Answer:
The quadratic equation 3x² - 12x + 9 = 0 has two real roots when solved:
x₁ = 1 and x₂ = 3
Step-by-step explanation:
✍ An equation of type ax² + bx + c = 0, can be solved, for example, using the quadratic formula:
[tex]\boldsymbol{x=\dfrac{-b\pm\sqrt{b^{2}-4ac } }{2a} }[/tex]
either
[tex]\boldsymbol{x=\dfrac{-b\pm\sqrt{\Delta} }{2a} }[/tex]
where
[tex]\bf{\Delta=b^{2}-4ac }[/tex]
Identify the coefficients
a = 3, b = -12 and c = 9
Calculate the discriminant value
Δ = b² - 4ac
Δ = (-12)² - 4.3.9 = 144 - 12.9
Δ = 144 - 108 = 36
Enter the values of a, b and the discriminant value in the quadratic formula
[tex]\boldsymbol{x=\dfrac{-b\pm\sqrt{\Delta } }{2a} }[/tex]
[tex]\boldsymbol{x=\dfrac{-(-12\pm\sqrt{36} }{2\cdot3 } }[/tex]
[tex]\boldsymbol{x=\dfrac{(12\pm\sqrt{36} }{6 } \ ==== > \ (Solution \ general) }[/tex]
As we can see above, the discriminant (Δ) of this equation is positive (Δ> 0) which means that there are two real roots (two solutions), x₁ and x₂.
[tex]\boldsymbol{x_{1}=\dfrac{-12-\sqrt{36} }{6}=\dfrac{12-6}{6}=\dfrac{6}{6}=1 \ === > (First \ solution) }[/tex]
To find x₁, just choose the positive sign before the square root. Later,
[tex]\boldsymbol{x_{1}=\dfrac{12+\sqrt{36} }{6}=\dfrac{12+6}{6}=\dfrac{18}{6}=3 \ === > (Second \ solution) }[/tex]
In a cave, a stalactite gets 44 millimeters longer each year. this year it is 72 centimeters long. how many years until it is 1 meter long?
Answer:
6.36 years (the .36 repeats), so rounded is 6.4.
Step-by-step explanation:
First, I'm going to convert all measurements to meters.
Since 1000 mm = 1 m, divide 44 by 1000.
44/1000 = 0.044 m
Since 100 cm = 1 m, divide 72 by 100.
72/100 = 0.72 m
Next, I'm going to setup an equation using y = mx + b.
b is the beginning amount which is 0.72.
m in the equation represents the rate which is 0.044.
y represents the total which is 1.
Plugin the numbers and solve for x.
1 = 0.044x + 0.72
1 - 0.72 = 0.044x + 0.72 - 0.72
0.28 = 0.044x
0.28/0.044 = 0.044x/0.044
6.36 = x
can someone help me with this problem? Thank you!
Answer:
152
Step-by-step explanation:
If you substitute the 6, the expression will look like this-
8( 3(6) + 1 )
You can then times 3 by 6, this makes 18. You then add the one, this equals 19.
19 times 8 = 152.
I hope this helps, have a lovely day.
wyatt has a square garden. he adds 3 feet to one side and 1 foot to the other and now it is a rectangle with an area of 24 ft^2 what are the new dimensions of the garden
The new dimension of the rectangle are as follows;
6 ft4 ftHow to find the new dimension of the rectangle?The new dimension of the rectangle can be found as follows;
The sides of a square are equal.
Therefore,
let the side of the square = x
Hence,
length = x + 3
width = x + 1
area of rectangle = lw
where
l = lengthw = widthTherefore,
24 = (x + 3)(x + 1)
24 = x² + x + 3x + 3
24 = x² + 4x + 3
x² + 4x + 3 - 24
x² + 4x - 21 = 0
(x - 3)(x + 7)
Therefore,
x = 3 or -7
It can't be negative. Hence, x = 3 ft
Therefore, the new dimension of the rectangle are as follows;
x + 3 = 3 + 3 = 6 ftx + 1 = 3 + 1 = 4 ftlearn more on rectangles here: https://brainly.com/question/17562456
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What is the answer of 32% of 125
Answer:
40
Step-by-step explanation:
32÷100= 0.32
0.32×125= 40
convert your percentage into a number first, so that you are working with normal numbers only, and then multiply that by your total.
The illustration below shows the graphs of fourteen functions.
Two of them have equations
y = (z+6)³-2
y=-(z-9)³ +3
Due to length restrictions, we kindly invite to check the explanation of this question to understand the derivation of the polynomic expressions.
How to determine a family of cubic functions
Cubic functions are polynomials of grade 3. In this case, we have pairs of cubic functions of the following form:
y = (x - h)³ + k (1)
y = - (x - h)³ + k (2)
a) Where (h, k) are the coordinates of the vertex of each cubic function. There is a translation of (x, y) = (3, 1) between each two consecutive pairs of cubic functions. Hence, we have the following fourteen cubic functions:
y = (x + 9)³ - 3y = - (x + 9)³ - 3y = (x + 6)³ - 2y = - (x + 6)³ - 2y = (x + 3)³ - 1y = - (x + 3)³ - 1y = x³ y = - x³y = (x - 3)³ + 1y = - (x - 3)³ + 1y = (x - 6)³ + 2y = - (x - 6)³ + 2y = (x - 9)³ + 3y = - (x - 9)³ + 3b) Another family of functions with a similar pattern is shown below:
y = (x + 9)² - 3y = - (x + 9)² - 3y = (x + 6)² - 2y = - (x + 6)² - 2y = (x + 3)² - 1y = - (x + 3)² - 1y = x² y = - x²y = (x - 3)² + 1y = - (x - 3)² + 1y = (x - 6)² + 2y = - (x - 6)² + 2y = (x - 9)² + 3y = - (x - 9)² + 3To learn more on cubic functions: https://brainly.com/question/25732149
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Please help!!
The two-way frequency table below shows data on playing a sport and playing a musical instrument for in a class.
Answer: Check out the screenshot below for the answers
==========================================================
Explanation:
For each column, we divide the values by the sum of each column.
The sum of the first column is 6+8 = 14, and the sum of the second column is 7+3 = 10.
Column 1:
6/14 = 0.438/14 = 0.57Those values are approximate to two decimal places.
Column 2:
7/10 = 0.703/10 = 0.30Those values are exact.
The values in bold are then filled in the table as you see below.
A result like 0.43 means that 43% of those who play a sport also play an instrument. The other values are interpreted in a similar fashion.
If a fair die is rolled 3 times, what is the probability, to the nearest thousandth, of getting exactly 2 ones?
[tex]\displaystyle|\Omega|=6^3=216\\A=\binom{3}{2}\cdot5=3\cdot5=15\\|A|=\dfrac{15}{216}=\dfrac{5}{72}\approx0.069[/tex]
A hypothesis regarding the weight of newborn infants at a community hospital is that the mean is 22.1 pounds. a sample of seven infants is randomly selected and their weights at birth are recorded as 21.1, 25.1, 26.1, 27.1, 24.1, 31.1, and 31.1 pounds. if α = 0.010, what is the critical value? the population standard deviation is unknown.
The critical value at the 0.010 significance level with 6 degrees of freedom are ±3.7074
How to determine the critical value?The dataset is given as:
21.1, 25.1, 26.1, 27.1, 24.1, 31.1, and 31.1
The sample mean is:
[tex]\bar x = 22.1[/tex]
The significance level is
α = 0.010
Start by calculating the degrees of freedom using:
df = n - 1
Where n represents the sample size (n = 7)
So, we have:
df = 7 - 1
df = 6
Using the table for t critical values for a two-tailed test, we have:
Critical value = ±3.7074
Hence, the critical value at the 0.010 significance level with 6 degrees of freedom are ±3.7074
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Find the values of X and Y in the parallelogram.
Answer: x=-5, y=20
Step-by-step explanation:
Opposite sides of a parallelogram are congruent, so AB=CD.
[tex]x^{2}-15=-2x\\x^{2}+2x-15=0\\(x+5)(x-3)=0\\x=-5, 3[/tex]
But as distance must be positive, as CD=-2x, x must be negative, so x=-5.
Opposite angles of a parallelogram are congruent, so [tex]m\angle D=m\angle B[/tex].
[tex]4y+26=106\\4y=80\\y=20[/tex]
Use vectors to find the interior angles of the triangle with the given vertices. (Enter your answers as a comma-separated list. Enter your answers in terms of degrees. Round your answers to two decimal places.) (−6,-9), (2,7),(7,2)
Answer:
23.20°, 71.57°, 85.24°
Step-by-step explanation:
The angle between two vectors can be found by making use of the definition of the dot product. The computed angle is the angle between the vectors when they are placed tail-to-tail.
__
setupvector definition
The second attachment shows the given points plotted as A, B, C in the order given. In the diagram, the vectors a, b, c are defined as BA, CB, AC, respectively. That is, the vector a is ...
a = A -B = (-6, -9) -(2, 7) = (-6-2, -9-7) = (-8, -16)
The vectors are defined in counterclockwise order around the triangle, though that makes no difference to the calculation.
dot product definition
The definition of the dot product of vectors A and B is ...
A·B = |A|×|B|×cos(θ) . . . . . where θ is the angle between the vectors
Solving for the angle, we find it to be ...
[tex]\theta=\arccos\left(\dfrac{\vec{A}\cdot\vec{B}}{|\vec{A}|\,|\vec{B}|}\right)[/tex]
__
computationBecause of the way the vectors in this solution are defined, the angle between any given pair of vectors will be an exterior angle of the triangle. In order to find the measure of the interior angle, we must reverse one of the vectors. That is, the dot product used in our computation will be the opposite of the dot product of the vectors we have found.
The spreadsheet shown in the first attachment does the necessary computations.
Each vector is the difference of successive points. (x1 -x2, y1 -y2)Each dot product shown is the opposite of the dot product of successive vectors. -(x1x2 +y1y2)The magnitude is computed in the usual way: the root of the sum of the squares of the components of the vector. √(x²+y²)The cosine is the dot product of successive vectors, divided by each of the vector magnitudes.The spreadsheet shows the angle in degrees, having converted it from the radian value produced by the ACOS function.As a check, the spreadsheet shows the sum of the angle values. (The rounded values add up to 180.01°.)
The interior angles of the triangle are 23.20°, 71.57°, 85.24°.
_____
Additional comment
Another way to find the interior angles from coordinates is to consider the slope of each side of the triangle as being the tangent of the angle it makes with the x-axis. The difference of these angles can be used to find the interior angles of the triangle. Less work is involved because there is no dot-product or magnitude computation.
What is the length of the missing leg? If necessary, round to the nearest tenth.
Answer:
is that the ixl
website ????????????????????????????????
Of the following three statements, choose the best option and write a viable argument showing how it is reflected by the cost function for wichita's factory
A viable argument showing how it is reflected by the Cost Function for Wichita’s factory is; The cost of production per unit of item produced is 1$ as long as total units produced is less than 2500. If more than 2500 units are produced than the cost per unit actually comes down to 0.8$per unit.
How to Interpret Cost Functions?
From the given options, the best statement is;
The plant’s production processes are performed primarily by robots that are able to work longer hours, when needed, at no additional cost.
A cost function is a formula used to predict the cost that will be experienced at a certain activity level. Thus, the cost function for the best option is;
The cost of production per unit of item produced is 1$ as long as total units produced is less than 2500. If more than 2500 units are produced than the cost per unit actually comes down to 0.8$per unit.
The missing options are;
___ The electricity contract with the utility company is structured so that higher daily energy usage is charged at a lower rate.
___ The plant’s production processes are performed primarily by robots that are able to work longer hours, when needed, at no additional cost.
___ Overtime wages were required to produce at levels above 25,000 units.
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Please help me with this question, it says:
Solve the equation
x to the power of 3, take away x equals to 120
Using a trial and improvement method.
You must show working out.
Answer:
x = 5.
Step-by-step explanation:
x^3 - x = 120
We first guess the value of x and calculate the left side.
Take x = 4:-
4^3 - 4 = 64-4 = 60
This is < 120 so try x = 5:
5^3 - 5 = 125 - 5 = 120.
So one solution is x = 5.
Determine, to the nearest tenth of a year, how long it would taken an investment to double at a 3 3/4% interest rate, compounded continuously
Based on the interest rate and continuous compounding, the investment would double in value after 18.5 years.
We have given that,
investment to double at a 3 3/4% interest rate,
When will the investment double in value?The future value using continuous compounding is:
= Amount x e ^ (rate x time)
Interest is
= 3.75%
What is the formula of an exponential function?2 = e ^ (0.0375 x time)
In2 = 0.0375 x time
t = In2 / 0.0375
t= 18.5 years
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PLEASE HELP!
Which of the following statements is/are true? (5 points)
I. If f '(c) = 0, then f has a local maximum or minimum at x = c.
II. If f is continuous on [a, b] and differentiable on (a, b) and f '(x) = 0 on (a, b), then f is constant on [a, b].
III. The Mean Value Theorem can be applied to f(x) = 1/x2 on the interval [-1, 1].
I only
II only
I and III only
I and II only
I especially need help with the first statement... I think it's true in some situations, but there are some counter-examples. I don't know if this means it's true or false!!!
Both the first and second statements are true and therefore Option D is the correct answer .
What is a Function ?A function can be defined as the equation that identifies relation between a dependent variable and an independent variable.
A function always has a defined range and domain.
In the following statements
I. If f '(c) = 0, then f has a local maximum or minimum at x = c.
f'(c) = 0
It means that the slope of f(x) = 0 at x = c
If f has a local maximum or minimum at x = c It is a True statement
II. If f is continuous on [a, b] and differentiable on (a, b) and f '(x) = 0 on (a, b), then f is constant on [a, b].
If f'(x) = 0 on (a, b),
then f is neither increasing nor decreasing
and therefore is constant on [a, b]
So it is a True statement.
III. The Mean Value Theorem can be applied to f(x) = 1/x² on the interval [−1, 1].
No the theorem cannot be applied as, f is neither continuous nor differentiable at x = 0
It is a False Statement.
Therefore Option D is the correct answer.
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Calculate the mean of 1,2,3,4,5,6
Answer:
mean = 3.5
Step-by-step explanation:
the mean is calculated as
mean = [tex]\frac{sum}{count}[/tex] = [tex]\frac{1+2+3+4+5+6}{6}[/tex] = [tex]\frac{21}{6}[/tex] = 3.5
Answer:
The mean of 1 , 2 , 3 , 4 , 5 , 6 is 3.5
Given :
The observations 1 , 2 , 3 , 4 , 5 , 6
To find :
The mean of 1 , 2 , 3 , 4 , 5 , 6
Solution :
Step 1 of 2 :
Calculate sum of the observations
Here the given observations are 1 , 2 , 3 , 4 , 5 , 6
Number of observations = 6
Sum of the observations
= 1 + 2 + 3 + 4 + 5 + 6
= 21
Step 2 of 2 :
Find the mean
The required mean
A local public library decides to track the number of hours that a certain computer is being used. The table represents the number of hours, y, which is dependent on the number of days, x.
x
y
3
21
5
37
7
53
What is the linear equation that represents this situation?
y = 8 x + 45
y = 8 x minus 3
y = 2 x + 16
y = 2 x minus 32
That statistic is based on a pinterest quote, the actual statistic is closer to 8-10%
What is the answer (6√2)(-3√5)
Solution:
[tex] - 18 \sqrt{10} [/tex]
Step-by-step explanation:1. Apply rule : a(-b) = -ab
[tex] = - 6 \sqrt{2} \: \: • \: \: 3 \sqrt{5} [/tex]
2. Apply radical rule : [tex] \sqrt{a} \sqrt{b} = \sqrt{ab} [/tex] , [tex]a \geqslant 0[/tex], [tex]b[/tex]
[tex] = 6 \sqrt{2 \: • \: 5} \: • \: 3[/tex]
3.Multiply the numbers : 2 • 5 = 10=[tex]6 \sqrt{10} •3[/tex]
4.Multiply the numbers : - 6 • 3= - 18[tex] = - 18 \sqrt{10} [/tex]
Fawzia randomly selected 50 plants at a nursery to test for aphids. she discovered 7 plants that had the pests. if the nursery has 780 plants. what could be the predicted number of plants with aphids?
The predicted number of plants with aphids is 109.
What is the predicted number of plants with aphids?The predicted number of plants with aphids is the ratio of the random plants with aphids to total number of random plants selected multiplied by the number of plants in the nursery.
Predicted number = (7/50) x 780 = 109.
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The points (4,2) and (-1, y) are √√74 units apart. What is the
value of y?
Please help!!!
Answer:
y = - 5 or y = 9
Step-by-step explanation:
using the distance formula
d = [tex]\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }[/tex]
with (x₁, y₁ ) = (4, 2 ) and (x₂, y₂ ) = (- 1, y )
d = [tex]\sqrt{(-1-4)^2+(y-2)^2}[/tex]
= [tex]\sqrt{(-5)^2+(y-2)^2}[/tex]
= [tex]\sqrt{25+(y-2)^2}[/tex]
equating to [tex]\sqrt{74}[/tex] , that is
[tex]\sqrt{25+(y-2)^2}[/tex] = [tex]\sqrt{74}[/tex] ( square both sides )
25 + (y - 2)² = 74 ( subtract 25 from both sides )
(y - 2)² = 49 ( take square root of both sides )
y - 2 = ± [tex]\sqrt{49}[/tex] = ± 7 ( add 2 to both sides )
y = 2 ± 7
Then
y = 2 + 7 = 9 or y = 2 - 7 = - 5
What is another name for the Sierpinski triangle?
Sierpinski Pyramid
Sierpinski Gasket
Sierpinski Fractal
Fractal Triangle
Answer:
The Sierpinski triangle is also called the Sierpinski gasket.
Step-by-step explanation:
The Sierpinski triangle is a fractal and attractive fixed set named after the Polish mathematician Wacław Sierpiński who described it in 1915.
Given that ABCD is a rhombus, what is the value of x?
Answer:
E. 18.6
Step-by-step explanation:
The diagonals of a rhombus bisect each other at right angles. The means all of the triangles shown are right triangles, and the two marked angles are complementary.
__
solve for xComplementary angles total 90°.
(x°) +(4x -3)° = 90° . . . . . the marked angles are complementary
5x = 93 . . . . . . . . . . . . divide by °, add 3
x = 93/5 = 18.6 . . . . . divide by the coefficient of x
The value of x is 18.6.
In her free time, isabelle likes to read. she kept track of the number of pages in each book she read last summer. this box plot shows the results. what fraction of the books has 200 or fewer pages?
200 is the median (50%) of the data represented in the box plot. The fraction of the books has 200 or fewer pages is: 1/2.
What is the Median of a Data in a Box Plot?In a box plot, the median is indicated as the value marked by the vertical line that divides the rectangular box of the box plot.
The median is the middle of the data and represents the 50th percentile. That is 1/2 the data.
In the box plot given, 200 pages is the median. Therefore, the fraction of the books that have 200 or fewer pages is: 1/2.
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Which equation is equivalent to StartRoot x squared + 81 EndRoot = x + 10?
The given equation is equivalent to Option D .
The mission option is
A. x+10= x+10
B. x+9 =x² + 20x + 100
C. x² +81 = x² + 100
D. x² + 81 = x² +20x + 100
What is an Equation ?An equation is a mathematical statement formed when two algebraic expression is equated by an equal sign .
An equation is given in the question
[tex]\rm \sqrt{x^2 + 81} = x +10[/tex]
[tex]\rm {x^2 + 9^2} = (x +10)^2[/tex]
[tex]\rm {x^2 + 81} = x^2 +20x + 100[/tex]
Therefore the given equation is equivalent to Option D .
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*Please Help*
F(x) = (x+2)(-x+3)(x-2)^2
Please put F(x) in standard form. Show how you figured it out.
Answer:
F(x) = (x+2)(-x+3)(x-2)^2. I think answer should be like this
Need Help ASAP
Consider the circles A, B, C, and D.
Four circles A, B, C, and D are shown. Circle A has a diameter of 8 feet. Circle B has a diameter of 10 inches. Circle C has a diameter of 2 feet. Circle D has a radius of 50 inches.
a. Without calculating, circle
has the greatest circumference.
Question 2
Explain.
Answer:
Circle D
Step-by-step explanation:
The circumference of a circle can be calculated by multiplying the diameter by π. Therefore, the circle with the greatest circumference will be the circle with the greatest diameter.
The only circle for which we have been told the radius is Circle D. The diameter of a circle is twice the radius, therefore the diameter of Circle D is 50 in × 2 = 100 in
There are 12 inches in 1 ft.
Therefore, the circumferences of the circles in order from smallest to largest is: B < C < A < D
Proof
Diameters
Circle A = 8 ft = 96 inCircle B = 10 inCircle C = 2 ft = 24 inCircle D = 2 × 50 in = 100 inFind the area of the following rhombus: A m? 52m 8m A=?m2
The area of the rhombus is 832 square m if the length of the diagonals are 104 m and 16 m.
What is a rhombus?A rhombus is a two-dimensional shape having four parallel opposite pairs of straight, equal sides. This shape resembles a diamond and is what you'd find on a deck of cards to represent the diamond suit. Rhombuses can be encountered in a variety of common situations.
We have given a rhombus with diagonals.
The length of the long diagonal p = 2×52 = 104 m
The length of the short diagonal q = 2×8 = 16 m
Area = pq/2 = (104×16)/2 = 832 square m
Thus, the area of the rhombus is 832 square m if the length of the diagonals are 104 m and 16 m.
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the first container of milk has twice as many gallons as the second if john uses 2 gallons from the second and 3 gallons from the first if the first one has 4.5 times more than the second how much did they each have originally
Answer:
SORRY I DONOT KNOW THE ANSWER
Step-by-step explanation:
i am sorry i do not know the amswer
Read the statement.
Doubling the dimensions of a rectangle increases the area by a factor of 4.
If p represents doubling the dimensions of a rectangle and q represents the area increasing by a factor of 4, which are
true? Select two options.
Op q represents the original conditional statement.
~p~q represents the inverse of the original conditional statement.
q- p represents the original conditional statement.
-q- ~p represents the converse of the original conditional statement.
Op q represents the contrapositive of the original conditional statement.
-
Mark this and return
Submit
Next
The options that are true as a result of doubling the rectangle dimensions are;
p → q represents the original conditional statement.
p → ~q represents the contrapositive of the original conditional statement.
What are conditional statements?We are told that;
Doubling the dimensions of a rectangle increases the area by a factor of 4.
Now, p represents doubling the dimensions of a rectangle and q represents the area increasing by a factor of 4.
Thus, using conditional statements notation, we can say that;
p → q represents the original conditional statement.
p → ~q represents the contrapositive of the original conditional statement.
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triangle png What is the angle measure of angle B?
Answer:
[tex]m\angle B = 28^{\circ}[/tex]
Step-by-step explanation:
All angles in a triangle must add up to [tex]180^{\circ}[/tex]
We can solve for [tex]\angle B[/tex] using this knowledge
[tex]\angle B + 40 + 112 = 180\\\angle B + 152 = 180\\\boxed{\angle B = 28}[/tex]
That is the answer
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