Let x = p+qi be the complex root and x = r be the the real root. The p and q are real numbers.
If x = p+qi, then x-p-qi = 0 which indicates (x-p-qi) is a factor.If x = r, then it leads to x-r = 0. This makes (x-r) the other factor.Multiply the two factors like so:
(x-p-qi)(x-r)
y(x-r) ..... let y = x-p-qi
xy - ry ..... distribute
x( y ) - r( y )
x( x-p-qi ) - r( x-p-qi ) ........ plug in y = x-p-qi
x^2 - px - (qi)x - rx + pr + qr*i ........ distribute
x^2 + (-px-qi*x-rx) + (pr+qr*i)
x^2 + (-p-qi*-r)x + (pr+qr*i)
x^2 + ((-p-r)-qi*)x + (pr+qr*i)
We end up with an expression in the form ax^2 + bx + c where the coefficients are:
a = 1 = real numberb = (-p-r) - qi = complex numberc = pr + qr*i = complex numberThe coefficients b and c are complex numbers. So it is possible to have one real root and exactly one complex root; however, as shown above, it leads to some of the coefficients being complex numbers. I'm assuming your teacher wants all coefficients to be real numbers. If that's the case, then a quadratic equation cannot have one real root and one complex root. You would need to have the complex roots come in conjugate pairs. Meaning that if a quadratic equation with real coefficients has a complex root, then there's another complex root as its nearly "twin" pair so to speak.
Recall that quadratic equations have at most 2 roots according to the fundamental theorem of algebra. Having 2 complex conjugate pairs for the roots will then mean there isn't enough room for that third real root. In other words, here are all the possible scenarios:
A) There are two different real rootsB) There is exactly one real rootC) There are no real roots. Instead, there are two complex roots that come in conjugate pairs.Again this only applies if all coefficients a,b,c are real numbers. For scenario B, the root repeats itself and we consider it a double root. For instance, x^2+10x+25 = 0 has the root x = -5 repeat itself. In scenarios A and B, we don't have any complex roots happen.
Because always that we have a discriminant different than zero, we have two solutions.
When a quadratic equation has complex roots?
A quadratic equation has complex roots when its discriminant is negative, where for the general quadratic:
y = a*x^2 + b*x + c
The discriminant is:
D = b^2 - 4ac
And the solutions of the roots are:
[tex]x = \frac{-b \pm \sqrt{b^2 - 4ac} }{2a}[/tex]
Because we have two possible signs for the square root of the discriminant, we always will have two solutions when the discriminant is different than zero.
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A cone has a slant height that measures 24 feet and a radius that measures 13.5 feet. What is the surface area of the cone?
Surface area of the cone is 1589.625 feet²
What is cone?A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called an apex or vertex. A cone is formed by a series of line segments, half-lines, or lines connecting apex that is common to all points on the base that lie in a plane that does not contain the apex.
Given,
Slant height of the cone l = 24 feet
Radius of the cone r = 13.5 feet
Surface area of cone
= πr² + πrl
= 3.14(13.5)² + 3.14×13.5×24
= 3.14(182.25) + 3.14(324)
= 572.265 + 1017.36
= 1589.625 feet²
Hence, 1589.625 feet² is the surface area of the cone.
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what size is this angle it goes up to 360 I need this asap
Answer:
its 145 degrees
Step-by-step explanation:
please help me out and if you can get it right please and thank you
Consider the sequence 3/4, 4/5, 5/6, 6/7, . . .
Which statement describes the sequence?
A. The sequence diverges.
B. The sequence converges to 0.
C. The sequence converges to ∞.
D. The sequence converges to –∞.
Answer:
b is a correct answer
Step-by-step explanation:
that is correct and right answer.
The sequence 3/4, 4/5, 5/6, 6/7, . . . is diverges option (A) The sequence diverges is correct.
What is the convergent of a series?
A series is convergent if the series of its partial sums approaches a limit; that really is, when the values are added one after the other in the order defined by the numbers, the partial sums get closer and closer to a certain number.
The sequence can be defined as the systematic way of representing the data that follows a certain rule of arithmetic.
It is given that:
The sequence:
3/4, 4/5, 5/6, 6/7, . . .
AS we know, the convergent sequence occurs when, as n approaches infinity, you arrive at a final and constant term through a number of terms.
Divergent sequences are those in which the terms never stabilize; instead, they constantly increase or decrease as n approaches infinity, approaching either infinity or -infinity.
The nth term of the sequence
a(n) = 2+n/3+n
Applying divergence test:
The sequence diverges
Thus, the sequence 3/4, 4/5, 5/6, 6/7, . . . is diverges option (A) The sequence diverges is correct.
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the charge for a microwave repair visit was $81.21, including tax. if the tax was $6.70, then how much was repair visit?
Answer:
i belive the answer would be 74.51$?
Step-by-step explanation:
An experiment consists of spinning the spinners shown and adding the numbers together. Complete the array.
Answer:
[tex]\begin{array}{ccccc}{} & {1} & {2} & {3} & {4} & {1} & {2} & {3} & {4} & {5} & {2} & {3} & {4} & {5} & {6} & {3} & {4} & {5} & {6} & {7} & {4} & {5} & {6} & {6} & {8} \ \end{array}[/tex]
Step-by-step explanation:
Given
[tex]Spinner = \{1,2,3,4\}[/tex]
Array:
[tex]\begin{array}{ccccc}{} & {1} & {2} & {3} & {4} & {1} & {} & { } & { } & {5} & {2} & { } & {} & { } & { } & {3} & {4} & { } & {} & { } & {4} & { } & {6} & { } & {} \ \end{array}[/tex]
Required
Complete the blanks
To do this, we simply add up corresponding row and column elements.
[tex]\begin{array}{ccccc}{} & {1} & {2} & {3} & {4} & {1} & {} & { } & { } & {5} & {2} & { } & {} & { } & { } & {3} & {4} & { } & {} & { } & {4} & { } & {6} & { } & {} \ \end{array}[/tex]
So, we have:
[tex]\begin{array}{ccccc}{} & {1} & {2} & {3} & {4} & {1} & {1+1} & {1+2 } & {1+3 } & {4} & {2} & {2+1 } & {2+2} & {2+3 } & {2+4 } & {3} & {4} & {3+2 } & {3+3} & {3+4 } & {4} & {4+1 } & {6} & {4+3 } & {4+4} \ \end{array}[/tex]
Add
[tex]\begin{array}{ccccc}{} & {1} & {2} & {3} & {4} & {1} & {2} & {3} & {4} & {5} & {2} & {3} & {4} & {5} & {6} & {3} & {4} & {5} & {6} & {7} & {4} & {5} & {6} & {6} & {8} \ \end{array}[/tex]
Which representation does not show a proportional relationship between x and y?
Answer:
Table A
Step-by-step explanation:
looking at the two tables, we have the observations as follows;
For table B, if we divide x by y; we have a ratio of 2/3
This happens throughout the table
What this means is that x = 2/3 * y
But for table A, we notice a pattern for the first two lines
The pattern here is that x = 2y
But as we move to the next two rows, we notice this fails and thus, we fail to establish a pattern that works for all the rows;
Hence table B has a pattern for all its rows
Kailynn has a bag with an equal number of red, yellow, and blue marbles. She
makes a tree diagram to show the sample space if she randomly selects two
marbles out of the bag.
What is the probability that both marbles selected are different colors?
Answer:
6/9
Step-by-step explanation:
Probability is the likelihood or occurrence of an event. Probability is the ratio of the total number of possible outcomes to the total outcomes. That is:
Probability = total possible outcomes / total outcomes
From the tree diagram we can see that:
total outcomes = {RR, RY, RB, YR, YY, YB, BR, BY, BB} = 9
getting different colors = {RY, RB, YR, YB, BR, BY} = 6
Therefore, the probability that both marbles selected are different colors = number of different colors / total outcomes
The probability that both marbles selected are different colors = 6 / 9
help asap ill give brainliest
Answer :
This figure consists of two shapes that are a rectangle and a square . We will first find the area of rectangle then the square then add them . Let's work on this together ,
Breadth of rectangle = 4 cmLength of rectangle = 4 + 5 + 3 Length = 12 cmArea of rectangle = 12 × 448 cm²Area of square ,
Side × Side5 × 525 cm²Total area of the figure ,
48 + 25 73 cm²what number line is the right one?? HELP
Answer:
B
Step-by-step explanation:
x/-5 < 9
×-5 ×-5
x < -45
x is less than -45
so, it's diagram B
write the product as a mixed number: 2 x 2 1/3
Answer:
14/3 or 4.666666667
Step-by-step explanation:
choose 1
hey! i’ll give brainliest please help.
Answer:
shah
:) hope this helps!
When Eric multiplied two binomials together, his result was a trinomial. An example is
(+2) G + 3) = x2 + 5x+6. Eric concluded that the product of any two binomials is a
trinomial. The product of which pair of binomials disproves Eric's conclusion?
Answer:
(x + 4)(x - 4)
Step-by-step explanation:
There are actually quite a lot of pairs of binomials the disproves Eric's conclusion, but they all model after the same special product: a^2 - b^2.
The special product a^2 - b^2 can be factored into (a + b)(a - b) and for all real a and b, it will come out as a binomial.
Here is an example:
(x + 4)(x - 4)
We can use the distributive property to get:
x^2 - 4x + 4x - 16
which is the same as
x^2 - 16
This would disprove Eric's conclusion.
The dot plots show the heights of boys and girls at a summer camp.
Heights of Boys and Girls at Camp
2 dot plots with number lines going from 40 to 60. A plot is titled Boy's Heights. There are 0 dots above 40, 1 above 41, 3 above 44, 3 above 46, 2 above 48, 3 above 50, 4 above 52, 4 above 54, and 0 above 56, 58, and 60. A plot is titled Girl's Heights. There are 0 dots above 40 and 41, 2 dots above 44, 3 above 46, 1 above 48, 3 above 50, 4 above 52, 3 above 54, 4 above 56, and 0 above 58 and 60.
Which is a true statement for most of the data in each plot?
Most of the data in each plot are greater than 48.
Most of the data in each plot are less than 48.
Most of the data in each plot are around 52.
Most of the data in each plot are around 54.
Answer:
A.
Step-by-step explanation:
Most of the data in each plot are greater than 48, EDG 2021
Answer:
a
Step-by-step explanation:
4x + 22 = 86
(what's the value of x?)
Answer:
x = 16
Step-by-step explanation:
4x + 22 = 86
4x + 22 - 22 = 86 - 22
4x = 64
4x/4 = 64/4
x = 16
Answer:
16
Step-by-step explanation:
86-22= 64 64/4=16
plz mark branliest
A crayon has a mass of 10.0 g and volume of 5.0 mL. What is the density?
Answer:
2000 g/L
Step-by-step explanation:
(10.0 g) / (0.005 L) = 2000 g/L
f(x) = -x squared + 2x + 8
guys i’m gonna fail math
Final answers:
Vertex: (1,9)Domain: all real numbers, or (-infinity, infinity)Range: (-infinity, 9]Maximum: 9; there is no minimumy-intercept: (0,8)Interval of increase: (-infinity, 1)Interval of decrease: (1, infinity)Average rate of change: 3 or -3Explanations
Vertex: this is the maximum or minimum point on the graph, which is (1,9).Domain: this is everywhere that the function is defined. Since there are no values of x for which -x^2 + 2x + 8 is not defined, the domain is all real numbers, or [tex](-\infty, \infty)[/tex].Range: this is all y-values the function takes. We can see from the graph that the function takes on all values less than or equal to 9, so the range is (-infinity, 9].Maximum or minimum: the maximum value is 9. There is no minimum value.y-intercept: this is the point where the graph intersects the y-axis, which is (0,8) or simply 8. You can think of this as the value of f(x) when x is zero.Interval of increase: f(x) is increasing on (-infinity, 1).Interval of decrease: f(x) is decreasing on (1, infinity).Average rate of change: the question is asking us to find the slope between the vertex and one of the x-intercepts – I will use (-2,0). The vertex is (1,9). The average rate of change is the slope between these points, which is (9-0)/(1-(-2)), or 3. If you instead used the x-intercept of (4,0), you would get an answer of -3.Please help me using product rule
Using the product rule, then (A) is the answer.
Part A Anna took a taxi home from the airport. The taxi fare was \$ 2.1$2.1 per mile, she gave the driver a tip of \$ 5$5, and Anna paid a total of \$ 49.10$49.10. Write an equation to determine the distance in miles \left(x\right)(x) between the airport and Anna's home. Answer: Part B Find the distance between the airport and Anna's home. Answer: miles.
Answer:
44.10 = 2.1x
x = 21 miles
Step-by-step explanation:
Cost per mile = $2.1
Driver tip = $5
Total amount paid = $49.10
Let
x = number of miles
Write an equation to determine the distance in miles \left(x\right)(x) between the airport and Anna's home.
Total cost of miles = Total amount paid - Driver tip
= $49.10 - $5
= $44.10
Equation to determine distance in miles
Total cost of miles = cost per mile * number of miles
44.10 = 2.1x
Find the distance between the airport and Anna's home
44.10 = 2.1x
Divide both sides by 2.1
44.10 / 2.1 = 2.1x / 2.1
21 = x
x = 21 miles
what the highest common factor for 12 and 20
Answer:4
Step-by-step explanation:
4 NH3 + 5 O2 → 4 NO2 + 6 H2O
The balanced equation above shows that 4.00 moles of NH3 requires ___ mole(s) of O2
0.57
0.80
1.25
5
Answer:
0.57 PROOOOOOOOOOOOOOOOOOO
Step-by-step explanation:
Answer:
since this question is answered
Step-by-step explanation:
don't mine if I take the points
Find the value of x in the triangle shown below.
5.7
20
6
6
56
Answer:
62 degrees
Step-by-step explanation:
the base angles are congruent and the sum of angles is 180 degrees
56 + 2x = 180
2x = 124
x = 62 degrees
What two numbers multiply to -40 and add to 10
Answer:
5 + sqrt(65), 5 - sqrt(65)
Step-by-step explanation:
x + y = 10
x×y = -40
so, two equations with 2 variables. using one equation to express one variable by the second. and then using the second equation to solve and calculate the second variable. and then by knowing the second variable we calculate the first.
=> x = 10 - y
=> (10-y)×y = -40
=> -1×(10-y)×y = 40
=> (y-10)×y = 40
=> y^2 - 10×y - 40 = 0
remember the solution for a quadratic equation
a×x^2 + b×x + c
is
x = (-b ± sqrt(b^2 - 4×a×c)/(2×a)
in our case
a = 1
b = -10
c = -40
and we used "y".
so,
y = (--10 ± sqrt((-10)^2 - 4×1×(-40))/(2×1) =
= (10 ± sqrt(100 + 160))/2 = (10 ± sqrt(260))/2 =
= (10 ± sqrt (4×65))/2 = 10/2 ± 2×sqrt(65)/2 =
= 5 ± sqrt(65)
=>
for y = 5 + sqrt(65), x = 10 - 5 - sqrt(65) = 5 - sqrt(65)
for y = 5 - sqrt(65), x = 10 - 5 + sqrt(65) = 5 + sqrt(65)
4. For what event must theoretical probability be determined to solve this problem?
Answer:
40
Step-by-step explanation:
1/5*200=40
ASAP LAST DAY TO TURN IN*
Determine the surface area of the three-dimensional composite figure after rotation about the x-axis. Show all work to receive credit. Bubble answer on griddable. Round to the nearest hundredth if necessary.
Answer:
72.85 sq units
Step-by-step explanation:
SA = (½×4×3.14×2²)+(2×3.14×2×2) +(3.14×2×3.6)
= 25.12+25.12+22.61
= 72.85 sq units
will give brainlyist if quick Which of the equations is graphed below?
Answer:
c
Step-by-step explanation:
A line is parallel to y = 2x + 1 and
intersects the point (3, 2). What is
the equation of this parallel line?
Start by entering this into point-slope form.
y 52 = [?](x-[])
Answer:
y= 2x-4
Step-by-step explanation:
y-2= 2(x-3)
y-2=2x-6
y= 2x-6+2
y= 2x-4
If f(x)=2x and g(x)=−2x3+2x, then what is f(x)+g(x)?
x3+2x
−4x3+2x
−2x3+4x
2x
Answer:
[tex]-2x^3+4x[/tex]
Step-by-step explanation:
I am assuming that the functions are:
[tex]f(x)=2x\\\\g(x)=-2x^3+2x[/tex]
-----------------------
[tex]f(x)+g(x)\\\\2x+-2x^3+2x\\\\2x+2x+-2x^3\\\\4x+-2x^3\\\\\boxed{-2x^3+4x}[/tex]
Hope this helps you!
Round 367.8633 to nearest thousandth.
Answer:
367.863
Step-by-step explanation:
- 1/3 x + 12 = -2x + 2 ?
Answer:
x = -6
Step-by-step explanation:
- 1/3 x + 12 = -2x + 2
(- 1/3 x + 1/3) + 12 = (-2x + 1/3) + 2
12 = - 1 2/3x + 2
12 -2 = - 1 2/3x (+ 2 - 2)
10 = - 1 2/3x
x = -6
Answer:
- 30/7
step by step explanation:
i dont really know how to explain it sorry