Mathematics High School

Find all complex solutions of 4x^2-5x+2=0.

(If there is more than one solution, separate them with commas.)

Answers

Answer 1
we use quadratic formula for that one
remember that √-1=i

so

for
ax^2+bx+c=0
[tex]x=\frac{-b+/-\sqrt{b^2-4ac}}{2a}[/tex]
so
given
4x^2-5x+2=0
a=4
b=-5
c=2

[tex]x=\frac{-(-5)+/-\sqrt{(-5)^2-4(4)(2)}}{2(4)}[/tex]
[tex]x=\frac{5+/-\sqrt{25-32}}{8}[/tex]
[tex]x=\frac{5+/-\sqrt{-7}}{8}[/tex]
[tex]x=\frac{5+/-i\sqrt{7}}{8}[/tex]

so in complex form (a+bi)
[tex]x=\frac{5}{8}+\frac{i\sqrt{7}}{8}, \frac{5}{8}-\frac{i\sqrt{7}}{8}[/tex]

Related Questions

Use the quadratic formula to solve the equation. If necessary, round to the nearest hundredth. −6y2 − 9y = −1

Answers

[tex]-6y^2 - 9y = -1 [/tex]

[tex]y= \dfrac{-9+ \sqrt{9^2-4*-6} }{2*6}, \dfrac{-9-\sqrt{9^2-4*-6}}{2*6} [/tex]


[tex]y= \dfrac{-9+ \sqrt{105} }{12}, \dfrac{-9- \sqrt{105} }{12} [/tex]

Find the coordinates of point Q that lies along the directed line segment from R(-2, 4) to S(18, -6) and partitions the segment in the ratio of 3:7.
Please help!!

Answers

check the picture below

thus then

[tex]\bf \qquad \textit{internal division of a line segment}\\\\ R(-2,4)\qquad S(18,-6)\qquad ratio1=3\qquad ratio2=7\qquad 3:7\\ \quad \\ \quad \\ \cfrac{RQ}{QS}=\cfrac{ratio1}{ratio2}\implies \cfrac{R}{S}=\cfrac{3}{7} \implies 7R=3S \\\\\\ 7(-2,4)=3(18,-6)[/tex]

[tex]\bf {{ Q=\left(\cfrac{\textit{sum of "x" values}}{ratio1+ratio2}\quad ,\quad \cfrac{\textit{sum of "y" values}}{ratio1+ratio2}\right)}}\\ \quad \\ \qquad thus\qquad \\ \quad \\ Q=\left(\cfrac{(7\cdot -2)+(3\cdot 18)}{3+7}\quad ,\quad \cfrac{(7\cdot 4)+(3\cdot -6)}{3+7}\right)[/tex]

Joe the trainer has two solo workout plans that he offers his clients: Plan A and Plan
b. Each client does either one or the other (not both). On Monday there were 2 clients who did Plan A and 3 who did Plan
b. On Tuesday there were 4 clients who did Plan A and 8 who did Plan
b. Joe trained his Monday clients for a total of 7 hours and his Tuesday clients for a total of 17 hours. How long does each of the workout plans last?

Answers

Let workout Plan A last a hours, and Plan B last b hours.

we are assuming personal training for each client.

i)
"On Monday there were 2 clients who did Plan A and 3 who did Plan"

the total time spent is : 2*a + 3*b =2a+3b

ii)
"On Tuesday there were 4 clients who did Plan A and 8 who did Plan B"

the total time spent was 4*a+8*b=4a+8b

iii) "Joe trained his Monday clients for a total of 7 hours"

so 2a+3b = 7

iv)

"Joe trained his Tuesday clients for a total of 17 hours"

so 4a+8b=17

v) thus we have the following system of equations:

2a+3b = 7
4a+8b=17

multiply the first equation by -2, and then add both equations, to eliminate a:

-4a-6b=-14
4a+8b=17
-------------------
2b=3, so b=3/2

2a+3b = 7
2a+3(3/2)=7
2a+9/2=7
multiply by 2:
4a+9=14
4a=5
a=5/4

Answer :

Plan A lasts 5/4=1.25 h
Plan B lasts 3/2=1.5 h


Assume the birth of a boy or a girl is equally likely. The probability that a single child is born a girl is 1/2. What is the probability that the next child born to the same familiy will also be a girl?

Answers

for both children to be girls, the probability is 1/4

probability is 1/4 (b)

Step-by-step explanation:

Suppose the vertex of a parabola is in the first quadrant and the parabola opens upwards. What can be determined about the value of a and the discriminant?

Answers

A parabola is the graph of a quadratic function, 

that is the graph of [tex]f(x)=a x^{2} +bx+c[/tex], where a is not 0.

from a, b and c we can derive the following informations about the shape of a parabola:

if a>0, the parabola opens upwards.
if a<0, the parabola opens downwards.

Consider the discriminant [tex]D= b^{2} -4ac[/tex]

If D>0, the parabola intersects the x-axis at 2 points.
If D=0, the parabola intersects the x-axis at 1 point.
If D<0, the parabola does not intersect the x axis.

"the vertex of a parabola is in the first quadrant and the parabola opens upwards."

the vertex is in the first quadrant means that the vertex is above the x-axis, and it opens upwards, so the parabola does not intersect the x-axis.

This means that:

Answer: a>0, the discriminant D<0

Final answer:

A parabola in the first quadrant opening upwards implies a positive 'a' value and a discriminant that, if not negative, yields real roots with positive values.

Explanation:

When a parabola has its vertex in the first quadrant and it opens upwards, we can determine specific values for a and the discriminant. The coefficient 'a' in the quadratic equation ax²+bx+c = 0 must be positive for the parabola to open upwards. Concerning the discriminant (calculated as b²-4ac), if the vertex is in the first quadrant, the parabola either does not intersect the x-axis at all (discriminant < 0), or it intersects the x-axis at one point (discriminant = 0) or two points (discriminant > 0) that both have positive x values.

The discriminant plays a key role in determining the nature of the roots of the quadratic equation. For quadratic equations constructed on physical data, they usually have real roots. Practical applications often deem the positive roots significant.

K is the midpoint of line segment lm. the coordinates of k are (5, 12) and the coordinates of l are (2, 6), find the coordinates of m.

Answers

midpoint (K) is (5,12)

L = (2,6)
M = (x2,y2)

midpoint = (x1 + x2)/2 , (y1 + y2) / 2
m = (2 + x2) / 2 , (6 + y2)/2

(2 + x) / 2 = 5
2 + x = 5 * 2
2 + x = 10
x = 10 - 2
x = 8

(6 + y) / 2 = 12
6 + y = 12 * 2
6 + y = 24
y = 24 - 6
y = 18

so M = (8,18) <==

What is the answer to this?

Answers

Hello there!

5/8 = x+7/ 2
We need to start by doing cross multiply
(8)(2) = (x+7)*5
16 = 5x + 35
5x = 35 - 16
5x =-19
x = -19/5

The correct answer is -19/5

--------------------------------------------------------------------------
The mistake is easy to see.
Instead of doing 35 - 16 { they did 35+16 which is wrong because you are transferring 16 to the opposite so you must change the positive sign as well.}

I hope that helps, please let me know if you have further questions about my answer.

The population of current statistics students has ages with mean muμ and standard deviation sigmaσ. samples of statistics students are randomly selected so that there are exactly 4242 students in each sample. for each​ sample, the mean age is computed. what does the central limit theorem tell us about the distribution of those mean​ ages?

Answers

We are told that the population has a mean of μ and standard deviation of σ.

From each sample of 42, we can compute a mean value of [tex]\Bar{x}[/tex].
We do not know the distribution for any sample.

According to the Central Limit Theorem, the distribution of the sample means will follow a normal distribution, regardless of the distribution of the individual samples.

Answer:
The distribution of sample means is normally distributed, and [tex]\Bar{x} \Rightarrow \mu[/tex],

Of five letters (a, b, c, d, and e), two letters are to be selected at random. how many possible selections are there

Answers

You can use C(5,2) which is [tex]5*4/2[/tex] which makes 10!

Prism M and pyramid N have the same base area and the same height. Cylinder P and prism Q have the same height and the same base perimeter. cone Z has the same base area as cylinder Y, but its height is three times the height of cylinder Y. Which two figures have the same volume?
Choices:
Prism M
Cylinder p
Cone Z
And
Pyramid N
Prism Q
Cylinder Y

Answers

V=1/3*π*R²*H
Сone Z and Cylinder Y have the same volume. 

A carnival game allows a group of players to each draw and keep a marble from a bag. The bag contains 5 gold marbles, 25 silver marbles, and 70 red marbles.

A player wins a large prize for drawing a gold marble and a small prize for drawing a silver marble. There is no prize for drawing a red marble.

At the start of the game, the probability of winning a large prize is 0.05 and the probability of winning a small prize is 0.25.

1. Suppose that the first player draws a silver marble and wins a small prize. What is the probability that the second player will also win a small prize?

2. If a group of four plays the game one at a time and everyone wins a small prize, which player had the greatest probability of winning a large prize?

3. How could the game be made fair for each player? That is, how could you change the game so that each player has an equal chance of winning a prize?

Answers

5 gold marbles,
25 silver marbles, and
70 red marbles.
------------------------------

100 total marbles

large prize: drawing a gold marble
small prize: drawing a silver marble.

At the start of the game,

probability of winning a large prize = positive outcoumes / total possible outcomes = 5 gold marbles / 100 total marbles =  0.05

probability of winning a small prize = positive outcomes / total possible outcomes = 25 silver marbles / 100 total marbles = 0.25.

1. Suppose that the first player draws a silver marble and wins a small prize. What is the probability that the second player will also win a small prize?

Answer:

numer of silver marbles / number of total marbles = (25 -1 ) / (100 - 1) = 24 / 99 ≈ 0.24

2. If a group of four plays the game one at a time and everyone wins a small prize, which player had the greatest probability of winning a large prize?

Answer:

First player: 0.25

Second player: 5 gold marbles / ( 100 - 1) total marbles = 5 /99 ≈ 0.0505

Third player: 5 gold marbles / (99 - 1) total marbles = 5 / 98 ≈ 0.051

Fourth player: 5 gold marbles / ( 98 - 1) total marbles = 5 / 97 ≈ 0.0515

So, the probability of winning a big prize increases as more balls different of gold marbles are extracted from the bag, and so, in this case, the fourth player has a greater chance to win a large prize.

3. How could the game be made fair for each player? That is, how could you change the game so that each player has an equal chance of winning a prize?

Answer: All the players would have equal chance of winning a prize if the balls were replaced in the bag after each play.

A spherical scoop of ice cream is placed on top of a hollow ice cream cone. the scoop and cone have the same radius. the ice cream melts completely and it fills the cone to the top. how many times greater is the height of the cone than the radius of the cone?

Answers

The figure shown below illustrates the problem.

The volume of the empty cone is
V₁ = (1/3) π r²h

The volume of the sphere is
V₂ = (4/3) π r³

Because the melted ice cream completely fills the cone, therefore
V₁ = V₂
(1/3) π r² h = (4/3) π r³
Divide each side by (1/3) π r².
h = 4r

Answer:
The height of the cone is 4 times greater than the radius f the cone.

Determine the slope and y-intercept of the line.
y =  5x + 4

a.
Slope = 4, y-intercept is (0, 5)
c.
Slope =  5, y-intercept is (0, 4)
b.
Slope = -5, y-intercept is (0, 4)
d.
Slope =  4, y-intercept is (0, -5)




 

Please select the best answer from the choices provided

A
B
C
D

Answers

The answer is C. Slope = 5, y-intercept is (0,4)

How can an expression or process be determined for an arithmetic sequence?

Answers

an aritmetic sequcne is represented as

[tex]a_n=a_1+d(n-1)[/tex]

where
an=nth term
a1=first term
d=common differnce, or how much each term increases by
n=which term



example

1,3,5, etc
first term is 1
common difference is 2 because it increases by 2 each time
so the formula would be
[tex]a_n=1+2(n-1)[/tex]

if we had
5,3,1
first term s 5
common difference is -2 since it goes up by -2 each time
so formula is
[tex]a_n=5-2(n-1)[/tex]



hope this helps

Solve the system by the elimination method.
x + y - 6 = 0
x - y - 8 = 0
When you eliminate y , what is the resulting equation?

Answers

Since we have a positive y in one equation and a negative y in the other, we can simple add the two together to eliminate y...

2x-14=0

2x=14

x=7, which makes x+y=6 become:

7+y=6

y=-1, so the solution to the system of equations is the point:

(7, -1)

Answer: 2x = 14

Step-by-step explanation:

Solving the equation us in elimination method,

x + y - 6 = 0...1

x - y - 8 = 0...2

From 1,

x+y = 6...3

x-y = 8...4

To eliminate y, we will add equation 3 and 4 since both the signs attached to y are different.

2x=6+8

2x = 14 (This will be the resulting equation)

To get the variables x, we will divide both sides of the resulting equation by 2

x = 14/2

x = 7

Substituting x = 7 into eqn 3

7 + y = 6

y = -1

A system of linear equations includes the line that is created by the equation y=0.5x-1 and the line through the points (3, 1) and (–5, –7), shown below.

What is the solution to the system of equations?

a. (–6, –4)
b. (0, –1)
c. (0, –2)
d. (2, 0)

Answers

The answer is (2,0). If you graph the lines, it shows that the intersection occurs right there.

Answer: Solution is,

d. (2, 0)

Step-by-step explanation:

Since, the equation of line that passes through points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex] is,

[tex](y-y_1)=\frac{x_2-x_1}{y_2-y_1}(y-y_1)[/tex]

Thus, the equation of line through the points (3, 1) and (–5, –7) is,

[tex](y-1)=\frac{-7-1}{-5-3}(x-3)[/tex]

[tex](y-1)=\frac{-8}{-8}(x-3)[/tex]

[tex]y - 1 = x - 3[/tex]

[tex]\implies y = x - 2------(1)[/tex],

Equation of second line is,

[tex]y = 0.5x - 1 -----(2)[/tex],

By equation (1) and (2),

x - 2 = 0.5x - 1 ⇒ 0.5x = 1 ⇒ x = 2,

From equation (1),

We get, y = 0,

Hence, the solution of line (1) and (2) is (2,0).

Write the equation in vertex form

f (x)= x^2-10x+16

Answers

You complete the square. There is actually a trick to it. If everything is on the same side, you make x^2-10x into (x-5)^2. You divide b/2 and insert that into the perfect square. You then subtract 25 because FOIL tells you that you need to add 25 when in fact you subtract 25 because you are on the same side. 

In final terms, it becomes (x-5)^2+16-25=(x-5)^2-9.

________ probability is used when we know the number of possible outcomes of the event of interest and the total number of possible outcomes in the sample space.

Answers

The answer in the blank is classical.

If a car is $27,000 and loses 15% of its value each year what will be the value in 5 years

Answers

$27,000

10%- $2,700.
5%- $1,350.

$2,700+ $1,350 = $4,050.

$4,050 × 5 = $20,250.

$27,000 - $20,250 = $6,750.

the value will be $6,750.

1. A ladybug lands on the end of a pinwheel that is spinning at a steady speed. The equation y = –7 sin (0.4πx) + 11 gives the height of the ladybug y, in centimeters, above the ground x seconds after landing. What is the diameter of the pinwheel

Answers

That question isn't nearly as hard as they want it to be.  The "a" in the sin function is the altitude of the function...how high it goes up and how low it goes down.  The negative in front of the 7 just means that it starts going down before it goes up (a classic sine wave goes up and then down; the negative just reverses that).  From the wave's midpoint on the y axis, which is 11 (hence, the +11 in the equation), the wave goes up 7 (to a y value of 18) and then goes down 7 (to a y value of 4).  Because it goes up 7 and then down 7, the diameter of the pinwheel is 14. The sin wave mimics the round and round motion of the pinwheel.  You could have been able to tell that it was 14 just by looking at what the altitude was.  The altitude is always the number the wave goes up and down from its midpoint. 7 up and 7 down makes 14 altogether.

A junior basketball has a diameter of approximately 7 in., and a regulation basketball has a diameter of approximately 9.5 in. about how many times as great is the volume of the regulation basketball as the volume of the junior basketball?

Answers

scale factor  = 9.5 / 7
the ratio of the volumes will be 9.5^3 / 7^3

=  2.5  to nearest thousandth

So the volume the regulation ball is  2.5 times the volume of the junior one.

50 POINTS!!!!!!!!!
Find a number that is between 7/11 and 0.75, theres a line on top of 75
77/99
23/33
25/44
76/99

To convert 0.9 to a fraction, Lauren wrote n=0.9 as her first step. For her second step, what should she multiply both sides of that equation by?
10
100
1000
10000

Which number is between 1/7 and 0.2 ?
9/35
6/35
2/9
1/9

Which fraction shows 5.2 as the quotient of two integers?
11/2
21/5
26/5
51/10

Answers

22/33, 10, 6/35, 26/5 is your answers in order from top to bottom.
1) 25 / 44
2) 10
3) 6 / 35
4) 26 / 5

(15 POINTS) A card is drawn from a deck of 52. What is the probability of drawing either a diamond or a seven?
A) 6/13
B) 17/52
C) 19/52
D) 4/13

Answers

that would be a 6/13

Answer:

The correct answer is 4/13

Step-by-step explanation:

The events "drawing a diamond or a seven" are inclusive events since there is a seven of diamonds. Follow the rule for inclusive events.

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Hope this helps! :)

WHICH ONE IS IT?////

Answers

The answer is:  [B]:  " (3.69/10) = 12/x " .
______________________________________________



Kite DCFE is inscribed in circle A shown below:



If the measure of arc DEF is 266°, what is the measure of ∠DEF?
Numerical Answers Expected!

Answers

check the pictures below.

the arcDEF is the intercepted arc.

Answer:

133  degrees

Step-by-step explanation:

Given is a picture of a kite DCFE inscribed in circle A

The vertices of the kite D, C, E and F lie on the circle.

A is the centre.

Arc DEF = 266 degrees

This means the subtended angle of arc DEF at the centre = Angle DAF = 266

By theorem on circles, we have central angle of any arc is twice the angle subtended by the arc on the remaining part of the circumference

Hence

Angle DEF = 1/2 angle DAF

i.e. DEF =[tex]\frac{266}{2} =133[/tex]

133 degrees

How to factor out the greatest common factor in a polynomial?

Answers

Hello,

To find something COMMON, we must at least have 2 things!

So , we can find the greatest common factor of 2 polynomials!

Final answer:

To factor out the GCF in a polynomial, identify the highest common factor, write it outside the parentheses, divide each term by the GCF, and write the quotients inside the parentheses.

Explanation:

To factor out the greatest common factor (GCF) in a polynomial, follow these steps:

First, identify the highest common factor that is present in each term of the polynomial.Write down this factor outside of a set of parentheses.Divide each term of the polynomial by the GCF, and place the resulting quotient inside the parentheses. This step can be seen as dividing both sides by the same factor to turn polynomial terms into integers, if that is easier to understand.Check your answer to see if it simplifies further and whether it is reasonable.

For example, for the polynomial 6x³ + 9x², the GCF is 3x2. Factoring out the GCF gives us:

3x²(2x + 3)

The products inside the parentheses are the result of dividing the original terms by the GCF. Remember, by finding the GCF, we simplify the algebra and may check the work by expanding the factored form back out to verify it equals the original polynomial.

You invest $500 in an account with an annual interest rate of 1.1%, compounded continuously. How much money is in the account after 15 years? Round your answer to the nearest whole number.

Answers

The formula is
A=p e^rt
A future value?
P present value 500
E constant
R interest 0.011
T time 15 years
A=500×e^(0.011×15)
A=589.7

Use the pythagorean theorem to find the distance between x(7,11) and y(-1,5)..

Answers

so, if the legs are a and b and the hypothuse is c then
a²+b²=c²

the distance bewteen the points is the hyptohuse

the legs are the distances between th x and y values

so
(7,11) and (-1,5)
distances between the x values are 1 leg
that is from 7 and -1, a distance of 8 units

distances betweeen the y values are the other leg
that is from 11 to 5
that is 6 units

a²+b²=c²
8²+6²=c²
64+36=c²
100=c²
sqrt both sides
10=c
the distance between them is 10 units

How do you solve equations for indicated variables?

ax+r=7
Solving for x?

Answers

You would subtract R from both sides so it would look like

ax=-r+7
then you would divide both sides by A
so then X = A/(-r+7)
and now you can plug in for A and R to solve

The solution is x = (7 - r) / a.

To solve the equation ax + r = 7 for x, follow these steps:

Subtract r from both sides of the equation:

ax + r - r = 7 - r

This simplifies to:

ax = 7 - r

Divide both sides by a:

x = (7 - r) / a

Substitute the value of x back into the original equation to ensure it is correct.

Use the graph below for this question:

graph of parabola going through negative 3, negative 3 and negative 4, negative 1.

What is the average rate of change from x = −3 to x = −4?

3
4
−3
−2

Answers

average rate of change is just the slope betwen the points
that is just the slope between (-3,-3) and (-4,-1)

slpe between (x1,y1) and (x2,y2) is (y2-y1)/(x2-x1)

so slope between (-3,-3) and (-4,-1) is (-1-(-3))/(-4-(-3))=(-1+3)/(-4+3)=2/-1=-2

the average rate of change is -2
I believe it's going to be -2
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