Solve the equation.
x^2 − 6 + 34 = 0
Answers
Answer:
[tex]x=6-\sqrt{ -100}/2=3-5i=3.0000-5.0000i\\x=6+\sqrt{-100} )/2=3+5i=3.0000+5.0000i[/tex]
Step-by-step explanation:
Step 1: Trying to factor by splitting the middle term
The first term is, x² its coefficient is 1 .
The middle term is, -6x its coefficient is -6 .
The last term, "the constant", is +34
Step-1 : Multiply the coefficient of the first term by the constant 1 • 34 = 34
Step-2 : Find two factors of 34 whose sum equals the coefficient of the middle term, which is -6 .
-34 + -1 = -35
-17 + -2 = -19
-2 + -17 = -19
-1 + -34 = -35
1 + 34 = 35
2 + 17 = 19
17 + 2 = 19
34 + 1 = 35
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step 1:
x² - 6x + 34 = 0
Parabola, Finding the Vertex:
Find the Vertex of y = x^2-6x+34
Parabolas have a highest or a lowest point called the Vertex . Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) . We know this even before plotting "y" because the coefficient of the first term, 1 , is positive (greater than zero).
Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.
Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex.
For any parabola,Ax2+Bx+C,the x -coordinate of the vertex is given by -B/(2A) . In our case the x coordinate is 3.0000
Plugging into the parabola formula 3.0000 for x we can calculate the y -coordinate :
y = 1.0 * 3.00 * 3.00 - 6.0 * 3.00 + 34.0
or y = 25.000
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = x2-6x+34
Axis of Symmetry (dashed) {x}={ 3.00}
Vertex at {x,y} = { 3.00,25.00}
Function has no real roots
Solve Quadratic Equation by Completing The Square
Solving x2-6x+34 = 0 by Completing The Square .
Subtract 34 from both side of the equation :
x2-6x = -34
Now the clever bit: Take the coefficient of x , which is 6 , divide by two, giving 3 , and finally square it giving 9
Add 9 to both sides of the equation :
On the right hand side we have :
-34 + 9 or, (-34/1)+(9/1)
The common denominator of the two fractions is 1 Adding (-34/1)+(9/1) gives -25/1
So adding to both sides we finally get :
x2-6x+9 = -25
Adding 9 has completed the left hand side into a perfect square :
x2-6x+9 =
(x-3) • (x-3) =
(x-3)2
Things which are equal to the same thing are also equal to one another. Since
x2-6x+9 = -25 and
x2-6x+9 = (x-3)2
then, according to the law of transitivity,
(x-3)2 = -25
We'll refer to this Equation as Eq. #2.2.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(x-3)2 is
(x-3)2/2 =
(x-3)1 =
x-3
Now, applying the Square Root Principle to Eq. #2.2.1 we get:
x-3 = √ -25
Add 3 to both sides to obtain:
x = 3 + √ -25
In Math, i is called the imaginary unit. It satisfies i2 =-1. Both i and -i are the square roots of -1
Since a square root has two values, one positive and the other negative
x2 - 6x + 34 = 0
has two solutions:
x = 3 + √ 25 • i
or
x = 3 - √ 25 • i
Solve Quadratic Equation using the Quadratic Formula
2.3 Solving x2-6x+34 = 0 by the Quadratic Formula .
According to the Quadratic Formula, x , the solution for Ax2+Bx+C = 0 , where A, B and C are numbers, often called coefficients, is given by :
- B ± √ B2-4AC
x = ————————
2A
In our case, A = 1
B = -6
C = 34
Accordingly, B2 - 4AC =
36 - 136 =
-100
Applying the quadratic formula :
x = 6 ± √ -100/2
In the set of real numbers, negative numbers do not have square roots. A new set of numbers, called complex, was invented so that negative numbers would have a square root. These numbers are written (a+b*i)
Both i and -i are the square roots of minus 1
Accordingly,√ -100 =
√ 100 • (-1) =
√ 100 • √ -1 =
± √ 100 • i
Can √ 100 be simplified ?
Yes! The prime factorization of 100 is
2•2•5•5
To be able to remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. second root).
√ 100 = √ 2•2•5•5 =2•5•√ 1 =
± 10 • √ 1 =
± 10
So now we are looking at:
x = ( 6 ± 10i ) / 2
Two imaginary solutions :
x =(6+√-100)/2=3+5i= 3.0000+5.0000i
or:
x =(6-√-100)/2=3-5i= 3.0000-5.0000i
Two solutions were found :
x =(6-√-100)/2=3-5i= 3.0000-5.0000i
x =(6+√-100)/2=3+5i= 3.0000+5.0000i
Related Questions
PLEASE HELP I AM TRYING TO STUDY FOR A CALCULUS EXAM!! WORTH A TOTAL OF 100 POINTS!!!!!!!!!
Find the mass of a sphere of radius 7 if δ= kρ, k a constant.
Answers
Presumably, both δ and ρ represent density of some object. Considering the context, ρ is likely a function of 3 variables, ρ = ρ(x, y, z). Then the mass of the sphere with the prescribed density δ(x, y, z) = k ρ(x, y, z) is
[tex]\displaystyle \iiint_B k \rho(x,y,z) \, dx\, dy\, dz[/tex]
where B is the set
[tex]B = \left\{(x,y,z) ~ : ~ x^2+y^2+z^2 \le 7^2\right\}[/tex]
If you have all the details at hand, you can compute the integral by converting to spherical coordinates, substituting
[tex]\begin{cases}x = u \cos(v) \sin(w) \\ y = u \sin(v) \sin(w) \\ z = u \cos(w)\end{cases} \text{ and } dx\,dy\,dz = u^2 \sin(w) \, du \, dv \, dw[/tex]
The integral then transforms to
[tex]\displaystyle k \int_0^{2\pi} \int_0^\pi \int_0^7 \rho(u\cos(v)\sin(w),u\sin(v)\sin(w),u\cos(w)) \, u^2 \sin(w) \, du \, dw \, dv[/tex]
Without any additional information, there's not much more to say...
I SUCK AT MATH PLS HELP
WORTH 1 POINT
GIVING 10
Answers
Answer:
34 degrees
Step-by-step explanation:
Assuming that ADJ is a right angle, we can add 4x and 3x-8 and set equal to 90 degrees to simply to get get x.
7x-8=90
Add 8
7x = 98
x= 14
Now that we know x, we can plug in the value to find angle CDJ.
3x-8
3(14)-8
34 degrees
40% if what number is 46?
Answers
Answer:
40 percent of 46 is 18.4.
Express with fractional exponents instead of radicals.
6√x² y³
Answers
Answer:
using of fractional exponents give us the value 6(x^2y^3)^1/2 or 6xy^3/2.
Find the value by using logarithms
25.753
0.5341
Answers
Step-by-step explanation:
589292 is the correct answer:-)
What is the mean of this data set? 12, 14, 16, 17, 18, 25
Answers
Does this graph show a function? Explain how you know.
• A. No; there are y-values that have more than one x-value.
B. No; the graph fails the vertical line test.
• c. Yes; the graph passes the vertical line test.
D. Yes; there are no y-values that have more than one x-value.
Answers
A graph that shows a function must pass the vertical line test
The true statement is (b) No; the graph fails the vertical line test.
How to determine if the graph shows a function?
For a graph to show a function, the graph must pass the vertical line test
The given graph do not pass the vertical line test;
This is so because, a line drawn from the x-axis would touch the graph at at least two different points
Hence, the true statement is (b) No; the graph fails the vertical line test.
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Given the functions f(x) = –4x + 5 and g(x) = x3 + x2 – 4x + 5, 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.)
Answers
The f(x) has a degree of 1, hence it is linear in nature while g(x) is a polynomial function because of the third degree.
The key features that they have in common are the y-intercept, domain, and range
Linear and quadratic equationsLinear equations are equations that has a degree of 1 while polynomial equations are equations that has a degree of 2 and above.
Given the following functions
f(x) = –4x + 5 and;
g(x) = x^3 + x^2 – 4x + 5
Since the f(x) has a degree of 1, hence it is linear in nature while g(x) is a polynomial function because of the third degree.
The key features that they have in common are the y-intercept, domain and range
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For this question, I solve Q.1, but I cannot solve Q2.
Q1. Suppose that the balance of a person’s bank account in Superior, WI is normally distributed with mean $580 and standard deviation $125. Find the amount of money which would guarantee a person has more money in their account than 80% of Superior, WI residents.
Q2. For the same setup as in Problem 2, find the probability a random person from
Superior has less than $400 or more than $1000 in their bank account.
Answers
The amount of money which would guarantee a person has more money in their account than 80% of Superior, WI residents is $680 approx. The probability a random person from Superior has less than $400 or more than $1000 in their bank account is 0.9247
How to get the z scores?If we've got a normal distribution, then we can convert it to standard normal distribution and its values will give us the z score.
If we have
[tex]X \sim N(\mu, \sigma)[/tex]
(X is following normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex])
then it can be converted to standard normal distribution as
[tex]Z = \dfrac{X - \mu}{\sigma}, \\\\Z \sim N(0,1)[/tex]
(Know the fact that in continuous distribution, probability of a single point is 0, so we can write
[tex]P(Z \leq z) = P(Z < z) )[/tex]
Also, know that if we look for Z = z in z tables, the p-value we get is
[tex]P(Z \leq z) = \rm p \: value[/tex]
Let we take:
X = The balance of a person’s bank account in Superior, WI
Then, by the given data, we have:
[tex]X \sim N(\mu = 580, \sigma = 125)[/tex]
Let the amount of money which would guarantee a person has more money in their account than 80% of Superior, WI residents be X = x dollars. This is the money, below which lie 80% of Superior, WI residents, and so as their income, or values of X.
Symbolically, we have:
[tex]P(X < x) = 80\% = 0.8[/tex]
Converting X to standard normal variate Z, we have:
[tex]Z = \dfrac{X - \mu}{\sigma} = \dfrac{X - 580}{125}\\[/tex]
Thus, the probability statement can be rewritten as:
[tex]P(X < x) =0.8\\P\left( Z < z = \dfrac{x -580}{125} \right) = 0.8[/tex]
From the z-tables, the value of Z for which the p-value comes out as between 0.84 and 0.85, let it be their average 0.845.
Thus, we get:
[tex]P(Z < z = 0.845) \approx 0.8[/tex]
Thus, we get:
[tex]z = \dfrac{x - 580}{125} \approx 0.8\\\\x \approx 0.8 \times 125 + 580 = 680[/tex]
Thus, the amount of money which would guarantee a person has more money in their account than 80% of Superior, WI residents is $680 approx.
Now, for second question, we need:
The probability a random person from Superior has less than $400 or more than $1000 in their bank account.
So, income of a random person from Superior is a random value of X.
So, this probability is written as:
[tex]P(400 < X < 1000)[/tex]
Rewritting it, we get:
[tex]P(400 < X < 1000) = P(X < 1000) - P(X \leq 400)[/tex]
Converting X to Z (standard normal variate), we get:
[tex]P(400 < X < 1000) = P(X < 1000) - P(X \leq 400)\\\\P(400 < X < 1000) = P\left(Z < \dfrac{1000-580}{125} \right) - P\left(Z \leq \dfrac{400-580}{125} \right)\\\\P(400 < X < 1000) = P( Z \leq 3.36) - P(Z \leq -1.44)[/tex]
The p-value for Z = 3.36 is 0.9996
The p-value for Z = -1.44 is 0.0749
Thus, we get:
[tex]P(400 < X < 1000) = P( Z \leq 3.36) - P(Z \leq -1.44)\\\\P(400 < X < 1000) = 0.9996 - 0.0749 = 0.9247[/tex]
Thus, the probability a random person from Superior has less than $400 or more than $1000 in their bank account is 0.9247
Thus, the amount of money which would guarantee a person has more money in their account than 80% of Superior, WI residents is $680 approx. The probability a random person from Superior has less than $400 or more than $1000 in their bank account is 0.9247
Learn more about z-score here:
https://brainly.com/question/21262765
#SPJ1
When K is negative the parabola _____
Answers
Answer:
Is shifted downwards
Step-by-step explanation:
K determines where the parabola is on the y axis basically. If the k value was positive it would be shifted upwards and if it was negative it would be shifted downwards.
Find the numbers of faces, edges, and vertices of the solid.
Answers
Answer:
7 faces15 edges10 verticesStep-by-step explanation:
This is a counting problem. As with many counting problems, it is helpful to adopt a strategy that helps ensure you count everything only once.
__
FacesThere are two pentagonal faces and 5 rectangular faces for a total of ...
7 faces
__
EdgesThere are 5 edges around each of the pentagonal faces, and 5 edges connecting the top face to the bottom faces, for a total of ...
15 edges
__
VerticesThere are 5 vertices on the top face, and 5 on the bottom face, for a total of ...
10 vertices
A triangle is inside of a rectangle. The triangle has a height of h and a base of b. Section 1 is to the left of the triangle, and section 2 is to the right.
Look at the drawing of a triangle within a rectangle. The triangle and the rectangle have the same base, b, and height, h.
If the area of the triangle is 48 square units, what is the total area of sections 1 and 2?
square units
Answers
Answer:
I think its 40
Step-by-step explanation:
Answer:
48
Step-by-step explanation:
Got it right on the assignment.
Hope this helps!!
Which expression is a factored form of
3n² - 4n - 4
Answer choices.............................
................................................
ANSWER
for reference of other who need it. cuz I needed it and it wasn't on here
A: (n+2) (3n +1)
Answers
Answer:
The answer is (n – 2) (3n + 2)
Step-by-step explanation:
3n² – 4n – 4
3n² – 6n + 2n – 4
3n(n – 2) + 2(n – 2)
(n – 2) (3n + 2)
Thus, (n – 2) (3n + 2) is a factored of
3n² - 4n - 4
i need this done help please asap i have a meeting with my teacher in 30 minutes
Answers
The answer to the given permutation and combination are
3024015585009252To solve the given problems, we have to use permutation and combination
Permutation and CombinationThis is a topic in mathematics that deals with the probability or number of ways an event can occur.
a)10P5
[tex]_1_0P_5 = 30240[/tex]
b) 6C2
[tex]_6C_2 = 15[/tex]
c) This will be a multiplication of permutation of number. This becomes
[tex]_2_6P_2 * _1_0P_2 = 58500[/tex]
This can happen in 58,500 ways.
d) To arrange 9 blocks in a row, she can do it in
[tex]_9C_1 = 9[/tex]
she can do this in 9 ways
e) This can be solved using combination by writing it as
[tex]_1_0C_5 = 252[/tex]
She can do this in 252 ways.
From the above calculations, the answer to the given permutation and combination are
3024015585009252Learn more on permutation and combination here;
https://brainly.com/question/9764004
The percentage error in measuring a clothe is 40% if the error committed is 10cm what is the original Value of the measurement??
Answers
cross multiply:
40% - 10
100% - x
so 40x = 1000
x=25
The mass of 1 tablespoon of syrup is as heavy as 5 teaspoons of cooking oil. The mass of 2 teaspoons of cooking oil is 12 g lighter than 1 tablespoon of syrup.find the mass of 1 teaspoon of cooking oil
Answers
Answer:
1 tsp of oil = 4 g
Step-by-step explanation:
1 Tbsp of syrup = 5 tsp of oil
2 tsp of oil = 1 Tbsp of syrup - 12 g
2 tsp of oil = 5 tsp of oil - 12 g
-3 tsp of oil = -12 g
1 tsp of oil = 4 g
The best way to convince employers of your skills and abilities is to _____. a. provide a written list, such as in your resume b. offer proof by example in answering interview questions c. have the employer contact your references d. none of the above
Answers
Answer:
A.
Step-by-step explanation:
B and C seem to stray away from what the question originally asked.
Answer:
the answer is B ( offer proof by example in answering interview questions)
Step-by-step explanation:
hope this helps:)
ed 2023
Find the chance of having a face card if a card is drawn randomly from a deck of cards.
Answers
Answer:
3/13
Step-by-step explanation:
P(Choosing a Face card) =
12/52= 3/13
pls mark as brainliest
The mean of the ages of 5 brothers is 13 years.
12, 16, O, 14, 8
Answers
Answer:
O = 15
Step-by-step explanation:
.mean ' is the average
(12 + 16 + O + 14 + 8) / 5 = 13
50 + O = 65
O = 15
If a chain is on 24 tooth front gear and 8 tooth rear gear determine whether this combination will result in high speed than a 2 to 1 gear ratio, assuming the speed you are pedalling is the same
Answers
24/8 = 3
the 24 to 8 gear = 3:1 ratio which is greater than 2:1
Is this statement true or false in the ordered pair (4,3) the y coordinate is 4
Answers
false
Question 1
Suppose a person is standing on the top of a building and that she has an instrument that allows her to
measure angles of depression. There are two points that are 100 feet apart and lie on a straight line that is
perpendicular to the base of the building. Now suppose that she measures the angle of depression from the
top of the building to the closest point to be 35.5º and the angle of depression from the top of the
building to the furthest point to be 28.8º. Determine the height of the building. (Round your answer to the
nearest tenth of a foot.)
Answers
Answer:
239.8 feet to the nearest tenth.
Step-by-step explanation:
We have 2 right triangles one inside the other.
Let the height of the builing be x ft and the distance from foot of thebuiding to the closest point be y feet.
So we have the system:
tan 28.8 = x/(y + 100)
tan 35.5 = x/y
From second equation:
x = tan35.5 y
Sustituting in the first equation:
tan 28.8 = tan35.5 y / (y + 100)
tan35.5 y = tan 28.8 y + 100 tan 28.8
y(tan35.5 - tan28.8) = 100 tan 28.8
y = 100 tan 28.2 / (tan35.5 - tan28.8)
y = 336.16 ft
So x = height of the buiding = tan35.5 * 336.16
= 239.78 ft.
please show me how you also got the answer as well, thanks a bunch!!
Answers
Answer:
Step-by-step explanation:
BRAINLIEST?! :D
You look up at a 70°
angle and see a plane directly above a building that is 30 meters away from you. How high is the plane flying? Round your answer to the nearest tenth.
Answers
Answer:
tan 70 times thirty gives you -1.70
What is the name given to a polynomial with 10 terms
Answers
Answer:
It should be called A decic polynomial
Step-by-step explanation:
*50 POINTS* Pls HELP. ANSWER THE QUESTION IN THE PIC!
Answers
Answer:
D
Step-by-step explanation:
Here's why I think it's D.
y = x + 44 is the equation the car wash owner uses
x represents the number of days
y represents the number of cars (washed)
so if we plug in one of the given numbers (from the chart)
example:
Day 2 Numbers of Cars Washed 44
2 will represent the "number of days" also known as x
44 will represent the "number of cars" also known as y
so if we plug in those values to the equation the car wash owner uses
y = x + 44, it doesn't match
__________________________________
(44) = 2 + (44)
44 [tex]\neq[/tex] 46
Therefore, I believe the answer is D
Answer:
D
Step-by-step explanation:
It is not a good fit for the dataThere should equal number of points below and above the x-axis in the residual plotThe given equation : y = x + 44 does not match the given dataat a basketball game, a vender sold a combined total of 166 sodas and hot dogs. The number of hot dogs sold was 54 less than the number of sodas sold. Find the number of sodas sold and the number of hot dogs sold.
Answers
The best way to solve these sorts of problems that involve two factors of unknown quantities but with certain properties is to use equations:
Let's set some variables
h: # of hot dogs solds: # of sodas soldNow based on the information given by the question, let's set up some equations:
vender sold a combined total of 166 sodas and hot dog⇒ [tex]s + h = 166[/tex]
number of hot dogs sold was 54 less than the number of sodas sold⇒ h = s - 54
Now lets put the equation together
s + h = 166 -- equation 1
h = s - 54 -- equation 2
Solve:
Plug the value of h from equation 2 into equation 1
s + (s - 54) = 166
2s - 54 = 166
2s = 220
s = 110 -- equation 3
Plug the value of s from equation 3 into equation 2
h = 110 - 54 = 56
h = 56
Therefore:
number of sodas sold: 110number of hot dogs sold: 56Hope that helps!
If Alex did 3/6 and then 1/4 of his homework, how much more does he need in order to finish ALL of his homework?
Answers
1) Zero helps mark directions in value Choose ALL contexts that correctly describes zero.
A) elevation:0 feet is called the ground floor
B bank account below 5D is a
negative balance
C)electricity an object that has 0 charge is neutral
D) buildings on an elevator, o represents the sea level
temperature temperature above is a negative temperature
E) temperature : temperature above 0 is a negative temperature
Answers
Answer: A,C,D
Step-by-step explanation:
Answer:
bank account and electricity
Step-by-step explanation:i tried the other one and it was all incorrect
Suppose f(x) = x². Find the graph of f(x-4)
4
Click on the correct answer.
1
2 3
Click on each graph to enlarge it.
graph 1
graph 2
graph 3
graph 4
What’s the answer ?
Answers
Answer:
1
Step-by-step explanation:
Translation of a function h units to the right is accomplished by replacing x in the function definition with (x-h).
Here, x has been replaced by x-4, so we expect to see the graph look like the original function graph shifted 4 units to the right.
That matches Graph 1.
