The value of m + n for the given quadratic equation is 8.
What is a quadratic equation?
A quadratic equation is a polynomial equation of the form:
[tex]ax^{2} +bx+c=0[/tex]
where x is the unknown variable, and a,b and c are constants. The highest power of the variable x in a quadratic equation is 2, hence the term "quadratic". The constants a,b and c are known as the coefficients of the quadratic equation.
According to the given information:
Let's denote the roots of the quadratic equation [tex]2x^{2} -mx+n=0[/tex] as r1 and r2, with r1 and r2 being the roots of the equation.
Given that the sum of the roots is 6 and the product of the roots is 10. This gives us the following equations:
r1 + r2 = 6 ------ (1)
r1*r2 = 10 ------- (2)
We can use these equations to solve for the values of m and n.
The sum of the roots r1 + r2 is also equal to [tex]\frac{-m}{2}[/tex] by Vieta's formulas, which state that the sum of the roots of a quadratic equation [tex]ax^{2} +bx+c=0[/tex] is given by [tex]\frac{-b}{a}[/tex]. Therefore, we can write:
[tex]\frac{-m}{2}[/tex] = 6
Solving for m, we get:
m = -2*6 = -12 ----------(3)
The product of the roots r1*r2 is also equal to n/2 by Vieta's formulas, which states that the product of the roots of a quadratic equation [tex]ax^{2} +bx+c=0\\[/tex] is given by [tex]\frac{c}{a}[/tex]. Therefore, we can write:
n/2 = 10
Solving for n, we get:
n = 2 * 10 = 20 ------------(4)
Now, we can find the value of m+n by substituting the values of m and n from equations (3) and (4) respectively:
m + n = -12 + 20 = 8
So, the value of m+n is 8.
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Find the length of the arc, s, on a circle of radius r intercepted by a central angle θ. Express the arc length in terms of π. Then round your answer to two decimal places.
Radius, r = 16 inches; Central angle θ = 75°
(Simplify your answer. Type an exact answer in terms of π. Use integers or fractions for any numbers in the expression.)
Answer in form of Pi and Inches
The length of the arc of the circle that has a radius of 16 inches and a central angle of 75 degrees is calculated as 6.66π inches (to two decimal places).
What is the Length of an Arc?The length of an arc is calculated using the formula below:
arc length = θ/360 * 2 * π * r
Given the variables:
Central angle (θ) = 75 degrees
radius (r) = 16 inches
Plug in the values:
arc length = 75/360 * 2 * π * 16
length of arc ≈ 6.66π inches (to two decimal places).
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find the discriminant x^2-5x+22=7x+2
[tex]x^2-5x+22=7x+2\implies x^2-12x+20=0 \\\\[-0.35em] ~\dotfill\\\\ \qquad \qquad \qquad \textit{discriminant of a quadratic} \\\\\\ \stackrel{\stackrel{a}{\downarrow }}{1}x^2\stackrel{\stackrel{b}{\downarrow }}{-12}x\stackrel{\stackrel{c}{\downarrow }}{+20}=0 ~~~~~~~~ \stackrel{discriminant}{b^2-4ac}= \begin{cases} 0&\textit{one solution}\\ positive&\textit{two solutions}\\ negative&\textit{no solution} \end{cases} \\\\\\ (-12)^2-4(1)(20)\implies 144-80\implies 64[/tex]
Every page of the newspaper is a rectangle measuring 43 cm by 28 cm, both correct to the nearest
centimetre.
Calculate the upper bound of the area of a page.
Answer: 1204 cm²
Step-by-step explanation:
The upper bound of the area of a page would be the area of a rectangle with the upper bound of the length and width measurements.
Given that each page of the newspaper is a rectangle measuring 43 cm by 28 cm, both correct to the nearest centimetre, the upper bound of the length would be 43.5 cm (since we round up to the nearest centimetre) and the upper bound of the width would be 28.5 cm.
Therefore, the upper bound of the area of a page would be:
43.5 cm x 28.5 cm = 1240.25 cm²
Rounding this value to the nearest whole number, we get:
1204 cm²
So, the upper bound of the area of a page is 1204 cm².
Answer:
1204 cm 2
Step-by-step explanation:
Adjacent angles .help me solve
BGC or AGF
Adjacent means next to or adjoining something else and both of the mentioned angles are next to angle AGB.
1. Undemeath each number write I for Irrational or R for Rational
3.2 x 10³
4/3
1.423...
TT/6
The numbers are characterized as rational and irrational numbers as below
1. 3.2 x 10³ - R (Rational)
2. 4/3 - R (Rational)
3. 1.423... - I (Irrational)
4. TT/6 - I (Irrational)
What are rational and irrational numbers?Rational numbers harmonized with a fraction of two integers where the denominator must not be zero. In simpler terms, it can be conveyed as p/q whereby both "p" and "q" are integers while "q" is unequal to zero.
As opposed to rational numbers, irrational ones cannot be expressed in a fraction or quotient of two integers. It's not possible to write them down in terms of the figure p/q whereby "p" and "q" are integers, differing from one another, and "q" is also inequivalent to zero.
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The country of Scotstats requires the people in their country to have license tags on their
car such that the first 3 characters are English letters but no letter may repeat. The last 3
characters must each be a number 0 - 9 and again no numbers can be repeated. How
many license tags are possible?
The licenses have space for 6 characters.
We need to note that there are 26 alphabets and 10 numbers to pick from.
So, for the first character, any of the 26 alphabets can take this spot.
For the second character, 25 alphabets are now available for that space. (Since repetition is not allowed)
For the third character, 24 alphabets are available for that.
For the fourth character, any of the 10 numbers can take up that spot.
For the fifth character, only 9 numbers can take this spot now. (No repetition rule too)
For the sixth character, 8 numbers can take that spot.
So, mathematically, the number of license tags possible will be
[tex]\bold{26 \times 25 \times 24 \times 10 \times 9 \times 8 = \underline{11,232,000} \ possible \ license \ tags}[/tex]
Answer:
11,232,000
Step-by-step explanation:
26 choices for the first letter.
25 choices left for the second letter (can't repeat letters)
24 choices left for the third letter
26 x 25 x 24 = 15,600 = possible first 3 characters.
10 choices for the first number
9 choices left for the second number (can't repeat numbers)
8 choices left for the third number
10 x 9 x 8 = 720 = possible last 3 characters.
chatgpt
What is the value of x in this triangle?
Answer:
83 degrees
Step-by-step explanation:
See attachment below:
What does this problem mean |x+3| if x>5
Answer:
x+3 . . . . if x > 5
Step-by-step explanation:
You want the meaning of |x+3| if x > 5.
Absolute valueThe absolute value function is a piecewise defined function:
[tex]y=|x|=\begin{cases}-x&\text{for }x < 0\\x&\text{for }x\ge0\end{cases}[/tex]
ApplicationThe expression (x+3) is zero when x=-3. So, your absolute value function will have the piecewise definition ...
[tex]|x+3|=\begin{cases}-(x+3)&\text{for }x < -3\\x+3&\text{for }x\ge-3\end{cases}[/tex]
You are interested in the value for x > 5, which is definitely greater than -3. That means only the second part of this definition is applicable.
x+3 . . . . if x > 5
__
Additional comment
The attache graph shows the given absolute value expression (dashed red line). The blue line is (x+3) for x > 5. You can see that it matches the given expression in that region.
The table below shows the costs for
different numbers of folders.
Folder Costs:
•Number of Folders
1 = $0.17
3 = $0.51
5 = $0.85
7 = $1.19
Based on this information, what is the constant of proportionality?
A0.15
B 0.17
C 0.51
D 0.85
Question 8 (Essay Worth 10 points)
(07.04 MC)
Given the functions f(x)=x²+x²-3x+4 and g(x)=2"-4, what type of functions are f(x) and g(x)? Justify your answer. What key feature(s) do f(x) and g(x) have in common?
(Consider domain, range, x-intercepts, and y-intercepts.)
The prominent similar feature between f(x) and g(x) is the present y-intercept.
How to explain the informationSince the discriminant is negative, there are no possible solutions to the given equation, which implies that f(x) has no x-intercepts.
It should be noted that to identify the y-intercept, we set x = 0 and evaluate f(x):
f(0) = 2(0)² - 3(0) + 4
f(0) = 4,
The y-intercept of f(x) can be determined as (0, 4).
Moreover, the function g(x) = 2x-4 is a power function having a negative exponent, which makes it a reciprocal function. It's domain comprises all real numbers but 0, while its range includes all real numbers excluding 0.
In order to obtain the x-intercept of g(x), we set g(x) equal to zero and solve for x:
0 = 2x-4
2x = 4
x = 2
Consequently, the x-intercept of g(x) is (2, 0), together with the y-intercept (0, -4).
The prominent similar feature between f(x) and g(x) is the present y-intercept. On the other hand, f(x) does not contain any x-intercepts, whereas g(x) has a single x-intercept. Also, f(x) is categorized as a second-degree polynomial function, whereas g(x) is classified as a reciprocal function.
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In a triathlon, Farhan swims a distance of 1.5 km at an average speed of 2.5 km/h,
hours.
cycles 40 km in 12 hours and runs at an average speed of 9 km/h for 1
Find his
average speed for the entire competition.
Answer:
We can use the formula:
average speed = total distance / total time
To find the total distance, we add the distances for each event:
total distance = 1.5 km (swimming) + 40 km (cycling) + 1 km (running) = 42.5 km
To find the total time, we add the times for each event:
time for swimming = distance / speed = 1.5 km / 2.5 km/h = 0.6 hours
time for cycling = 40 km / 12 km/h = 3.33 hours
time for running = 1 km / 9 km/h = 0.11 hours
total time = 0.6 hours + 3.33 hours + 0.11 hours = 4.04 hours
Now we can calculate the average speed:
average speed = total distance / total time = 42.5 km / 4.04 hours ≈ 10.52 km/h
Therefore, Farhan's average speed for the entire competition is approximately 10.52 km/h.
The menu board at a local pizza restaurant shows all the ordering options. Customers can choose up to one item from each category for a one-topping pizza.
Size Sauce Cheese Toppings
Small
Medium
Large
Garlic
Marinara
Mozzarella
Provolone
Parmesan
Pepperoni
Sausage
Ham
Pineapple
Onions
Peppers
What is the probability that a randomly selected one-topping pizza will not have onions on it? Express your answer as a simplified fraction.
Answer: 4/5
Step-by-step explanation: In decimal form: 0.8
A random number generator picks a number from 7 to 69 in a uniform manner. Round answers to 4 decimal places when possible.
Find the maximum for the lower quartile
(100PTS)
Answer:
The lower quartile corresponds to the 25th percentile of the data set. To find the maximum value for the lower quartile, we need to find the largest number that is at or below the 25th percentile.
The range of the numbers that the random number generator can pick is 69 - 7 + 1 = 63. The 25th percentile corresponds to the value that is 25% of the way through this range, which is:
0.25 * 63 = 15.75
Since we can't pick a fractional number, we need to round down to 15. Therefore, the maximum value for the lower quartile is 15.
400 adults,96 of the 400 have high blood pressure 116 report being willing to change. an adult choosen at random what is probability they will be willing to change diet
solution is what
The probability that an adult chosen at random will be willing to change their diet is 0.29 or 29%.
what is probability?
In general terms, probability is the measure of the likelihood or chance of an event occurring. It is a mathematical concept that is used to quantify uncertainty or randomness in various situations.
In formal terms, probability is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes in an event. It is usually represented as a value between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.
The probability of an adult being willing to change their diet can be calculated using the given information as follows:
Probability of an adult being willing to change diet = Number of adults willing to change / Total number of adults
Total number of adults = 400 (as given in the question)
Number of adults willing to change = 116 (as given in the question)
Therefore, the probability of an adult being willing to change their diet can be calculated as:
Probability = 116/400
Probability = 0.29 or 29%
So, the probability that an adult chosen at random will be willing to change their diet is 0.29 or 29%.
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Adult men have heights with a mean of 69.0 inches and a standard deviation of 2.8 inches. Find the z-score of a man who is 66.7 inches tall. (to 4 decimal places)
Answer:
!
Step-by-step explanation:
A circle is centered on point
B points a c and d lie on its
circumference. if
The measure of the angle ADC is: 20°
How to find the angle at the circumference?The parameters are given as:
∠ABC = 40°
∠ADC = ?
We know from circle geometry that:
Angle subtended at the center is twice the angle subtended at the circumference.
Therefore:
∠ABC = 2 x ∠ADC
On substituting the value we get:
40° = 2 * ∠ADC
∠ADC = 20°
Therefore, the measure of the angle ADC is 20°
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Complete question is:
A circle is centered on point B. Points A, C and D lie on it's circumference.
If ABC measures 40 degrees, what does ADC measure?
Select any of these that are LINEAR. (Select MORE THAN ONE answer !)
Responses
A
A
B
B
C
C
D
D
The linear equations from the options given are expressed as: y = 5 and y = -2/5x + 4.
How to Determine an Equation that is Linear or Non-linear?An equation is linear can be expressed in the following forms:
y = mx + b
x = b
y = b
In the above, y is the dependent variable and x is the independent variable, m = slope, and b = y-intercept.
Note that in a linear equation the exponent of the variable is always 1. On the other hand, a nonlinear equation has an exponent that is not equal to 1 and can be expressed as:
y = x²
Therefore, the equations that are linear are:
y = 5 and y = -2/5x + 4.
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help with calculus. determine wheter each labeled point is an absolute maximum or minumum OR a relative maximum or minimum or neither
The curve is defined as follows
A. absolute minima
B. absolute maxima
C. neither
D. relative minima
E. relative maxima
F. neither
G. neither
What is absolute minima and maxima?
When examining a function, its absolute minimum refers to the lowest possible value it can reach throughout its entire domain.
In a similar manner, absolute maximum pertains to the highest value that said function is capable of attaining within the same specified range, without consideration of separate intervals or regions.
To reiterate, these values denote the smallest and largest outcomes the function can yield over its complete set of input values, regardless of where those values may be located.
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Gracie walks around the edge of a circular lake exactly 15 times. If she walks 2178 m in total, what is the diameter of the lake? If your answer is a decimal, give it to 1 d.p.
The diameter of the lake is 6.2 m.
What is the diameter?The diameter is the distance from one edge of the circle to the next across the center.
The diameter can be determined given the circumference and pi, since circumference is pi x diameter.
The number of times Gracie walks around the circular lake = 15 times
The total distance covered by the walk = 2,178 meters
The circumference (the distance around the edge of the circular lake) is 145.2 m (2,178 ÷ 15).
Circumference = C = 2πr = πd
Diameter = C/π
145.2 ÷ 3.143
= 46.2 m
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which of the following rational functions is graphed below? A. F(x) = 1/(x-1)² B. F(x) = 1/(2+1)² C. F(x) = 1/x2 D. F(x) = 1/x2+1
Answer:
c c c c c c c c c c c c c c c c c c c c c c c c c c c c c
Determine the number of triangles ABC possible with given parts
The number of triangles formed by triangle ABC is 1
How to determine the number of angles
In order to calculate the possible number of triangles ABC that can be made with the specified parts, it is essential to employ conditions for triangle congruence, which includes the Triangle Inequality Theorem.
If sides a, b, and angle A can create a valid triangle, then they must follow these requirements:
a + b > c
a + c > b
b + c > a
By replacing given values with assigned variables, we get:
38 + 77 > c,
38 + c > 77,
77 + c > 38.
After solving these inequalities, we determine that c should be
39 < c < 115.However, if the Triangle Inequality Theorem is not observed, an unfeasible configuration results. Therefore, it would be impossible to generate a triangle using the following: AB = 38, AC = 77, and angle BAC = 74 degrees.
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Use synthetic division to find the quotient and remainder when is divided by by completing the parts below.
After using synthetic division, the answer are given as follows;
first row = -4, 4, 12
Second Row = -2 2 -6 -5
Answer = 6 + [-5/x-2]
What is the full calculation?-2x^3+6x^2-10x+7 / x-2
Coefficient of the numerator polynomial are:
- 2, 6, -10 , 7
the zeros of the denominator are:
x -2 = 0
x = 2
X³ X³ X¹ X⁰
-2 6 -10 7
2
X³ X³ X¹ X⁰
-2 6 -10 7
2 ↓ -4 4 -12
-2 2 -6 -5
So the quotient is -2x² +2x - 6 remainder -5
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The owner of a new pizzeria in town wants to study the relationship between weekly revenue and advertisi dollars. The summary output for the regression model is given below.
R2, since the model only has one independent variable.
How to solveFind R2:
R^{2} = [tex]\frac{SSR}{SST}[/tex]
R^{2} = [tex]\frac{18.26347892}{26.24578040}[/tex]
R^{2} = 0.6958634
[tex]R^{2} = 0.6959}}[/tex]
Part 2 of 3:
Find Ra2:
[tex]R^{2}_{a}= 1-\frac{(1-R^{2}) \\times (n-1)}{n-k-1}[/tex]
where
n = 9
since dftotal = n -1 = 8
thus n = 8+1=9
and k = number of independent variables = 1
Thus
[tex]R^{2}_{a}= 1-\frac{(1-0.6958634 ) \\times (9-1)}{9-1-1}[/tex]
[tex]R^{2}_{a}= 1-\frac{0.3041366 \\times 8}{7}[/tex]
[tex]R^{2}_{a}= 1-0.3475846[/tex]
[tex]R^{2}_{a}= 0.6524154[/tex]
[tex]R^{2}_{a}= 0.6524}}[/tex]
Part 3 of 3:
R2, since the model only has one independent variable.
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PLEASE HELP QUICKLY(25 points)!!
Match the description with the correct answer.
(questions)
———————
y-intercept -
slope-
domain-
range-
is this graph increasing, decreasing or both-
x-intercept-
———————
(answer choices)
(4,0), (0,4), (-2,0), (0,-2), +2, -4,
input values, increasing, decreasing,
both increasing and decreasing, output values
———————
please help quickly its worth 10.34 points on my test
Answer: In that picture the slope is increasing
Step-by-step explanation: positive rise and positive run (uphill from left to right)
What is the median
9,7,7,7,8,8,6
39. Firefighters must reach a fire anywhere
in the city within 7 minutes. They are
currently reaching fires in 3.5 minutes
less than the maximum time allotted.
Which equation shows how much time
it takes them to reach a fire?
A. t + 3.5=7
B. t - 3.57
C. t - 7 = 3.5
D. t + 7 = 3.5
PLEASE HELP QUICKLY(20 POINTS!!)
what is the slope of this function
A. 1/2
B. -2
C. 2
D. -1/2
E. -3/2
F. -3
Answer:
A
Step-by-step explanation:
calculate the slope m using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = (- 6, - 6) and (x₂, y₂ ) = (6, 0) ← 2 points on the line
m = [tex]\frac{0-(-6)}{6-(-6)}[/tex]= [tex]\frac{0+6}{6+6}[/tex] = [tex]\frac{6}{12}[/tex] = [tex]\frac{1}{2}[/tex]
Evaluate the function for f(2)
f(x) = 8x + 7
Answer:To evaluate the function for f(2), we simply substitute x = 2 into the function
:f(2) = 8(2) + 7
Simplifying the expression:
f(2) = 16 + 7f(2) = 23
Therefore, f(2) = 23.
Step-by-step explanation:
14. As shown in the figure, the area of square ABCD is known to be 1320 square centimeters, E is the midpoint of AB, F is a quarter point of BC close to point B, and the intersection points of CE, DB and DF are P and Q , find the area of the quadrilateral BPQF.
The coordinates of all four vertices of quadrilateral BPQF, we can use the shoelace formula to find its area. The shoelace formula states that the area of a polygon with vertices (x1, y1), (x2,
What is the triangle?A triangle is a closed, two-dimensional geometric shape that has three sides and three angles. It is one of the basic shapes in geometry and can be formed by connecting three non-collinear points. Triangles can be classified based on their side lengths and angles.
To solve this problem, we can use the fact that the area of a triangle is equal to half the product of its base and height. Let's first find the length of the sides of the square.
Since the area of the square is 1320 square centimeters, we can find its side length by taking the square root of 1320:
√(1320) ≈ 36.32
So the side length of the square is approximately 36.32 cm.
Next, let's find the coordinates of points E and F. Since E is the midpoint of AB, its coordinates are:
E = ((0+36)/2, (36+0)/2) = (18, 18)
Similarly, since F is a quarter point of BC close to point B, its coordinates are:
F = (36, (3/4)36) = (36, 27)
Now let's find the equation of line CE. Since we know two points on the line (C and E), we can use the point-slope form of a linear equation to find its equation:
y - y1 = m(x - x1)
where m is the slope of the line and (x1, y1) is one of the points on the line. We can find the slope of the line by taking the difference in y-coordinates and dividing by the difference in x-coordinates:
m = (0 - 36)/(36 - 18) = -2
Using the point-slope form with point C, we get:
y - 0 = -2(x - 36)
y = -2x + 72
Similarly, the equation of line DB can be found using points D and B:
m = (0 - 36)/(0 - 18) = 2
y - 0 = 2(x - 0)
y = 2x
Finally, we can find the coordinates of point P by solving the system of equations:
y = -2x + 72
y = 2x
Setting the two expressions for y equal to each other, we get:
-2x + 72 = 2x
4x = 72
x = 18
Substituting x = 18 into either equation, we get:
y = 2x = 36
So the coordinates of point P are (18, 36). To find the coordinates of point Q, we can use the fact that it lies on line DF. Since we know two points on the line (D and F), we can again use the point-slope form of a linear equation:
m = (27 - 0)/(36 - 0) = 3/4
y - 0 = (3/4)(x - 0)
y = (3/4)x
To find the intersection point of lines DF and CE, we need to solve the system of equations:
y = (3/4)x
y = -2x + 72
Setting the two expressions for y equal to each other, we get:
(3/4)x = -2x + 72
(11/4)x = 72
x = 64/11
Substituting x = 64/11 into either equation, we get:
y = (3/4)(64/11) ≈ 14.73
So the coordinates of point Q are (64/11, 14.73).
Hence, Now that we have the coordinates of all four vertices of quadrilateral BPQF, we can use the shoelace formula to find its area. The shoelace formula states that the area of a polygon with vertices (x1, y1), (x2,
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Supplementary angles. I will be giving out points
Answer: Angle aeb
Step-by-step explanation: To be supplementary, the angle(s) must add up to 180. Angle AEB and angle DEA add to 180. Another way to find this is to find the angle that the angle is adjacent to in a straight line.