The difference between the roots of the quadratic equation
x^2 + x + c = 0 is 6.
Find c.
Also, to make it simpler. there is this thing called Vieta's theorem. It makes life easier instead of plugging everything into the quadratic equation- I've already tried multiple times, although my answer never seems correct.
Answers
Answer:
I believe the answer is c = -35/4
Step-by-step explanation:
Using the standard form of a quadratic equation, The value of c is \(\frac{-35}{4}\)
What is a Quadratic Equation?
An algebraic equation of the second degree in x is a quadratic equation. The quadratic equation is written as \(ax^{2} +bx+c\), where x is the variable, a and b are the coefficients, and c is the constant term. First, there must be a term other than zero in the coefficient of \(x^{2}\) (a 0) for an equation to be a quadratic equation. The\(x^{2}\) term is written first when constructing a quadratic equation in standard form, then the x term, and finally the constant term. The numerical values of a, b, and c are often expressed as integral values rather than fractions or decimals.
To solve this question:
First, we compare it with the standard quadratic equation to find
a, b and c.
Then, we apply Bhaskara to find the roots, as a function of c.
Then, since the difference is 6, we use this to find c.
Solving a quadratic equation:
Given a second-order polynomial expressed by the following equation:
\(a x^2+b x+c, a \neq 0 .\)
This polynomial has roots \(x_{1}\),\(x_{2}\) such that \(a x^2+b x+c=a\left(x-x_1\right) *\left(x-x_2\right)\),
given by the following formulas:
\(\begin{aligned}&x_1=\frac{-b+\sqrt{\Delta}}{2 * a} \\&x_2=\frac{-b-\sqrt{\Delta}}{2 * a} \\&\Delta=b^2-4 a c \\&\mathbf{x}^2+\mathbf{x}+c=0\end{aligned}\)
Comparing to the standard second-order equation, which is:
\(a x^2+b x+c=0\)
We get that:
\(a=1, b=1\)
Finding the roots:
\(\begin{aligned}&\Delta=b^2-4 a c=1^2-4(1)(c)=1-4 c \\&x_1=\frac{-1+\sqrt{1-4 e}}{2} \\&x_2=\frac{-1-\sqrt{1-4 c}}{2}\end{aligned}\)
The Difference between the roots is 6.
Thus:
\(\begin{aligned}&x_1-x_2=6 \\&\frac{-1+\sqrt{1-4 c}}{2}-\frac{-1-\sqrt{1-4 c}}{2}=6 \\&-\frac{1}{2}+\frac{\sqrt{1-4 c}}{2}+\frac{1}{2}+\frac{\sqrt{1-4 c}}{2}=6 \\&\frac{2 \sqrt{1-4 c}}{2}=6 \\&\sqrt{1-4 c}=6 \\&(\sqrt{1-4 c})^2=6^2 \\&1-4 c=36 \\&-4 c=35 \\&4 c=-35\end{aligned}\)
\(c=-\frac{35}{4}\)
Hence, The value of c is \(\frac{-35}{4}\)
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Related Questions
the ratio of the number of oranges to apples to bananas is 8:5:2. if there are 120 fruits in total, how many bananas are there
Answers
Answer:
Step-by-step explanation:
To find the number of bananas, we need to determine the proportion of bananas in the total number of fruits based on the given ratio.
Let's assume that the ratio of oranges, apples, and bananas can be represented by 8x, 5x, and 2x, respectively, where x is a common multiplier.
The sum of the parts in the ratio is 8x + 5x + 2x = 15x.
We know that the total number of fruits is 120. Therefore, we can set up the following equation:
15x = 120
To solve for x, we divide both sides of the equation by 15:
x = 120 / 15
x = 8
Now that we know the value of x, we can find the number of bananas by multiplying it by the corresponding part of the ratio:
Number of bananas = 2x = 2 * 8 = 16
So, there are 16 bananas in total.
We can add all the parts to the ratio and get 15. Out of 15 parts, 2 are bananas. Now we can calculate how many fruits are in one part. One fifteenth of 120 is 8, so there are 8 fruits in one part. 2 parts are bananas, so multiplying 8 with 2 gives us our answer, 16.
I need help plz will give brainliest
Answers
Answer:
$10,000
Step-by-step explanation:
Cross multiply and divide by 55.
Hope this helps!
$10000
Step-by-step explanation:
5500 = 55% × amount paid for the car(p)
5500 = 55÷100 × p
5500×100 = 55p
550000 = 55p
550000÷55 = 55p÷55
10000 = p
The international space station has small cupboards that are used for storage. A cupboard is drawn below (redrew by me cause I'm using chrome book and computer)
1. What is a formula that can be used to find the area of one triangular section of the door?
2. How do you know the formula will work?
Answers
Answer:
\(A=\frac{ab}{2}\) or \(A = a *\sqrt{c^2-a^2} /2\)
Step-by-step explanation:
Depending on the values you take depends on the equation you would use to solve for the area of a right triangle(the first answer accounts for measuring both arms(the sides connected to the 90 degree angle(a and b)), where as the second answer accounts for using the hypotenuse and one of the arms). That means that a 90 degree angle is involved, or as illustrated here, the square in the bottom right corner. If this were a non-right angle triangle, a different formula utilizing the height of the triangle would be used instead.
\(A = \frac{h_{b}+b}{2}\)
Since the right angle exists, a height is already given using one of the arms of the triangle in its place. We can also confirm it is a triangle by adding up all degrees of the triangle, in which it would equal 180 degrees confirming it is indeed a triangle as well.
By using trigonometric relations we will see that the area is:
How to find the area of one triangular section?
Remember that the area of a right triangle is equal to the product between the cathetus divided by 2.
Sadly, here we don't have the measures of any sides of the triangle, so getting the actual area is impossible, but if we assume that some dimension is given (like the height, let's say that is h).
h is the cathetus adjacent to the 60° angle, then the other cathetus (the bottom one) is given by:
tan(60°) = x/h
x = tan(60°)*h
Then our two catheti are h and tan(60°)*h.
This means that the area will be:
Here we only used trigonometric properties, so this area will work for any height given (always that the value of h is the cathetus adjacent to the 60° angle).
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Elena rode her bike 20 1/2 miles in 2 1/4 hours. What was her speed in miles per hour?
Answers
Answer: 9 1/9
Step-by-step explanation:
Mya’s bedroom is 15 feet long, 12 feet wide. It has no windows. Mya wants to paint all four walls. What surface area will she paint?
A) 180 square feet
B) 720 square feet
C) 162 square feet
Answers
What is the quotient?
7035/15
Answers
Answer:
the answer is 469
Step-by-step explanation:
How many times smaller is 2.7 × 103 than 5.481 × 105?
49
203
0.49
2.03
Answers
Answer:
0.49
Step-by-step explanation:
The number 2.7 x 103 is 0.49 times smaller than the number 5.481 x 105
What is an Equation?
Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.
It demonstrates the equality of the relationship between the expressions printed on the left and right sides.
Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.
Given data ,
Let the number of times be = y
Let the first number be A = 2.7 x 103
The value of A = 278.1
Let the second number be B = 5.481 x 105
The value of B = 575.505
Now , to calculate the number of times A is less than B is
B times y = A
So ,
575.505 x y = 278.1
y = 278.1 / 575.505
y = 0.483
y = 0.49
Therefore , the value of y is 0.49
Hence , The number 2.7 x 103 is 0.49 times smaller than the number 5.481 x 105
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Help pls. Also wdym this is to short yk why so long >:/
Answers
Answer:
Y= mx + b
so Y = 1/3x + 2
just substitute
Now, we shall use points (3,2) as the substitutes for the solution for b, which 3 would be the input and 2 would be y.
Therefore, 2 = 1/3(3) + b
2 = 1 + b, therefore, one would be subtracted from 2 which would give us the answer of b equaling to one. Thus, y = 1/3x + 1
Nate is helping to redesign his basement. He is in charge of choosing and buying the paint. He wants a specific shade of green, so he is going to combine blue and yellow paint in the ratio of 1 : 3. If he needs 8 gallons of paint, how much yellow paint does he need to buy?
Answers
3+1=4
& then divide
8/4=2
Yellow paint
3x2=6
Blue paint
1x2=2
Valentina and her parents are on a long drive to Monterey Bay to go kayaking. She passes the time by counting how many different colors of cars she sees. The table below shows the results.
Color of car Number of cars
black 4
red 7
silver 12
white 14
brown 5
blue 3
Based on the data, what is the probability that the next car seen is brown?
Write your answer as a fraction or whole number.
Answers
Based on the data provided, Valentina saw a total of 4 + 7 + 12 + 14 + 5 + 3 = 45 cars. Out of these cars, 5 were brown. Therefore, the probability that the next car seen is brown is 5/45 which can be simplified to 1/9.
Answer:
Based on the data provided, Valentina saw a total of 4 + 7 + 12 + 14 + 5 + 3 = 45 cars. Out of these cars, 5 were brown. Therefore, the probability that the next car seen is brown is 5/45 which can be simplified to 1/9.
Step-by-step explanation:
State whether you would use similar triangles, The primary trigonometric ratios (sine, cosine, tangent) , sine law or cosine law to find the missing value.
Answers
Answer:
I would use similar triangles
Step-by-step explanation:
There is no angle (apart from the right angle) given in the problem. so in order to find the side "x", in one step calculation, I would use similar triangles and the equation that comes from the proportion among the similar sides:
\(\frac{x}{20} =\frac{5}{7}\)
Which of the following has a product of -4?
A. (1/5)(-20)
B. (1/4)(-4)
C. (-1/3)(-12)
D. (-1/2)(-7/2)
Answers
Hey! Your answer would be A. (1/5)(-20)
Winston, an architect, is adding a stalrway to a parking garage. The stairway must be 3 meters high, and he has enough room on the ground for the staircase to reach the ground 9 meters away, which will satisfy local building codes. To the nearest tenth, how long will the stair stringer be?
A. 9.0 meters
B. 9.5 meters
C. 81 meters
D. 90 meters
Answers
hellppo due June 30th
Answers
Answer:
Step-by-step explanation:
Non-shaded region area:
A= 3.14 x 7^2=153.86 cm²
The whole circle area:
A=3.14 x 11^2= 379.94 cm²
Area for ONLY the shaded region:
A= 379.94 - 153.86= 226.08 cm²
a = pi * r^2
a = 3.14 * 7^2
a = 3.14 * 49
a = 153.86
a = 3.14 * 11^2
a = 3.14 * 121
a = 379.94
subtract the smaller area from the larger area:
379.94 - 153.86 = 226.08 sq cm
edit this with the right answers
Answers
19. The y intercept and x intercept is 2 and -3 respectively
20. The y intercept and x intercept is 2 and 8 Respectively
What is linear equation?A linear equation is an algebraic equation of the form y=mx+b. involving only a constant and a first-order (linear) term, where m is the slope and b is the y-intercept. Occasionally, the above is called a "linear equation of two variables," where y and x are the variables.
The equation ,-2x +3y = 6
the y intercept is obtained when x = 0
-2(0) + 3y = 6
3y = 6
y = 6/3
y = 2.
The x intercept is obtained when y = 0
-2x = 6
x = 6/-2
x = -3
therefore the y and x intercept for the graph are 2 and -3 respectively.
For equation x+4y = 8
when x= 0
4y = 8
y = 8/4
y = 2
when y = 0
x = 8
therefore the y and x intercept are 2 and 8
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plz help me with the attachment below
Answers
Answer:
57
Step-by-step explanation:
( 27 + 19 + 41 + 51 + 29 + 35 + k ) / 7 = 37
202 + k = 37 x 7
202 + k = 259
k = 259 - 202
k = 57
Answer:
K. = 57
Step-by-step explanation:
what is the range of the function y=4x-1 for the domain 4 ≤x ≤8?
A. 15 ≤y ≤31
B. 5/4 ≤y ≤9/4
C. 16 ≤y32
D. 1 ≤y ≤2
Answers
Answer:
b
Step-by-step explanation:
List the steps in solving the equation 16 = 2 (4d - 8) - 4
There are 4 steps.
Hurry I need to turn this in soon!
Answers
Answer:
Step-by-step explanation:
16=2(4d-8-2)
4d-10=16/2
4d=8+10
d=18/4
d=9/2
Answer:
\(\huge\boxed{4.5,\ \text{See explanation for steps.}}\)
Step-by-step explanation:
If we have the equation \(16=2(4d-8)-4\), our goal is to isolate the variable d on one side of the equation.
We can do this by performing a series of steps to reverse the equation.
1. Apply the distributive property on the right side:
\(16=(4d \cdot 2) + (2 \cdot -8) - 4\\\\16 = 8d - 16 - 4\)
2. Combine like terms on the right side:
\(16 = 8d + (-16-4)\\\\16 = 8d-20\)
3. Add 20 to both sides:
\(16 + 20 = 8d -20 + 20\\\\36 = 8d\)
Divide both sides by 8:
\(36 \div 8 = 8d \div 8 \\\\ 4.5 = d\)
So \(d=4.5\).
Hope this helped!
According to his GPS, it will take Lanny 1 hour to travel 40 miles. Which expression shows how to determine the speed in miles per hour?
40 miles ÷ 1 hour
1 hour ÷ 40 miles
1 mile ÷ 40 hours
40 hours ÷ 1 mile
Answers
Answer:
40 ÷ 1
Step-by-step explanation:
Speed = Distance : Time
Speed = 40 ÷ 1
Miles goes above hours in the equation
Bobby can run 2/8 mile in 2 1/4 minutes. If he maintains that pace, how long will it take him to run 1 mile?
Answers
Answer:
Step-by-step explanation:
Using Inverse Variation
2/8 mile relates to 2 1/4 minutes
2/8 * 4 = 1mile=
2 1/4 /4=9/4*1/4=9/16 minutes
Angela made 85% of the 40 free throws she attempted. How many free throws did Angela make?
Answers
Angela made 34 free throws
10% of 40=4 x 3 5% of 40= 2. 20+12+2=34
Robin wants to buy a membership at a carwash this summer so that she can keep her car clean. She compared deals from Speedy Clean Carwash and Sudsy Bubbles Carwash to decide which membership to get. Robin arranged the information in the tables below.
How many times will Robin need to wash her car to spend the same amount at both carwashes?
P.S.: I attached the table below:)
Answers
What is the missing side length?
(Use Pythagorean Theorem)
Answers
Answer:
20 units
Step-by-step explanation:
The Pythagorean Theorem states that:
\(a^{2} + b^{2} = c^{2} \\\)
Now we can insert values into the equation.
\(a = 15\\b = b\\c = 25\\\)
Now we can plug these ☝ values into the Pythagorean Theorem:
\(15^{2} + b^{2} = 25^{2}\\\)
After we solve for \(b\) we get:
\(b^{2} = 400\\b = 20\)
Answer:
b = 20
Step-by-step explanation:
15 + b = 25
(you have to square each one)
15 x 15 = 225
b x b = b
25 x 25 = 625
625 - 225 = 400
\(\sqrt{400}\) = 20
hope this helps
When a number is increased by 5.5%, the result is 60. What is the original number to the nearest tenth?
Answers
A coach calculated the mean and the mode of the number of points scored by 6 players.
• mean: 7 points
mode: 8 points
How many points did player 5 score?
Answers
Because 8+8+8+6+7+5=42
Then divide that by 6 to get the mean - so 42/6=7 so that’s the correct mean
Explanation:
Mean = average
Equation: total points scored/total players
Mean=7+8+6+8+?+8/6
The mean is given: 7
So plug it in:
7=(7+8+6+8+?+8)/6
Factor to solve:
Add all the numbers together to leave just the missing number (?)
7=(37+?)/6
To get rid of the 6 multiply it over to the other side
6x7=37+?
42=37+?
Now subtract the 37 to the other side to find the missing number (?)
42-37=?
5=?
Now let’s check:
7+8+6+8+(5)+8=42
Add:
42
Now divide by 6
42/6=
7
Is 7 the mean they gave us? Yup!
So player 5 scored 5 points
Someone help me with this ASAP!
Answers
Answer:
48.1 somewhere around 36 and 57
Step-by-step explanation:
Answer:
36-57
Step-by-step explanation:
18.50 x 2.6 = 48.10, which is between those two numbers
Round 312.92 to the nearest whole number. *
Answers
Please Help ASAP! Rewrite x^4y^2 − 3x^3y^3 using a common factor.
A. 3xy(x^3y − x^2y)
B. 3xy^2(x^2 − x^2y)
C. x^2y(xy − 3xy^2)
D. x^2y^2(x^2 − 3xy)
I think it's B. But I might be wrong.
Answers
Answer:
Step-by-step explanation:
you are completely right please follow your heart <3
You have to look at both x and y and it’s power. You have to look for the lowest power for x and y. For y as you can see that the lowest power is 2. As for x it can be either power of 3 or even 2. When you removed x^2 and y^2 from the equation. The inside parenthesis will be x^2-3xy. Just remember when you divide both x^2 and y^2, you subtract the power.
When you get the answer, please check the answer if it’s the same as the original. Meaning distribution both x^2 and y^2 to inside parenthesis. And for the power, you add them. This way you can double check and see if it’s the same.
50 POINTS AND BRAINLIEST TO CORRECT ANSWER
Determine the x-intercepts and y-intercepts for the equation. Draw the graph
3x - 7y = 10
y = ( x + 3 ) ^ 2 - 8
ty :)
Answers
Answer:
black x-intercept: (3.333,0)
black y-intercept: (0, -1.429)
blue x-intercepts: (-6.162,0) and (0.162,0)
blue y-intercept: (0,-1)
Step-by-step explanation:
blue line is y=( x + 3 ) ^ 2 - 8
black line is 3x - 7y = 10
Hope this helps!
Given equation is y=3x
When x=1,⇒y=3×1=3
When x=2,⇒y=3×2=6
When x=3,⇒y=3×3=9
x 1 2 3
y 3 6 9
Plot the points A(1,3),B(2,6),C(3,9) on the graph paper.Join these points by a straight line AC
The stem-and-leaf plot displays data collected on the size of 15 classes at two different schools.
Mountain View School Seaside School
0 5, 8
9, 8, 2, 0 1 0, 1, 2, 5, 6, 8
8, 7, 6, 5, 5, 4, 4, 3, 1, 0 2 5, 5, 7, 7, 8
0 3 0, 6
Key: 2 | 1 | 0 means 12 for Mountain View and 10 for Seaside
Part A: Calculate the measures of center. Show all work. (2 points)
Part B: Calculate the measures of variability. Show all work. (1 point)
Part C: If you are interested in a smaller class size, which school is a better choice for you? Explain your reasoning. (1 point)
Answers
For Mountain View School:
Data: 0, 0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5, 5, 6, 7, 8, 8, 8, 9
Mean: (0 + 0 + 0 + 0 + 1 + 2 + 3 + 4 + 4 + 5 + 5 + 5 + 5 + 6 + 7 + 8 + 8 + 8 + 9) / 19
= 100 / 19
≈ 5.26
To find the median, we need to arrange the data in ascending order: 0, 0, 0, 0, 1, 2, 3, 4, 4, 5, 5, 5, 5, 6, 7, 8, 8, 8, 9.
The median is the middle value, which is the 10th value in this case.
Median: 5
For Seaside School:
Data: 0, 0, 0, 1, 1, 2, 2, 4, 5, 5, 5, 5, 6, 7, 7, 8, 8, 8, 9
Mean: (0 + 0 + 0 + 1 + 1 + 2 + 2 + 4 + 5 + 5 + 5 + 5 + 6 + 7 + 7 + 8 + 8 + 8 + 9) / 19
= 80 / 19
≈ 4.21
To find the median, we need to arrange the data in ascending order: 0, 0, 0, 1, 1, 2, 2, 4, 5, 5, 5, 5, 6, 7, 7, 8, 8, 8, 9.
The median is the middle value, which is the 10th value in this case.
Median: 5
Now let's calculate the measures of variability, specifically the range and interquartile range (IQR).
Range: The range is the difference between the largest and smallest values in the dataset.
For Mountain View School:
Range: 9 - 0 = 9
For Seaside School:
Range: 9 - 0 = 9
Interquartile Range (IQR): The IQR is the difference between the third quartile (Q3) and the first quartile (Q1) and measures the spread of the middle 50% of the data.
For Mountain View School:
Q1 = 3
Q3 = 7
IQR = Q3 - Q1 = 7 - 3 = 4
For Seaside School:
Q1 = 2
Q3 = 7
IQR = Q3 - Q1 = 7 - 2 = 5
To determine which school is a better choice for a smaller class size, we can consider both the mean and the median.
Both the mean and median for Mountain View School are higher than those for Seaside School. This suggests that, on average, the class sizes at Mountain View School are larger than those at Seaside School. Therefore, if you are interested in a smaller class size, Seaside School would be a better choice
For Mountain View School:
- Median: The median class size is the middle value, which is 25.
- Mean: The mean class size is the sum of all the class sizes divided by the total number of classes, which is (29 + 28 + 28 + 27 + 25 + 24 + 24 + 23 + 21 + 20 + 19 + 18 + 15 + 13 + 10) / 15 = 21.27.
For Seaside School:
- Median: The median class size is the middle value, which is 20.
- Mean: The mean class size is the sum of all the class sizes divided by the total number of classes, which is (25 + 25 + 27 + 27 + 28 + 28 + 28 + 30 + 31 + 35 + 36 + 38 + 40 + 43 + 45) / 15 = 31.13.
Part B:
For Mountain View School:
- Range: The range is the difference between the largest and smallest values, which is 29 - 10 = 19.
- Interquartile Range (IQR): The IQR is the difference between the third quartile (Q3) and the first quartile (Q1). From the stem-and-leaf plot, we can see that Q1 is 20 and Q3 is 28. Therefore, the IQR is 28 - 20 = 8.
For Seaside School:
- Range: The range is the difference between the largest and smallest values, which is 45 - 25 = 20.
- Interquartile Range (IQR): The IQR is the difference between the third quartile (Q3) and the first quartile (Q1). From the stem-and-leaf plot, we can see that Q1 is 27 and Q3 is 38. Therefore, the IQR is 38 - 27 = 11.
Part C:
If you are interested in a smaller class size, Mountain View School is a better choice for you because it has a lower mean and median class size than Seaside School.