In geometry, a quadrilateral is a four-sided polygon. There are several special types of quadrilaterals, such as rectangles, squares, parallelograms, and trapezoids, each with their own set of defining properties.
To determine whether or not a given quadrilateral is a rectangle, we need to check if it satisfies the properties of a rectangle. There are several ways to do this, but one common method is to check the lengths of its sides and the angles between them.
In particular, a rectangle is a quadrilateral with four right angles and opposite sides that are equal in length. Therefore, to prove that quadrilateral ABCD is a rectangle, we need to show that,
All four angles are right angles.
Opposite sides are parallel.
Opposite sides are equal in length.
To check if quadrilateral ABCD satisfies these properties, First, we can calculate the length of each side using the distance formula:
AB = √[(2 − (−3))²+ (6 − 3)²] = √74
BC = √[(5 − 2)² + (1 − 6)²] = √34
CD = √[(−5 − 5)² + (−5 − 1)²] = 2√26
DA = √[(−3 − (−5))² + (3 − (−5))²] = 2√17
We can see that opposite sides AB and CD have the same length, and opposite sides BC and DA have the same length. This satisfies property 3 above.
we can calculate the slopes of each side using the slope formula
AB: (6 − 3)/(2 − (−3)) = 3/5
BC: (1 − 6)/(5 − 2) = −5/3
CD: (−5 − 1)/(−5 − 5) = −3/5
DA: (3 − (−5))/(−3 − (−5)) = −4/7
We can see that opposite sides AB and CD have the same slope, and opposite sides BC and DA have the same slope. This satisfies property 2
Finally, we can calculate the angle between each pair of adjacent sides using the dot product formula:
∠ABC = cos⁻¹[(AB⋅BC)/(√74⋅√34)] ≈ 91.9°
∠BCD = cos⁻¹[(BC⋅CD)/(√34⋅2√26)] ≈ 88.1°
∠CDA = cos⁻¹[(CD⋅DA)/(2√26⋅2√17)] ≈ 91.9°
∠DAB = cos⁻¹[(DA⋅AB)/(2√17⋅√74)] ≈ 88.1°
We can see that all four angles are approximately equal to 90⁰. This satisfies property 1.
Therefore, based on the calculations above, we can definitively prove that quadrilateral ABCD is a rectangle.
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A box contains hair clips of 3 diff sizes. -15 small hair clips -12 medium hair clips -6 large hair clips
If one hair clip is randomly selected from the box,what is the probability that the hair clip will be either small or medium?
A)9/11
B)3/7
C)2/9
D)1/6
The answer of the given question based on probability is , option (A) 9/11.
What is Probability?Probability is measure of likelihood of event occurring. It is number between 0 and 1, where 0 means that event is impossible, and 1 means that event is certain to occur. The probability of event can be expressed as fraction, a decimal, or a percentage. For example, if the probability of an event is 1/4, it means that there is a 25% chance of the event occurring.
The total number of hair clips in the box is:
Total number of hair clips = 15 (small) + 12 (medium) + 6 (large) = 33
The probability of selecting a small or medium hair clip is the sum of the probabilities of selecting a small hair clip and a medium hair clip:
P(small or medium) = P(small) + P(medium)
The probability of selecting a small hair clip is:
P(small) = number of small hair clips / total number of hair clips
= 15 / 33
The probability of selecting a medium hair clip is:
P(medium) = number of medium hair clips / total number of hair clips
= 12 / 33
Therefore, the probability of selecting a small or medium hair clip is:
P(small or medium) = P(small) + P(medium)
= 15/33 + 12/33
= 27/33
= 9/11
So the answer is (A) 9/11.
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Write the formula that shows the dependence of the volume V of a cube on the length a of its edge
a³ is the volume of a cube where a is the side of the cube.
We have,
The formula that shows the dependence of volume V of a cube on the length a of its edge is:
V = a^3
Where "a" is the length of one edge of the cube.
For instance,
If the length of the edge of a cube is 3 cm, then the volume of the cube would be:
V = 3^3 = 27 cm³
Similarly, if the length of the edge of a cube is 4 cm, then the volume of the cube would be:
V = 4^3 = 64 cm³
Thus,
a³ is the volume of a cube where a is the side of the cube.
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Four friends equally share 23 pounds of apples. How much does each
friend receive?
Answer: 5.75 lbs each
Step-by-step explanation:
We will divide 23 pounds of apples among the 4 friends.
23 / 4 = 5.75 lbs each
g it is estimated that 0.32 percent of the callers to the customer service department of dell incorporated will receive a busy signal.what is the probability that of today's 1,000 callers at least 5 received a busy signal? (round your answer to 4 decimal places.)
The probability that of today's 1,000 callers, at least 5 received a busy signal is approximately 0.0373.
Typically a binomial dissemination issue where we have n = 1000 trials, each with a victory likelihood of p = 0.0032 (0.32%).
We need to discover the likelihood that at the slightest 5 trials result in victory (accepting an active signal).
Utilizing the compliment to run the show, we will discover the likelihood of having less than 5 triumphs and subtract it from 1 to urge the likelihood of having at slightest 5 successes:
P(X ≥ 5) = 1 - P(X < 5)
To calculate P(X < 5), we will utilize the binomial likelihood equation or estimation:
P(X < 5) = Σ P(X = k) for k = to 4
Utilizing the binomial likelihood equation:
P(X < 5) = Σ (1000 select k) * (0.0032)[tex]^k[/tex] * (1 - 0.0032)[tex]^(1000 - k)[/tex]
for k =0 to 4
where (1000 select k) = 1000! / (k! * (1000 - k)!)
Stopping within the values and summing up:
P(X < 5) = 0.9627
In this manner, the likelihood of having at slightest 5 triumphs is:
P(X ≥ 5) = 1 - P(X < 5) = 1 - 0.9627 = 0.0373
So, the likelihood that at slightest 5 out of 1000 callers get an active flag is 0.0373, or roughly 0.0373.
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find the following arc measures. JL and JML
The measure of arc JML in the given circle is 233°.
By definition, the measure of an arc is equal to the measure of the central angle that intercepts it. A central angle forms when two rays have their origins at the center of the circle and it also serves as the vertex of the central angle. We can note from the diagram that
arc JL is formed by the endpoints of the central angle ∠JKL=127°.
Thus, the measure of arc JL is equal to the measure of the angle or
arc JL=127°.
To find arc JML, we can note from the diagram that is equal to the rest of the circle. We know that the measure of a complete circle is 360°, so we can subtract arc JL from it and we get:
arc JML=360°-arc JL
=360°-127°
=233°
Therefore, the measure of arc JML in the given circle is 233°.
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pls help me with this question
The discount expression is d > 400 and the temperature expression is t ≤ 40
The discount expressionWe understand that the shopper must spend more than $400 to get 10% discount
So, the expression for the amount spent is:
d > 400
The temperature expressionWe understand that the temperature is no more than 40F
So, the expression for the temperature is:
t ≤ 40
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Please help! Very easy to do!
Answer:
If the probability of landing on purple on a spinner is 3 out of 10, then we can expect to land on purple about 90 times if we spin it 300 times
Step-by-step explanation:
At the end of each year, Richard receives a
15% salary increase.
After two years, Richard's salary is £34,385.
What was his starting salary?
Give your answer in pounds (£).
Starting salary
Salary after one year
Salary after two years
4
£
£ 34,385
)+15%
+15%
let's approach this, this way
we have a principal amount P, with a annual rate of 15%, that after 2 years it became £34,385, what's P?
[tex]~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\dotfill & \pounds 34385\\ P=\textit{original amount deposited}\\ r=rate\to 15\%\to \frac{15}{100}\dotfill &0.15\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{annually, thus once} \end{array}\dotfill &1\\ t=years\dotfill &2 \end{cases}[/tex]
[tex]34385 = P\left(1+\frac{0.15}{1}\right)^{1\cdot 2} \implies 34385=P(1.15)^2 \\\\\\ \cfrac{34385}{(1.15)^2}=P\implies 26000=P[/tex]
after two years? well, we already know, is above, after one year? well is just 26000 + 15% of itself, so
[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{15\% of 26000}}{\left( \cfrac{15}{100} \right)26000}\implies 3900~\hfill \underset{ \textit{after one year} }{\stackrel{ 26000~~ + ~~3900 }{29900}}[/tex]
(x+5) x + 3x + 5 + x + 40 2 7x-x+ 25-5
Answer:
12x+72
Step-by-step explanation:
have a good day :)
Factor.
f(x) = x² + 4x - 12
(x + [?])(x - [])
Enter
What does 2x + 41 equal. Please help, I am timed
The answer to the expression, 2 x ? = 41 is 20.5
Why is this so?What is the sum of 2 and 41?
To put it another way, what do you multiply by 2 to get 41?
If "x" is "what" in the line "2 times what equals 41?" then the equation to solve the problem is
2x/2 = 41/2
2x/2 is x and 41/2 is 20.5 which means our equation will look like this:
x = 20.5
So the answer to "2 times what equals 41?" is 20.5.
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Full Question:
Although part of your question is missing, you might be referring to this full question:
2 times what equals 41?
Dorji asked Tenzin to help him sell his car at a price of Nu 200,000. Mean while Tenzin wants to make a profit of Nu 20,000 from the sale. Determine the percent markup?
Q12. A car is sold for ₹ at a loss of %. What is the cost price ()
of the car?
The cost price of the car sold at a loss of 18% is $872,222.22.
What is the cost price of a car sold?Let CP be the cost price of the car.
The selling price of the car:
= CP - 18% of CP
= 0.72 CP
Given:
Selling price of the car = $328,000
0.72 CP = $328,000
CP = $628,000 / 0.72
CP = $872222.222222
CP = $872222.22.
Full question "A car is sold for $628,000 at a loss of 18%. What is the cost price of the car?".
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How many triangles can be formed by the diagonals drawn from a fixed vertex in a convex polygon?
Answer: One Triangle
Step-by-step explanation:
The tables represent the points earned in each game for a season by two football teams.
Eagles
3 24 14
27 10 13
10 21 24
17 27 7
40 37 55
Falcons
24 24 10
7 30 28
21 6 17
16 35 30
28 24 14
Which team had the best overall record for the season? Determine the best measure of center to compare, and explain your answer.
Eagles; they have a larger median value of 21 points
Falcons; they have a larger median value of 24 points
Eagles; they have a larger mean value of about 22 points
Falcons; they have a larger mean value of about 20.9 points
The team that had the best overall record for the season is C. Eagles; they have a larger mean value of about 22 points Falcons; they have a larger mean value of about 20.9 points
Given is the table information about the scores of two teams,
In this case, the Eagles have a larger mean value of about 22 points compared to the Falcons' mean value of about 20.9 points. So, the correct answer would be Eagles; they have a larger mean value of about 22 points.
It's worth noting that using the median value in this case is not the most accurate, because this will give you a more robust representation of the center of the dataset in cases where data have outliers.
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I’m not sure how to answer?
The transformation of the graph f(x) = x² to the graph g(x) = -3(x+5)² + 12 involves a horizontal shift, a vertical stretch, and a vertical translation.
The graphs of f(x) = x² and g(x) = -3(x+5)² + 12 are the parabola.
The transformation of the graph is following as:
Firstly, the parabola has been shifted horizontally by 5 units to the left, which is reflected in the expression (x+5)².
Secondly, the parabola has been stretched vertically by a factor of -3, which is reflected in the coefficient in front of (x+5)².
Finally, the parabola has been raised up to 12 units. This means that the vertex of the parabola has been shifted upwards from the origin to the point (−5,12).
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Please help. It’s worth 4 marks
Answer:
[tex]f=\frac{3+4d}{3+d}[/tex]
Step-by-step explanation:
[tex]d= \frac{3(1-f)}{(f-4)} \\(f-4)d=3-3f\\f-4=\frac{3-3f}{d} \\f=\frac{3-3f+4d}{d} \\f+\frac{3f}{d} =\frac{3+4d}{d} \\fd+3f=3+4d\\f(d+3)=3+4d\\f=\frac{3+4d}{3+d}[/tex]
Which number describes the median of the data given in the table?
23
28
22
24.2
Which number describes the range of the data given in the table?
6
22
28
23
Answer:
The median is A. 23
To find the median Re-arrange the numbers.
22, 22, 23, 26, 28
The middle number which is 23 is A Median.
The Range is A. 6
To find the Range We subtract the lowest frequency to the highest frequency.
Range = 28 - 22
= 6
the New York kicks playing in Madison square garden, the Los Angeles Lakers play in the Staples centers and the Miami heat playing in American airlines arena Madison square garden is about 2.8 x 10^3 miles from the Staples center in about 1.4 x.10^3 miles from American airlines arena
how many times further does the Knicks travel when they play the Los Angeles Lakers compared to the Miami heat?
(divide the larger distance by the smaller distance)?!
Therefore, the Knicks travel two times further when they play the Lakers compared to the Miami Heat.
What is time ?Time is a fundamental concept used to describe the sequence of events and the duration between them. It is a continuous and irreversible flow of moments and intervals in which events occur in a chronological order. Time can be measured by various units, such as seconds, minutes, hours, days, weeks, months, and years, and is commonly divided into past, present, and future. Time is often perceived subjectively, with its perception varying depending on factors such as an individual's location, culture, and personal experience. The study of time is an essential part of various fields, including physics, philosophy, psychology, and sociology.
To find out how many times further the Knicks travel when they play the Lakers compared to the Miami Heat, we need to divide the larger distance (the distance between Madison Square Garden and the Staples Center) by the smaller distance (the distance between Madison Square Garden and the American Airlines Arena).
So, the calculation is:
[tex]2.8 x 10^3 miles / 1.4 x 10^3 miles = 2[/tex]
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How are the graphs of the functions f(x)=√16^x and g(x)=3√64^x
Option A is correct, the functions f(x) and g(x), f(x)=√16ˣand g(x)=∛64ˣ are equivalent
The given two functions are f(x)=√16ˣand g(x)=∛64ˣ
We have to find how the graphs of f(x) and g(x) functions are related.
Let us simplify the two functions
f(x)=√16ˣ
=√(4²)ˣ
[tex]=(4^2)^x^/^2[/tex]
=4ˣ
Now g(x) = ∛64ˣ
[tex]=(4^3)^x^/^3[/tex]
=4ˣ
f(x) and g(x) are equal
Hence, the functions f(x) and g(x), f(x)=√16ˣand g(x)=∛64ˣ are equivalent
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How are the graphs of the functions f(x) = sqrt 16^x and g(x) = ^3 sqrt 64 ^x related?
The functions f(x) and g(x) are equivalent.
The function g(x) increases at a faster rate.
The function g(x) has a greater initial value.
The function g(x) decreases at a faster rate
expand and simplify 2(x+3)-3(2x-5)
Answer:
Step-by-step explanation:
2(x+3)-3(2x-5) distribute the 2 and -3
2x + 6 - 6x + 15 simplify
-4x +21
Please help me with this homework
Answer:
A = 36 cm²
Step-by-step explanation:
the area (A) of a triangle is calculated as
A = [tex]\frac{1}{2}[/tex] bh ( b is the base and h the height )
here b = 9 and h = 8 , then
A = [tex]\frac{1}{2}[/tex] × 9 × 8 = 4.5 × 8 = 36 cm²
the top of a 21 foot ladder, leaning against a vertical wall, is slipping down the wall at the rate of 5 feet per second. how fast is the bottom of the ladder sliding along the ground when the bottom of the ladder is 5 feet away from the base of the wall?
The bottom of the ladder is sliding along the ground at a rate of 20 feet per second when it is 5 feet away from the base of the wall.
Let's denote the distance between the base of the ladder and the wall as x and the distance between the top of the ladder and the ground as y. Then we can use the Pythagorean theorem to relate x and y
x² + y² = 21²
We want to find how fast the bottom of the ladder (x) is sliding along the ground, which is the rate of change of x with respect to time. Let's call this rate dx/dt.
We are given that the top of the ladder is slipping down the wall at a rate of 5 feet per second, which means that the rate of change of y with respect to time (dy/dt) is -5. The negative sign indicates that y is decreasing.
We want to find the value of dx/dt when x = 5. To do this, we need to relate x, y, dx/dt, and dy/dt using implicit differentiation
2x(dx/dt) + 2y(dy/dt) = 0
Simplifying this expression, we get
dx/dt = -(y/x) × dy/dt
We can substitute the values we know into this equation
dx/dt = -(y/x) × (-5) = 5y/x
To find y when x = 5, we can use the Pythagorean theorem
5² + y² = 21²
y = √(21² - 5²) = 20
Substituting this into the equation for dx/dt, we get
dx/dt = 5y/x = 5(20)/5 = 20
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URGENT - Will also give brainliest to simple answer
-4X+7=15
how to solve this equation?
Answer:
to solve this, first you need to write the equation -4x + 7 = 15, then you take 7 off both sides like this -4x + 7-7=15-7 -4x=8, after than you divide both sides by -4 like this -4x/-4=8/-4 x=-2 and theres you're anwser
Step-by-step explanation:
Answer:
-4x+7=15
collect like terms
-4x=15-7
-4x=8
divide both sides by the co-efficient of x
-4x/-4=8/-4
x= -2
ans
Michael has a bag of marbles. The frequency of selecting each color is recorded in the table below.
Outcome Frequency
Green 4
Black 6
Orange 5
Based on the given frequency, determine the experimental probability of selecting a black marble.
0.27
0.33
0.40
0.60
The experimental probability of selecting a black marble is 0.40.
Probability = Number of favourable outcomes / Number of samples
Given that Michael has a bag of marbles.
The frequency of selecting each colour is recorded in the table below. Outcome Frequency
Green 4
Black 6
Orange 5.
Number of favourable outcomes = 6
Number of samples = 4+6+5 = 15
Probability = Number of favourable outcomes / Number of samples
Probability = 6 / 15
Probability = 0.40
Hence, the experimental probability of selecting a black marble is 0.40.
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A movie theater has a seating capacity of 363. The theater charges $5.00 for children, $7.00 for students, and $12.00 for adults. There are half as many adults as there are children. If the total ticket sales was $ 2630, How many children, students, and adults attended?
If the total ticket sales was $ 2630, the theater sold 178 children's tickets, 99 student tickets, and 89 adult tickets.
Let's represent the number of children as "C", the number of students as "S", and the number of adults as "A".
We know that:
A = 0.5C (there are half as many adults as students)
C + S + A = 363 (the total number of people attending)
5C + 7S + 12A = 2630 (the total ticket sales)
We can use the first equation to substitute for A in the second and third equations:
C + S + 0.5C = 363
1.5C + S = 363
5C + 7S + 12(0.5C) = 2630
5C + 7S + 6C = 2630
11C + 7S = 2630
Now we have two equations with two variables (1.5C + S = 363 and 11C + 7S = 2630). We can solve for one variable in terms of the other in the first equation for tickets:
S = 363 - 1.5C
Substitute this expression for S in the second equation:
11C + 7(363 - 1.5C) = 2630
Simplify and solve for C:
11C + 2541 - 10.5C = 2630
0.5C = 89
C = 178
Now we can substitute this value for C in the expression we found for S:
S = 363 - 1.5(178)
S = 99
And we can use the first equation we found to solve for A:
A = 0.5C
A = 0.5(178)
A = 89
Thus, the theater sold 178 children's tickets, 99 student tickets, and 89 adult tickets.
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the amount of income spent on housing is an important component of the cost of living. the total costs of housing for homeowners might include mortgage payments, property taxes, and utility costs (water, heat, electricity). an economist selected a sample of 20 homeowners in new england and then calculated these total housing costs as a percent of monthly income, 5 years ago and now. the information is reported in the table. is it reasonable to conclude the percent is less now than 5 years ago? homeowner five years ago now homeowner five years ago now 1 17 % 10 % 11 35 % 32 % 2 20 39 12 16 32 3 29 37 13 23 21 4 43 27 14 33 12 5 36 12 15 44 40 6 43 41 16 44 42 7 45 24 17 28 22 8 19 26 18 29 19 9 49 28 19 39 35 10 49 26 20 22 12
Yes ,it is reasonable to conclude that percent of income spent on housing for the sample of homeowners is less now than 5 years ago.
Sample size = 20
Condition to check it is reasonable to conclude percent of income spent on housing is less now than 5 years ago.
Compare the average percent of ,
Income spent on housing for sample of homeowners 5 years ago to Income spent on housing for same sample of homeowners now.
Calculating the mean percent of income spent on housing for the sample of homeowners 5 years ago,
(17+20+29+43+36+43+45+19+49+49+35+16+23+33+44+44+28+29+39+22)/20
= 33.15%
Calculating the mean percent of income spent on housing for the same sample of homeowners now,
=(10+39+37+27+12+41+24+26+28+26+32+21+12+40+42+22+19+35+12)/20 = 25.25%
It can be observed that the mean percent of income spent on housing for the sample of homeowners 5 years ago is higher .
Compare to mean percent of income spent on housing for the same sample of homeowners now.
Therefore, it is reasonable to conclude that percent of income spent on housing is less now than 5 years ago for sample of homeowners .
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The above question is incomplete, the complete question is:
The amount of income spent on housing is an important component of the cost of living. the total costs of housing for homeowners might include mortgage payments, property taxes, and utility costs (water, heat, electricity). an economist selected a sample of 20 homeowners in new En-gl-and and then calculated these total housing costs as a percent of monthly income, 5 years ago and now. the information is reported in the table. is it reasonable to conclude the percent is less now than 5 years ago?
Homeowner five years ago now
H-o-lt 17 % 10 %
Pi-e-rse 20 39
Me-re-ni-ck 29 37
La-n-oue 43 27
F-ag-an 36 12
Bo-bko 43 41
Ki-p-pert 45 24
Sa-n- Ro-man 19 26
Kur-im-sky 49 28
D-av-ison 49 26
Lo-zi-er 35 % 32 %
Cie-sli-nski 16 32
Row-a-tti 23 21
Kop-pel 33 12
Ru-m-sey 44 40
Mc-Gin-nis 44 42
Pie-rce 28 22
Ro-ll 29 19
La-ng 39 35
Mil-ler 22 12
Let f be the function with derivative f'(x)= x^3 - 3x - 2. Which of the following statements is true?
To determine the function f(x), we integrate the given derivative f'(x) as follows:
[tex]∫ f'(x) dx = ∫ (x^3 - 3x - 2) dxf(x) =[/tex]
[tex](1/4)x^4 - (3/2)x^2 - 2x + C[/tex]
where C is the constant of integration.
To find the value of C, we can use a point of the function. Let's say that f(0) = 5. Then we have:
[tex]5 = (1/4)(0)^4 - (3/2)(0)^2 - 2(0) + C[/tex]
[tex]C = 5[/tex]
Therefore, the function f(x) is:
[tex]f(x) = (1/4)x^4 - (3/2)x^2 - 2x + 5[/tex]
To answer the second question about retirement savings, we need more information about the retirement goals such as the desired retirement income, length of retirement, current age, and expected rate of return. Without this information, we cannot calculate the amount needed to be deposited each year.
Answer:
B. is true
Step-by-step explanation:
For turning points f'(x) = 0, so
x^3 - 3x - 2 = 0
f(-1) = -1 + 3 - 2 = 0
f(2) = 8 - 6 - 2 = 0
At = -1, f"(x) = 3x^2 - 3 = 3(1) - 3 = 0 so this is a point of inflection
and at x = 2 f"(x) = 3(-2)^2 - 3 = 9 so this is a minimum.
So there ia a point on inflection at x = 0 and a minimum at x = 2
g true or false? if a22 is a real matrix, then a can have two real eigenvalues, two non-real complex eigenvalues, or one real eigenvalue and one non-real complex eigenvalue. 1
The statement " if a 2×2 is a real matrix, then a can have two real eigenvalues, two non-real complex eigenvalues, or one real eigenvalue and one non-real complex eigenvalue" is false because a 2x2 real matrix cannot have one real eigenvalue and one non-real complex eigenvalue due to the fact that the eigenvalues of a real matrix always come in complex conjugate pairs.
A 2x2 real matrix can have two real eigenvalues or two non-real complex eigenvalues, but it is not possible for it to have one real eigenvalue and one non-real complex eigenvalue. This is because the eigenvalues of a real matrix always come in complex conjugate pairs.
This property can be proven by observing that the characteristic polynomial of a real matrix has real coefficients, and complex roots of a polynomial with real coefficients always come in complex conjugate pairs. Therefore, a 2x2 real matrix can have either two real eigenvalues or two non-real complex eigenvalues, but not one of each.
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