Good Evening. Please I really need help for this Further Math Question. Thanks
Answers
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]
Related Questions
I WILL MARK BRAINLY ASAP! Please Help its A Math Question (geometry) .
thank you alot!
Answers
Answer:
25 degrees
Step-by-step explanation:
There are 3 points on a line A, B and C.
If B is between A and C what is the length of BC.
AB= x²-6x
BC= x²+20x
AC= 60-12x
Answers
Given :
B is between A and C
AB= x²-6x
BC= x²+20x
AC= 60-12x
B divide into 2 equal parts
AB and AC are equal
AC = AB +BC
60-12x = x²-6x + x²+20x
[tex]60= 2x^{2} -6x +20x +12x[/tex]
[tex]60 = 2x^{2} -6x +32x\\[/tex]
[tex]60= 2x^{2} +26x[/tex]
[tex]30 = x^{2} +13x\\\\[/tex]
[tex]x^{2} +13x -30 = 0\\x^{2} +15x-2x-30 =0[/tex]
[tex]x(x+15) -2 (x+15) =0[/tex]
[tex](x-2) (x+15) =0\\[/tex]
x=2 , x=-15
x never be negative the x is 2
BC = x²+20x
BC = 2*2 +20*2
BC = 4 +40
BC = 44
length of BC =44
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A school is studying Math for Data and Financial Literacy standardized test scores. Which of the following displays could be used to represent the data, and why?
Bar chart, because test scores are numerical
Box plot, because test scores are categorical
Circle graph, because test scores are categorical
Stem-and-leaf-plot, because test scores are numerical
ik its not A bar charts
Answers
A school is studying Math for Data and Financial Literacy standardized test scores. A bar chart could be used to represent the data because test scores are numerical
This is further explained below.
What is a bar chart?Generally, A bar chart or bar graph is a kind of chart or graph that displays categorical data using bars of varying lengths and heights that are proportionate to the values that they represent. It is possible to lay out the bars either vertically or horizontally.
Column charts are occasionally used interchangeably with vertical bar charts.
In conclusion,Standardized exam results in Math for Data and Financial Literacy are now being analyzed at a school.
Due to the numerical nature of the test results, a bar chart could be an appropriate representation of the data.
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Write the slope-intercept equation for a line that passes through the points (-2, 4) and (3, -5)
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The slope intercept equation for a line that passes through the points (-2,4) and (3,-5) is y = -9/5x ₊ 2/5.
Given the points are:
P₁ = (-2,4) = (x₁ , y₁)
P₂ = (3,-5) = (x₂ , y₂)
Therefore, the equation of the line that passes through the above points is:
y = mx ₊ b , Will be the equation to arrive to at the very end, but we need m & b, the slope and the y-intercept.
m = ( y₂ - y₁ ) / ( x₂ - x₁ )
= ( -5 - 4 ) / ( 3 - ( -2) ) = -9 / 5
∴ m = - 9/5
Using either P₁ or P₂, plug in one of their respective x- and y-values into:
y = -9/5 x + b
substitute (3,-5)
₋5 = (₋9/5)(3) ₊ b
₋5 = ₋27/5 ₊ b
₋5₊27/5 = b
₋ 25 ₊ 27/5 = b
b = 2/5
slope intercept equation:
y = mx ₊ b
y = -9/5x ₊ 2/5
Therefore, y = -9/5x ₊ 2/5 is the equation of the line that passes through both points (-2, 4) and (3, -5).
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need help brainleist to first person who gets it right
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Answer:
Step-by-step explanation:
Answer: I believe A is 52 degrees.
Solution: Find angle c (D + C adds up to 180 degrees cuz its a line)
So do 180-116=64 degrees
Assuming this is an isosceles triangle because the sides on d and c look equal, we know that those 2 angles add up to 128 degrees(64 + 64).
Therefore, since the shape is a triangle(adds up to 180 degrees) you just do 180 minus 128 to get angle A which is 52 degrees.
The answer choices are the ones below. TYSM
Answers
Answer:
B (A reflection over the line y = x)
Step-by-step explanation:
I just think of it as imagining a positive diagonal line (like this --> /)
Hope this helps in some way
I need help.. but we are only doing question 34 please answer quickly
Answers
Answer: 33. r=13.6 34. m=12
Step-by-step explanation:
33.
-0.8+r=12.8
-0.8+r+0.8=12.8+0.8
r=13.6
34.
[tex]\displaystyle\\-\frac{m}{3}=-4\\\\\frac{m}{-3}=-4[/tex]
Multiply both parts of the equation by -3:
[tex]\displaystyle\\\frac{m}{-3}(-3)=(-4)(-3)\\\\ \frac{m(-3)}{-3} =12\\\\m=12[/tex]
Calculate the output for the function with the given input, explain why not.
Answers
Answer:
see explanation
Step-by-step explanation:
Firstly, dividing by zero is undefined.
given
f(x) = [tex]\frac{x}{x-1}[/tex] , then
f(1) = [tex]\frac{1}{1-1}[/tex] = [tex]\frac{1}{0}[/tex] = undefined
A logging company is funding a study to determine the best practice for reforesting a cut section of forest. the previous best practice has been to plant the same type of tree with equally spaced intervals. the proposed method of reforestry is to plant a variety of trees at differing intervals. in the study, some tracts of land are reforested using the previous best practice. what are these tracts of land in the study?
Answers
The tracts of land in the study is the Control group.
The tracts of land in the study are reforested using the previous best practice whose results are already known. Now, it is not known, how the proposed method of reforestrey to plant a variety of trees at different intervals would grow. So, the tracts of land of previous plant can serve as a standard to which comparisons can made as to whether the newly proposed method is performing better than the previous or not.
This study was conducted at the Agua Salud Project site located in the Panama Canal Watershed (9° 13 N′, 79° 47′ W, 33 m amsl) with an average rainfall of 2700 mm annually, 80% of which falls between May and mid-December. Mean daily maximum and minimum temperatures are 32 °C and 23 °C . The topography is characterized by short and steep slopes. Soils are silt clay to clay with pH values of 4.4 to 5.8 and have very low levels of both plant available and total phosphorus. The data were collected from the Agua Salud native species and T. grandis plantation, both established in 2008 and planted in the same 3 m by 3 m spacing configuration.
Despite this information, changing landowners’ perceptions and transitioning from one species to a lesser known one, is a risk for the landowner and may require financial incentives.
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1\b=4/2b-10 can you help me i dont know the answer so can you please help me to find the answer?
Answers
Solving the problem as per a linear equation, the value of b is found to be 1/10.
A linear equation is a form of algebraic equation in which the highest value of a variable is always equal to 1. Such an equation forms a straight line when carved on a graph.
As provided, the given equation is,
1/b = 4/2b - 10
or, 1/b = 2/b - 10 (as 4/2 = 2 )
or, 1/b - 2/b = - 10
or, 1/b ( 1 - 2) = - 10
or, - 1/b = - 10, ( we are using cross multiplication )
so, b = 1/ 10 ( as a negative sign on both sides will be nullified)
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Evaluate ^3 √216. Use pencil and paper. Explain how you can check that your result is correct. Show your work.
Answers
The evaluation of the expression given in surd form is 6.
What is the evaluation of the surd form?Surds are expressions that use a root symbol, such as a square root, or another root symbol. Since the decimals of irrational numbers do not finish or recur, they cannot be expressed accurately in decimal form, hence surds are used instead.
Surds, in mathematics, are square root values that cannot be factored into a whole number or an integer. Numbers that are surds are irrational numbers.
From the given expression, the cube root of 216 can be computed as:
[tex]\sqrt[3]{216}[/tex]
Factor the number 216 as:
= [tex]\sqrt[3]{6^3}[/tex]
= 6
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Solve each equation. 7x²+8 x=-6 .
Answers
7(-6)^2+8
-42^2
-1,764+8
-1,772
The grocery store sells beans in bulk the grocers sign above the beans says at this store you can buy any number of pounds of beans at the same rate and all prices include tax
Answers
Which expression is equal to 7(1+3)?
Answers
Answer: 7+21
Step-by-step explanation:
There are 6 bananas 4 apples and 3 oranges for every 1 orange there are how many bananas
Answers
Answer:
There are 18 bananas
Step-by-step explanation:
6 x 3 = 18
18 bananas.
|14c+5|>7
How can I graph this?
Answers
Answer:
Step-by-step explanation:
step 1: Absolute value inequality entered
|14c+5| > 7
step 2: Clear the absolute-value bars by splitting the equation into its two cases, one for the Positive case and the other for the Negative case.
The Absolute Value term is |14c+5|
For the Negative case we'll use -(14c+5)
For the Positive case we'll use (14c+5)
step 3: -(14c+5) > 7
Multiply
-14c-5 > 7
Rearrange and Add up
-14c > 12
Divide both sides by 14
-c > (6/7)
Multiply both sides by (-1)
Remember to flip the inequality sign
c < -(6/7)
Which is the solution for the Negative Case
step 4:
(14c+5) > 7
Rearrange and Add up
14c > 2
Divide both sides by 14
c > (1/7)
Which is the solution for the Positive Case
c < -6/7
c > 1/7
F=GmM/r^2 solve for G
Answers
Solution of the equation for G is given as G = Fr²/Mm
According to the question we have been given the formula for evaluating the value of F which is given as :
F = GmM/r²
Now we need to solve the equation for evaluating G. Which is given step by step as :
F = GmM/r²
Multiplying both the sides by r² we get
Fr² = GmM*r²/r²
Fr² = GmM
dividing both the sides by Mm we get
Fr²/Mm = GmM/Mm
Fr²/Mm = G
or, G = Fr²/Mm
Hence the solution for G is given as G = Fr²/Mm
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Write the ratio as a fraction in simplest form, with whole numbers in the numerator and denominator 45 to 9
Answers
It should be noted that the ratio in simplest form of 45 to 9 is 5:1.
How to illustrate the information?It should be noted that a ratio and fraction are simply used in mathematics to compare one thing with something else.
In this case, it should be noted that we want to write the ratio as a fraction in simplest form, with whole numbers in the numerator and denominator 45 to 9.
Therefore, the ratio of 45 to 9 will be:
= 45 / 9
= 5:1
Therefore, it should be noted that the ratio in simplest form of 45 to 9 is 5:1.
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. 4500 x ? = 3375, where ? is a proper fraction. Find the reciprocal of ?
Answers
Answer:
261111111/100000
Step-by-step explanation:
can someone please answer this
Answers
The linear equation of the table is y = 5x + 2.
when x = 20, y = 102
when y = 42, x = 8
How to find the linear equation that generates the table?Linear equation can be represented in slope intercept form as follows;
y = mx + b
where
m = slopeb = y-interceptTherefore,
m = 17 - 12 / 3 - 2
m = 5 / 1
m = 5
Therefore, let's find the y-intercept using (2, 12)
y = 5x + b
12 = 5(2) + b
12 = 10 + b
b = 12 - 10
b = 2
Therefore, the linear equation is y = 5x + 2.
when x = 20
y = 5(20) + 2 = 100 + 2
y = 102
when y = 42
42 = 5x + 2
42 - 2 = 5x
5x = 40
x = 40 / 5
x = 8
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A box of chocolates has 3 milk chocolates for every 2 white chocolate. There are 60 chocolates in a box. How many white chocolates are in the box?
Answers
Answer: so the answer will be 90 white chocolates that are in the box
Step-by-step explanation:
so first you have a box of chocolates has 3 milk chocolates for every 2 white chocolate. so first you need to divide 3 divided by 2 which is 1.5
and after you need to multiply 1.5 x 60= 90
Use the properties of logarithms to evaluate each expression. log₉27-log₉9
Answers
The result of the given logarithmic expression log₉27-log₉9 is log₉ 3.
What is termed as logarithms?A logarithmic term is one that embodies a number in logarithmic form. It can also be referred to as a log term.
A quantity is also the product of two or more quantities. As a result, a term could be the product of two or even more quantities, at least one of which can be in logarithmic form. In such cases, the terms are referred to as log terms.
A quantity is also the quotient of two quantities. So, if a term is also a quotient of quantities, and at least one of the quantities can be expressed in log form, the terms are mathematically referred to as log terms.
Quotient Rule: ㏒(A/B) = ㏒A - ㏒BProduct Rule : ㏒A + ㏒B = ㏒(AB)Power Rule: n㏒A = ㏒AⁿBase Rule = ㏒ₐb = ㏒ₓb/㏒ₓaExponent Rule = e∧lnx = xNow, as per the given question;
= log₉27-log₉9
So, applying the Quotient Rule in the given expression;
= log₉ (27/9)
= log₉ 3
Thus, the given expression simplified in the logarithmic expression as log₉ 3.
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A gardener can care for the greens property in 5 hours. if his helper assists him, they will be done in 4 hours. how many hours does it take for the helper to do it alone
Answers
If a gardener can care for the greens property in 5 hours and if his helper assists him, they will be done in 4 hours, then the number of hours it takes for the helper to do it alone is equal to 20.
The number of hours it takes for the helper to do it alone can be found as follows;
Time taken by both the gardener and the helper to complete the job=4
Work done by both the gardener and the helper in one hour = 1/4
Similarly,
Time taken by the gardener to complete the job = 5
Work done by the gardener in one hour = 1/5
The number of hours it takes for the helper alone can be calculated by subtraction. Subtracting the fraction done by the gardener in 1 hour from the fraction done by both the gardener and the helper in 1 hour;
1/4 - 1/5
As the common denominator is 20, we multiply 1/4 by 5 and 1/5 by 4.
5/20 - 4/20 = 1/20
The number of hours it takes for the helper to do it alone can be found by finding the reciprocal of 1/20 which is equal to 20/1
20/1 = 20
Hence, the helper takes 20 hours to do it alone.
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Triangle G is a scaled copy of triangle F
What scale factor takes triangle F to Trangle G?
Answers
The scale may be calculated by dividing the triangle side length by the length of the matching side in the original triangle. In this case, that's 36/48 which reduces to 3/4. Same thing with the other matching sides: 15/20 which reduces to 3/4.
2.
(01.06A LC)
Which rational number equals 0 point 3 with bar over 3 ? (5 points)
1 over 5
1 over 3
3 over 10
3 over 5
Answers
Answer:
1/3 or 1 over 3 but that is the correct answer
Solve each equation. ln (2-x)=1
Answers
The equation of solution ln (2-x)=1 is -0.718.
What is an equation ?ln (2-x)=1
2-x = e
x = 2-e
x= -0.718
A formula would be 3x - 5 = 16, for instance. When this equation is solved, we discover that the value of the variable x is 7.
The point-slope form, standard form, and slope-intercept form are the three main types of linear equations.
A formula that expresses the connection between two expressions on each side of a sign. Typically, it has a single variable and an equal sign.
Two expressions joined by the equals symbol ("=") form an equation. The "left hand side" and "right hand side" of the equation are the expressions on either side of the equals sign. The right side of an equation is frequently considered to be zero.
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hhhhhhhhhhheeeeeeeeeeeeeeeeeeeeellllllllllllllllllllllllllllllllpppppppppppppp please
Answers
Answer:
The correct answer is B
Step-by-step explanation:
It is A because all angles must be congruent to there opposing sides.
Hope that helps! Have an amazing rest of your day! :)
timmy is downloading movies from the internet. He can download 6 movies every 15 minutes. jonathan, who is also downloading movies, can download at a rate of 12 movies every 20 minutes
Answers
The unit rate of downloading per movie for each person is for Timmy it will take 2.5 minutes and for Jonathan it will take 1.66 minutes.
In question it is not given that what to do, so we will find the rate of downloading per movie for each person.
With the help of following steps as given below :
Timmy takes 15 minutes to download 6 movie.
6 Movies = 15 Minutes
1 Movie = 15/6
1 Movie = 2.5 Minutes
Jonathan takes 20 minutes to download 12 movies.
12 Movies = 20 Minutes
1 Movie = 20/12
1 Movie = 1.66 Minutes
Therefore, Timmy and Jonathan will take 2.5 and 1.66 minutes, respectively, to download one movie.
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The data below are the temperatures on randomly
chosen days during a summer class and the number
of absences on those days. Calculate the correlation
coefficient, r.
Temperature, x 70 83 89 88 86 96 73 98 78
Number of absences, y
7 11 14 14 12 19 8 19 9
Answers
Answer:
the data below are the tempretures on randomly
Step-by-step explanation:
Answer:
In plane Euclidean geometry, a rhombus is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal
Step-by-step explanation:
Work out the two numbers that
are multiples of 9
and
have a difference of 52
Answers
Answer:
AB and BC are the sides of a parallelogram, ABCD. AB = 5 cm, BC = 7 cm and ABC = 60 degrees. Draw the lines AD and CD to complete the parallelogram.
Step-by-step explanation: