by-step explanation:

The population mean can be estimated with the sample average, so add up all the ages and divide by how many there are.

(19 + 27 + 25 + 21 + 44 + 22 + 45 + 34) / 8

29.625

Rounded to the nearest whole number, the average age is 30.

Answer:

y = 105°

Step-by-step explanation:

In an isosceles trapezoid

• Any lower base angle is supplementary to any upper base angle

• The lower base angles are congruent

75 and x are supplementary, thus

x = 180° - 75° = 105°

x and y are lower base angles and congruent, so

y = x = 105°

Answer:

381.5 cubic inches

Step-by-step explanation:

Answer:

[tex]|5+13i|=\sqrt{194}[/tex]

Step-by-step explanation:

The given expression is [tex]|5+13i|[/tex].

This is a complex number expression.

The absolute value of the complex number means the magnitude of the complex number.

Recall that;

[tex]|a+bi|=\sqrt{a^2+b^2}[/tex]

This implies that:

[tex]|5+13i|=\sqrt{5^2+13^2}[/tex]

[tex]|5+13i|=\sqrt{25+169}[/tex]

[tex]|5+13i|=\sqrt{194}[/tex]

Answer:

B. [tex]\frac{x\sqrt{2}}{2y}[/tex]

Step-by-step explanation:

We want to divide [tex]\sqrt{9x^2}[/tex] by[tex]\sqrt{18y^2}[/tex].

This becomes:

[tex]\frac{\sqrt{9x^2}}{\sqrt{18y^2}}[/tex]

[tex]\frac{\sqrt{(3x)^2}}{\sqrt{2(3y)^2}}[/tex]

We remove the perfect squares to obtain

[tex]\frac{3x}{3y\sqrt{2}}[/tex]

Cancel out the common factors to get;

[tex]\frac{x}{y\sqrt{2}}[/tex]

Rationalize the denominator to get:

[tex]\frac{x}{y\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}[/tex]

[tex]\frac{x\sqrt{2}}{2y}[/tex]

The correct answer is B

Answer:

[tex]\large\boxed{B.\ \dfrac{x\sqrt2}{2y}}[/tex]

Step-by-step explanation:

[tex]\sqrt{9x^2}:\sqrt{18y^2}=\dfrac{\sqrt{9x^2}}{\sqrt{18y^2}}\qquad\text{use}\ \sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\\\\=\dfrac{\sqrt9\cdot\sqrt{x^2}}{\sqrt{18}\cdot\sqrt{y^2}}\qquad\text{use}\ \sqrt{a^2}=a\ \text{for}\ a\geq0\\\\=\dfrac{3\cdot x}{\sqrt{9\cdot2}\cdot y}=\dfrac{3x}{\sqrt9\cdot\sqrt2\cdot y}=\dfrac{3x}{3y\sqrt2}\qquad\text{cancel 3}\\\\=\dfrac{x}{y\sqrt2}\cdot\dfrac{\sqrt2}{\sqrt2}\qquad\text{use}\ \sqrt{a}\cdot\sqrt{a}=a\\\\=\dfrac{x\sqrt2}{2y}[/tex]

Mean: 20.5

Standard Deviation: 11.5

The mean is the total of the numbers divided by the amount of numbers. So, add 36 + 14 + 21 + 39 + 11 + 2 to get 123. Now, divide 123 by 6 to find that the mean is 20.5.

The standard deviation is the mean of the distances from the numbers to the mean. So, find the distance from the mean for each number. You get 15.5, 6.5, 0.5, 18.5, 9.5, and 18.5. Find the mean of these distances. Start by adding them together to get 69, then divide that by 6 to get a standard deviation of 11.5.

Answer with Step-by-step explanation:

Six samples are collected of the number of scanning errors: 36, 14, 21, 39, 11, and 2 errors

Mean=(Sum of all observations)/(Total number of observations)

        =(36+14+21+39+11+2)/6

        = 123/6

        = 20.5

Standard deviation is the square root of mean of squares of deviation around mean

Deviation around mean:

36-20.5, 14-20.5, 21-20.5, 39-20.5, 11-20.5, and 2-20.5

15.5,-6.5,0.5,18.5,-9.5 and -18.5

Square of deviations:

240.25,42.25,0.25,342.25,90.25 and 342.25

Mean of square of deviations

=(240.25+42.25+0.25+342.25+90.25+342.25)/6

=176.25

square root of mean of deviations= [tex]\sqrt{176.25}=13.28[/tex]

Hence, Standard deviation=13.28

and Mean=20.5

Answer:

f(x) = 3x - 3

Step-by-step explanation:

Plug into TI-83/84 calculator.

Hit STAT

Hit EDIT

X-Values go in for L1

Y-Values go in for L2

Hit STAT

Scroll over to CALC

Hit #4 for LinReg (Finds line of best fit)

Enter all the way through

A is your slope

B is your y-intercept

Answer:

[tex]f(x)=3x-3[/tex]  

Step-by-step explanation:

Data : (-2,-9), (0,-3), (1,0), (3,6), (5,12)

Using technology , Plot the points on the calculator .

(Refer the attached figure)

Or

First calculate the slope of given points

[tex]m = \frac{y_2-y_1}{x_2-x_1}[/tex]   ---A

[tex](x_1,y_1)=(-2,-9)[/tex]

[tex](x_2,y_2)=(0,-3)[/tex]

Substitute values in A

[tex]m = \frac{-3-(-9)}{0-(-2)}[/tex]

[tex]m = \frac{-3+9}{2}[/tex]

[tex]m = \frac{6}{2}[/tex]

[tex]m = 3[/tex]

[tex](x_1,y_1)=(1,0)[/tex]

[tex](x_2,y_2)=(3,6)[/tex]

Substitute values in A

[tex]m = \frac{6-0}{3-1}[/tex]

[tex]m = \frac{6}{2}[/tex]

[tex]m = 3[/tex]

Since the slopes are same .

So, the the given data is a linear function.

Now to obtain the equation for data we will use two point slope form.

[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]   ---B

[tex](x_1,y_1)=(-2,-9)[/tex]

[tex](x_2,y_2)=(0,-3)[/tex]

Substitute values in B

[tex]y+9= \frac{-3-(-9)}{0-(-2)}(x+2)[/tex]  

[tex]y+9= \frac{6}{2}(x+2)[/tex]  

[tex]y+9=3(x+2)[/tex]  

[tex]y+9=3x+6[/tex]  

[tex]y=3x+6-9[/tex]  

[tex]y=3x-3[/tex]  

So, [tex]f(x)=y=3x-3[/tex]  

Thus Option B is true .

Hence the required equation is  [tex]f(x)=3x-3[/tex]  

Answer:

19-(-8)-(-14) = 41

Step-by-step explanation:

First, we have to solve what is in parentheses

by law of signs ( - . - = +)

19 + 8 + 14 = ?

Then, we only have to sum the number to obtain the result

19 + 8 + 14 = 41

Answer:

41

Step-by-step explanation:

We must do multiplication before addition or subtraction here.

-(-8) = +8 and -(-14) = +14, and therefore:

19 - (-8) - (-14) becomes 19 + 8 + 14, or 19 + 22, or 41.

Answer:

50

Step-by-step explanation:

Divided $225 divided by $15 which equal $15 then divided $750 divided by 15 which equal $50.

Thelma and Laura need to mow at least 65 lawns to make a profit of $750 after accounting for the $225 cost of the lawnmower. They must charge $15 per lawn to meet this goal.

Let's define x as the number of lawns mowed. Thelma and Laura charge $15 per lawn mowed, so the total money they make through mowing lawns is $15x. They initially spent $225 for a lawnmower, and they want to make a profit of at least $750. Profit is calculated as revenue (money made) minus costs, so we have the inequality $15x - $225 >= $750. Simplify this inequality to get x >= 65. Therefore, they need to mow at least 65 lawns in order to make their desired profit.

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Answer:

infinite number of solutions

Step-by-step explanation:

Given

8(x - 3) = 8x - 24 ← distribute left side

8x - 24 = 8x - 24

The expressions on both sides are equal

Hence any real value of x is a solution

That is there are an infinite number of solutions