Please help me. Solve for x. ​

You should be aware of the fundamental theorem of algebra (FTCoA). If a quadratic has roots at [tex]r_1[/tex] and [tex]r_2[/tex], then we can factorize it as

[tex](x - r_1) (x - r_2)[/tex]

Expanding, this is equivalent to

[tex]x^2 - (r_1 + r_2) x + r_1 r_2[/tex]

Now, if we can also write this

[tex]x^2 + bx + c[/tex]

then we must have

[tex]x^2 + bx + c = x^2 - (r_1 + r_2) x + r_1r_2 \\\\ ~~~~ \implies \begin{cases} b = -(r_1 + r_2) \\ c = r_1r_2 \end{cases}[/tex]

(By the way, these are known as Vieta's formulas.)

We're told that the quadratic,

[tex]p(x) = 3x^2 - 6x + 8,[/tex]

has roots [tex]x=\alpha[/tex] and [tex]x=\beta[/tex], so by the FTCoA, we can write

[tex]p(x) = 3 (x - \alpha) (x - \beta)[/tex]

Expanding this last form, we have the identity and exact values

[tex]3x^2 - 6x + 8 = 3 (x^2 - (\alpha + \beta) x + \alpha\beta) \\\\ ~~~~ \implies x^2 - 2x + \dfrac83 = x^2 - (\alpha + \beta) x + \alpha\beta \\\\ ~~~~ \implies \begin{cases} \boxed{\alpha + \beta = 2} \\\\ \boxed{\alpha\beta = \frac83} \end{cases}[/tex]

(i) We want to construct a new quadratic [tex]p_{\rm i}(x)[/tex] such that its roots are [tex]x=\alpha^2[/tex] and [tex]x=\beta^2[/tex]. By the FTCoA, we would have

[tex]p_{\rm i}(x) = (x - \alpha^2) (x - \beta^2)[/tex]

Expanding,

[tex]p_{\rm i}(x) = x^2 - (\alpha^2 + \beta^2) x + \alpha^2\beta^2[/tex]

Now, notice that

[tex]\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = 2^2 - 2\cdot\dfrac83 = -\dfrac43[/tex]

[tex]\alpha^2\beta^2 = (\alpha\beta)^2 = \left(\dfrac83\right)^2 = \dfrac{64}9[/tex]

It follows that

[tex]p_{\rm i}(x) = x^2 + \dfrac43 x + \dfrac{64}9[/tex]

which we can rewrite with integer coefficients by scaling each term by a factor of 9 to get

[tex]p_{\rm i}(x) = \boxed{9x^2 + 12x + 64}[/tex]

(ii) Now we want a quadratic [tex]p_{\rm ii}(x)[/tex] with roots at [tex]2\alpha[/tex] and [tex]2\beta[/tex]. This means

[tex]p_{\rm ii}(x) = (x - 2\alpha) (x - 2\beta) \\\\ ~~~~~~~~ = x^2 - 2 (\alpha + \beta)x + 4\alpha\beta[/tex]

[tex]\implies p_{\rm ii}(x) = x^2 - 4x + \dfrac{32}3[/tex]

or more cleanly, scaling by 3,

[tex]p_{\rm ii}(x) = \boxed{3x^2 - 12x + 32}[/tex]

(iii) Now the roots are [tex]x=\frac\alpha2[/tex] and [tex]x=\frac\beta2[/tex], which gives

[tex]p_{\rm iii}(x) = \left(x - \dfrac\alpha2\right) \left(x - \dfrac\beta2\right) \\\\ ~~~~~~~~ = x^2 - \dfrac{\alpha + \beta}2 x + \dfrac{\alpha\beta}4[/tex]

[tex]\implies p_{\rm iii}(x) = x^2 - x + \dfrac8{12}[/tex]

[tex]\implies p_{\rm iii}(x) = \boxed{12x^2 - 12x + 8}[/tex]

(iv) If the roots are [tex]x=\frac\alpha{\alpha+1}[/tex] and [tex]x=\frac\beta{\beta+1}[/tex], we have

[tex]p_{\rm iv}(x) = \left(x - \dfrac\alpha{\alpha+1}\right) \left(x - \dfrac\beta{\beta+1}\right) \\\\ ~~~~~~~~~= x^2 - \left(\dfrac\alpha{\alpha+1} + \dfrac\beta{\beta+1}\right) x + \dfrac{\alpha\beta}{(\alpha+1)(\beta+1)}[/tex]

Note that

[tex]\dfrac\alpha{\alpha+1} + \dfrac\beta{\beta+1} = \dfrac{\alpha(\beta+1) + \beta(\alpha+1)}{(\alpha+1)(\beta+1)} \\\\ ~~~~~~~~~~~~~~~~~~~~ = \dfrac{2\alpha\beta + \alpha + \beta}{\alpha\beta + \alpha + \beta + 1} \\\\ ~~~~~~~~~~~~~~~~~~~~ = \dfrac{\frac{16}3 + 2}{\frac83 + 2 + 1} \\\\ ~~~~~~~~~~~~~~~~~~~~ = \dfrac{22}{17}[/tex]

[tex]\dfrac{\alpha\beta}{(\alpha+1)(\beta+1)} = \dfrac{\alpha\beta}{\alpha\beta + \alpha + \beta + 1} \\\\ ~~~~~~~~~~~~~~~~~~~~~ = \dfrac{\frac83}{\frac83 + 2 + 1} \\\\ ~~~~~~~~~~~~~~~~~~~~~ = \dfrac8{17}[/tex]

[tex]\implies p_{\rm iv}(x) = x^2 - \dfrac{22}{17}x + \dfrac8{17}[/tex]

[tex]\implies p_{\rm iv}(x) = \boxed{17x^2 - 22x + 8}[/tex]

Answer: 8

Explanation: Based on the given conditions, formulate: 580800 ÷ 363 ÷ 200

Reduce the fraction: 8

Answer: 8

Answer:

the ans is 8.00 the first one

Step-by-step explanation:

2^3x =10         //log

log(2^3x) = log(10)

3x*log(2) = log(10)

3x = log(10) / log(2)

3x = log₂ 10

Answer: log₂ 10 = 3x

Answer:

this cant be simplified further

Step-by-step explanation:

17 is prime

Sure I can help!!!!

Step by step explanation: - + - = -

2/3+1/2 (LCM is 6)

=2/3 (6) + 1/2 (6)

=4 + 3

=7

I really hope this helps!!!! have a great day!!!

Answer:

associative property

Step-by-step explanation:

because the variables match

Step-by-step explanation:

you can ask any questions for clarification

Answer:

12.56 km

Step-by-step explanation:

C = 2πr

= 2 x 3.14 x 2

= 12.56 km

Answer:

12.56km or 12,560m

Step-by-step explanation:

c = 2πr
c = (2)(3.14)(2)
c = (6.28)(2)
c = 12.56

200+10=210 this is the answer

please like it if it is correct

The estimated value of 197 + 10.3 is 210.

Although the term "rounding" is a general one, we typically use the terms "round up" or "round down" to indicate whether the number has gone up or down after being rounded. The supplied number is said to be rounded up when the rounded number is increased, and it is said to be rounded down when the rounded number is dropped.

Given:

197 + 10.3

Now, rounding off nearest 10

197---> 200

and, 10.3 ----> 10

So, 197+ 10.3 = 200+ 10 = 210

Hence, the estimated value is 210.

Learn more about round off here:

https://brainly.com/question/13391706

#SPJ2

In a figure we can see that 0.7 and0.70 are equivalent, we have shade in the figure.

Given that,

To illustrate that 0.7 and 0.70 are equivalent, shade the models.

We have 0.7

We can say 7/10

For example, take a square and divide it by 10 part in column and shade 7 column.

Draw horizontal lines connecting the model's dots using a ruler.

Now divide it into 100 equal pieces.

So,

70/100

Shade the 7 rows and column.

Shaded at 70 or 0.70.

Therefore, in a figure we can see that 0.7 and0.70 are equivalent.

To learn more about equivalent visit: https://brainly.com/question/14672772

#SPJ1

Answer:70

Step-by-step explanation:

70 divided by 5 equals 14

Yo what up?, Answer:

2/3 yards.

Best Explanation Possible:

Nothing lol just do the math.

1/6 x 4 = 4/6 = 2/3 of a yard per hour

LOL have a good day hope you do good in school kiddo.

The lengths of the sides of the quadrilateral are 10, 12, 14, 16, respectively.

Quadrilaterals are figures with four sides, the statement indicates that such lengths are four consecutive even integers. Two even numbers are consecutive when their difference is equal to 2.The mathematical definition of the perimeter is shown below:

52 = x + (x + 2) + (x + 4) + (x + 6)

52 = 4 · x + 12

4 · x = 40

x = 10

Then, the lengths of the sides of the quadrilateral are 10, 12, 14, 16, respectively.

To learn more on quadrilaterals: https://brainly.com/question/11936810

#SPJ1

482 rounded off to nearest 100 is 500 and 482 rounded off to nearest 1000 is zero.

By maintaining a number's value while moving it closer to the next number, rounding off makes the number easier to understand. It is done for entire numbers as well as decimals at different places such as hundreds, tens, and tenths. To maintain the major figures, numbers are rounded off.

At a concert there were 482 people in the audience:

(a) With a number line, 482 was rounded to the nearest hundred.

(b) Since 4 is the hundredth value of 482 and there is no previous thousand to 499. So when 499 is rounded to the nearest thousand, equals 0.

Learn more about the nearest 100 and 1000 here:

https://brainly.com/question/17193219

#SPJ9

Answer:

FG = 51

Step-by-step explanation:

2/5 of 85 = 34

The value of EF is 34

So the value of FG would be the value of EG - EF

85 - 34 = 51

Answer:

0.1 or 1/10

Step-by-step explanation:

-2/5 + 1/2 is also - 4/10 + 5/10 which is 1/10

Step-by-step explanation:

-2x2/5x2+1x5/2x5

-4/10+5/10

1/10

The answer to the formula are-

Part 1:  7√6 - 4√6

As both the digits have the same value √6. It means they can simply be added with sign to solve.

=  7√6 - 4√6

Take √6 common from both;

= √6(7 - 4)

= 3√6

Thus, the simplification of the expression is 3√6.

Part 2: -0.7√7 + 2.3√7

Here, also the common term is √7.

= √7( -0.7 + 2.3)

Simply add the number with sign.

= 1.6√7.

Thus, the simplification of the number is 1.6√7.

Part 3: √45 + √20

In thus, first write the number in its prime factors;

= √3×3×5 + √2×2×5

Take the root of 3 and 2 from the first and second digits of expression.

= 3√5 + 2√5

Now, both have the same term √5.

= √5(3 + 2)

Do the addition;

= 2√5

Thus, the simplification of given expression is 2√5.

To know more about the square roots, here

https://brainly.com/question/428672

#SPJ9

Answer:

3 i  √7

Step-by-step explanation:

sqrt (-63)  = sqrt ( -1 * 3 * 7 *3) =  sqrt (-1) * 3 * sqrt 7 =  3 i sqrt 7  

Reyna will need 0.048 quart of orange juice.

In this question, we have been given a recipe for a punch call for 12 fluid ounces of orange juice.

Reyna needs to make 4 batches of punch for a party.

We need to find number of quarts of orange juice Reyna will need.

From given conditions, we formulate:

= 4 × 12 ÷ 1000

= 12 ÷ 250                  .................(Separate out the common factor)

= 6/125

= 0.048 quart

Therefore, Reyna will need 0.048 quart of orange juice.

Learn more about the quarts here:

https://brainly.com/question/13798723

#SPJ1

The points for the ordered pairs of the linear function has been plotted in the graph shown in the image attached below.

A linear function is a straight line graph consisting of domain elements (x-values) and their corresponding range elements (y-values).

Given the function, y = 1/2x + 1 having the following domain elements (x-values):

D: {-8, -4, 0, 2, 6}

Find the corresponding y-value to each of the x-values by substitution

When x = -8, then:

y = 1/2(-8) + 1

y = -4 + 1

y = -3

(-8, -3)

When x = -4, then:

y = 1/2(-4) + 1

y = -2 + 1

y = -1

(-4, -1)

When x = 0, then:

y = 1/2(0) + 1

y = 0 + 1

y = 1

(0, 1)

When x = 2, then:

y = 1/2(2) + 1

y = 1 + 1

y = 2

(2, 2)

When x = 6, then:

y = 1/2(6) + 1

y = 3 + 1

y = 4

(6, 4)

Thus, the points for the ordered pairs of the linear function has been plotted in the graph shown in the image attached below.

Learn more about linear function graph on:

https://brainly.com/question/4025726

#SPJ1

We can state that, 5 represents the output of the function at x = 2.

A function is a relation whereby each input gives only one possible output for the function.

For example, "f(x)" is the output of a typical function that is given by an input of "x".

Given the function statement, f(2) = -5, the input is x = 2, the output is -5. This means that, the function that has an input of 2, will give an output of -5.

-5, therefore, represents the output of the function when the input, x = 2, that is, f(-2) = -5.

Thus, we can state that, 5 represents the output of the function at x = 2.

Learn more about function on:

https://brainly.com/question/17043948

#SPJ1

Answer:

At 10 hours both companies charge the same rate. So Company A is cheaper if the time is less than 10 hours and after 10 hours, company B is cheaper.

Step-by-step explanation:

It is given that Chloe is deciding between two landscaping companies for her place of business. Company A charges $70 per hour and a $150 equipment fee. Company B charges $55 per hour and a $300 equipment fee. Let A A represent the amount Company A would charge for t t hours of landscaping, and let B B represent the amount Company B would charge for t t hours of landscaping.

Now, we need to write an equation for each situation, in terms of t , t, and determine the number hours, t , t, that would make the cost of each company the same.

Now,

Let A(t) = 70t + 150

Let B(t) = 55t + 300

Further, on equating both the equations at same rate to determine the time t;

=> 70t + 150 = 55t + 300

=> 15t = 150

=> t = 10

Therefore, at 10 hours both companies charge the same rate. So Company A is cheaper if the time is less than 10 hours and after 10 hours, company B is cheaper.

To know more about this topic, click here:

https://brainly.com/question/24319495

#SPJ9

The equation that has no solution will be:

a. -2x + 4 = 2x + 4

b. 12x-2x = 10x -8

d. 3x - 12 = 3(x-4) +1

It should be noted that -2x + 4 = 2x + 4

-2x - 2x = 4 - 4

-4x = 0

It has no solution

Also, 12x-2x = 10x -8 and 3x - 12 = 3(x-4) +1has no solution.

The only equation with solution will be:

3x + 10 = 2x

3x - 2x = -10

x = -10

Learn more about equations on:

brainly.com/question/13763238

#SPJ1

The solution of the given equation by rounding off it to the nearest ten-thousandth is 55.5000.

What is a round-off of the digit?

In mathematics, a round-off is defined as an approximation of the given number to a fewer number of significant digits by taking into consideration some rules.

For eg: 13.58 ~ 13.6

Solving the given equation: 8 x = 444

We are given that 8 x = 444            —- 1

x = 444 / 8

= 55.5

Hence, the solution of the given equation rounded off to the nearest ten-thousandth is 55.5000.

To learn more about the round-off of a digit, visit the link given below:

https://brainly.com/question/4896544

#SPJ4

Question 15:

Option B is the correct answer

If the Patriots win their division, then they will go to the playoffs:

- Invalid conclusion was made.

Question 16:

Option D is the correct answer.

Conclusion: Ethan's boy scout dues are free:

- Drawn using the law of detachment.

The Law of Detachment states that we must detach ourselves from the outcome in order to allow what we desire to materialize in the physical universe.

Example:

If p equals q and p is also true. Then q is true.

We have,

- If the Patriots win their next game, then they will win their division.

- If the Patriots win their next game, then they will go to the playoffs.

Each statement's conclusion is drawn using the law of detachment.

Therefore, if the patriots win their division, then they will go to the playoffs is an invalid conclusion made.

Ethan sold $1000 worth of popcorn.

Conclusion: Ethan's boy scout dues are free.

This conclusion was drawn using the law of detachment.

Thus,

Therefore, if the patriots win their division, then they will go to the playoffs: - Invalid conclusion made.

Conclusion: Ethan's boy scout dues are free:

- Drawn using the law of detachment.

Learn more about the law of detachment here:

https://brainly.com/question/13966470

#SPJ1

Answer: I think its 4/16

Step-by-step explanation:

please let me know if I'm wrong.

Answer:

first row is 11

2nd row is 16

3rd row is 34,41,25

Step-by-step explanation:

45-18-16=11

55-25-14=16

18+16=34

100-41-34=25

The property with each of Tyrone's steps for solving the equation is;

4x = 2 - 4(2 - 3x)

4x = 2 - 8 + 12x

4x = - 6 + 12x

4x - 12x = -6

-8x = - 6

x = -6 / - 8

x = 3/4

Therefore, the solution to Tyrone's equation for x is 3/4

Learn more about property of equality:

https://brainly.com/question/1601404

#SPJ1