Answers
The solution of the differential equation y(3)=29/9.
Given a differential equation y'+(y/x)=x for y(1)=1.
A differential equation is an equation that contains at least one derivative of an unknown function, either a normal differential equation or a partial differential equation.
We will solve this differential equation by using the integrating factor method.
Firstly, we will find the integrating factors by using the formula, we get
[tex]I.F=e^{\int Pdx}[/tex]
Here P=1/x and substitute this in the formula, we get
[tex]\begin{aligned}I.F.&=e^{\int \frac{1}{x}dx}\\ &=e^{\log x}\\ &=x\end[/tex]
Now, we will substitute the values in the formula
[tex]y\times I.F.=\int Q\times I.F.dx[/tex]
Here Q=x an substitute this in the above formula, we get
[tex]\begin{aligned}y\times x&=\int x\times xdx\\ yx&=\int x^{2} dx\\ yx&=\frac{x^3}{3}+C\end[/tex]
Further, we will divide both sides with x, we get
y=(x²/3)+(C/x) ......(1)
Given that y(1)=1 means when x=1 then y=1 so substitute this in equation (1) to find the value of C, we get
1=(1/3)+C
1-(1/3)=C
2/3=C
Substitute the value of C in equation (1), we get
y=(x²/3)+(2/(3x))
Now find the value of y when x=3, we get
y=(9/3)+(2/9)
y=3+(2/9)
y=29/9
Hence, in the differential equation y'+(y/x)=x for y(1)=1 then y(3) is 29/9.
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Related Questions
A given triangle has vertices at (-3, 1) (2, 4) and (5, -3). If the triangle were rotated 360 degrees clockwise, the new coordinates would be:
Answers
New x-coordinate = x * cos(θ) - y * sin(θ)
New y-coordinate = x * sin(θ) + y * cos(θ)
Since we want to rotate the entire triangle 360 degrees, each point will be transformed using these formulas.
Let's calculate the new coordinates for each vertex:
For the first vertex (-3, 1):
New x-coordinate = -3 * cos(360°) - 1 * sin(360°) = -3 * 1 - 1 * 0 = -3
New y-coordinate = -3 * sin(360°) + 1 * cos(360°) = -3 * 0 + 1 * 1 = 1
For the second vertex (2, 4):
New x-coordinate = 2 * cos(360°) - 4 * sin(360°) = 2 * 1 - 4 * 0 = 2
New y-coordinate = 2 * sin(360°) + 4 * cos(360°) = 2 * 0 + 4 * 1 = 4
For the third vertex (5, -3):
New x-coordinate = 5 * cos(360°) - (-3) * sin(360°) = 5 * 1 - (-3) * 0 = 5
New y-coordinate = 5 * sin(360°) + (-3) * cos(360°) = 5 * 0 + (-3) * 1 = -3
Therefore, after rotating the triangle 360 degrees clockwise, the new coordinates would be (-3, 1), (2, 4), and (5, -3), which are the same as the original coordinates.
Use the given data to make a box-and-whisker plot.
5, 4, 16, 21, 10, 6, 15
Answers
Answer:
To make a box-and-whisker plot, we first need to order the data from smallest to largest:
4, 5, 6, 10, 15, 16, 21
Next, we can find the median, which is the middle value of the ordered data set. In this case, the median is 10.
Then, we can find the lower quartile (Q1) and the upper quartile (Q3). The lower quartile is the median of the lower half of the data set, and the upper quartile is the median of the upper half of the data set. To find Q1 and Q3, we can split the data set into two halves:
4, 5, 6, 10 and 15, 16, 21
The median of the lower half is (5+6)/2 = 5.5, and the median of the upper half is (16+21)/2 = 18.5. Therefore, Q1 = 5.5 and Q3 = 18.5.
Now we can draw the box-and-whisker plot. The box represents the middle 50% of the data, with the bottom of the box at Q1 (5.5), the top of the box at Q3 (18.5), and the line inside the box at the median (10). The whiskers extend from the box to the smallest and largest values inthe data set that are not outliers. To determine whether there are outliers, we can use the following formula:
Outlier = Q1 - 1.5 * (Q3 - Q1) or Q3 + 1.5 * (Q3 - Q1)
Plugging in the values, we get:
Outlier = 5.5 - 1.5 * (18.5 - 5.5) = -13.25 or Outlier = 18.5 + 1.5 * (18.5 - 5.5) = 37.25
Since there are no values in the data set that are less than -13.25 or greater than 37.25, there are no outliers.
Therefore, the box-and-whisker plot for the given data is:
| 4
___|___
| |
| |
| 5.5 |-----
| | |
| | |
|------------| 10 |
| | |
| 15.5 |------
| | |
___|___
| 21
The box spans from Q1 (5.5) to Q3 (18.5), with the median line inside the box at 10. The whiskers extend from the box to the minimum value of 4 and the maximum value of 21, since there are no outliers.
Hope this helps!
need help on this math question
Answers
The expression which gives the number of months passed in terms of p is √300 -0.5p
Subject of formulaWe make t the subject of the formula in the expression
p = 600 - 2t²
subtract 600 from both sides in other to isolate 2t²
p - 600 = 600 - 2t² - 600
p - 600 = -2t²
Rearrange the equation
2t² = 600 - p
Divide both sides by 2 in other to isolate t²
2t²/2 = 600/2 - p/2
t² = 300 - 0.5p
Take the Square root of both sides in other to isolate t
√t² = √300 - 0.5p
t = √300 - 0.5p
Hence, the correct expression which gives the number of months t in terms of price p is t = √300 - 0.5p
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what is an equation in point-slope form of the line shown in the graph, using the point (-2,-1)
Answers
The equation of the line in point-slope form, using the point (-2, -1), is: y + 1 = -2(x + 2)
How to Write the Equation of a Line in Point-slope Form?The equation of a line in point-slope form is given by y - y₁ = m(x - x₁), where (x₁, y₁) represents a point on the line and m represents the slope.
Given:
Slope (m) = rise/run = -2
Point (-2, -1)
Substituting the given slope and point into the equation, we have:
y - (-1) = -2(x - (-2))
y + 1 = -2(x + 2)
Therefore, the equation of the line is: y + 1 = -2(x + 2).
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The sum of two numbers is -6. One number is 14 more than the other one. Find the numbers.
Answers
Answer:
The two numbers are -10 and 4
Step-by-step explanation:
[tex]x+y=-6\\y=14+x\\\\x+(14+x)=-6\\2x+14=-6\\2x=-20\\x=-10\\\\x+y=-6\\-10+y=-6\\y=4[/tex]
2
Select the correct answer from each drop-down menu.
Triangle ABC is shown in the coordinate plane. It is translated 6 units left and 4 units down. Then it is dilated by a scale factor of 3 centered about the
origin. Complete the statements.
-6 -4
v.
6
4-
2-
lo
-2-
-4-
-6-
The translation of triangle ABC
A
2
с
6
B
Reset
The dilation of triangle ABC
Next
Answers
For the transformations of the triangle, we have that:
1. The translation of triangle ABC preserves side lengths and angles.
2. The dilation of triangle ABC preserves angles but not side lengths.
Hence the correct options are, respectively, C and B.
A translation involves shift left, shift right, shift down, or shift up, meaning that just the coordinates of the figure change, not the angles or the side lengths,
1. The translation of triangle ABC preserves side lengths and angles.
For a dilation, the coordinates of the vertices are multiplied by a constant, hence the side lengths change while the angles remain constant, thus:
2. The dilation of triangle ABC preserves angles but not side lengths.
Hence the correct options are, respectively, C and B.
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At a conference of educators, there ore 189 people total, and 128 are teachers. If the rest are principals, then how many principals are present?
Answers
Answer:
61 principals
Step-by-step explanation:
189 total - 128 teachers = 61 principals
If someone can explain this to me that would be amazing, I have a college algebra final tomorrow and I am rly confused. I don’t know how wide to make it, if someone can explain this I will give 100 points and brainliest
Answers
Move the entire graph 3 units to the right. This means you shift the curve horizontally to the right by 3 units.
Flip the graph upside down by reflecting it in the x-axis. This means the U-shaped curve will now open downward.
Squish or compress the graph vertically by a factor of 3. This means the curve will become narrower and steeper.
Move the graph 3 units up. This means you shift the curve vertically upward by 3 units.
This results in a graph that is a downward-opening parabola centered at the point (3, 3), with a vertical compression and vertical translation.
The first image is y=x^2 the second is how it should look
What is the area of the shaded region?
Answers
Answer:
80 mm²
Step-by-step explanation:
Even for slanted triangles, the area is base x half height! The base is 10 mm, half the height is 8 mm, so the area is 8x10=80 mm.
Zara's car depreciates at a rate of 14% per year.
By what percentage has Zara's car depreciated after 5 years?
Give your answer to the nearest whole number.
NO SPAM.
Answers
Answer:
53%
Step-by-step explanation:
You want to know the depreciation of car value after 5 years if it is 14% per year.
MultiplierThe multiplier of value each year is ...
1 + growth rate = 1 + (-14%) = 0.86
When the value has been multiplied by that 5 times, it has been multiplied by ...
0.86^5 ≈ 0.4704
This represents a change in value from the original value of ...
0.4704 -1 = -0.5296 = -52.96% ≈ -53%
Zara's car has depreciated by 53% after 5 years.
<95141404393>
Volume of Cone ___ mm3?
Answers
The volume of the cone is 301. 44mm³
How to determine the valueFirst, we need to know the formula
the formula that is used for calculating the volume of a cone is expressed as;
V = πr²h/3
Such that the parameters are;
V is the volume r is the radiush is the height of the coneSubstitute the values, we have;
Volume = 3.14 × 6² × 8/3
Multiply the values, we have;
Volume = 904.32/3
Divide the values
Volume = 301. 44mm³
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A triangle has sides with lengths of 3 miles, 4 miles, and 6 miles. Is it a right triangle
Answers
Yes, the given triangle is a right triangle.
We have,
This can be determined by applying the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
Now,
In this case, the squares of the side lengths are:
3² = 9
4² = 16
6² = 36
If the sum of the squares of the two shorter sides (9 + 16 = 25) is equal to the square of the longest side (36), then the triangle is a right triangle.
Thus,
In this case, since 25 is indeed equal to 36, the triangle with side lengths 3 miles, 4 miles, and 6 miles is a right triangle.
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hey here is my algebra 13 homework, can someone please help! heres screenshot. QUICK PLSS
Answers
The graph of the given exponential function f(x) = 4(0.5)ˣ + 2 is g(x) = 5(1/6)ˣ + 2
Hence, the correct option is A.
An exponential function is a mathematical function of the form f(x) = aˣ, where "a" is a positive constant known as the base, and "x" is the independent variable. The exponential function is defined for all real values of x, and the output values of the function are always positive.
The exponential function is characterized by its unique property of having a constant rate of change with respect to its own value. This means that the function increases (or decreases) by a constant factor each time the independent variable is increased by a fixed amount.
For the given exponential function g(x) = 5(1/6)ˣ + 2 as the graph of f(x) and g(x) are same.
Hence, the correct option is A.
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Can someone answer and provide an explanation for these problems? Find the midpoint of the line segment with the given endpoints.
Answers
52) The midpoint of the line segment is (-0.5, 9).
53) The midpoint of the line segment is (3, -8.5).
52) To find the midpoint of the line segment with endpoints (8,9) and (-9,9), we can use the midpoint formula. The midpoint formula states that the coordinates of the midpoint (x, y) are the average of the corresponding coordinates of the endpoints.
For the x-coordinate:
(x₁ + x₂) / 2 = (8 + (-9)) / 2 = (-1) / 2 = -0.5
For the y-coordinate:
(y₁ + y₂) / 2 = (9 + 9) / 2 = 18 / 2 = 9
Therefore, the midpoint of the line segment is (-0.5, 9).
53) To find the midpoint of the line segment with endpoints (8,-10) and (-2,-7), we apply the same process using the midpoint formula.
For the x-coordinate:
(x₁ + x₂) / 2 = (8 + (-2)) / 2 = 6 / 2 = 3
For the y-coordinate:
(y₁ + y₂) / 2 = (-10 + (-7)) / 2 = -17 / 2 = -8.5
Hence, the midpoint of the line segment is (3, -8.5).
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Please help I don’t understand thank you have a blessed day
Answers
By evaluation, the values of the rational function are f(11) = - 1 / 12 and f(- 2) = 1, respectively.
How to evaluate a rational function
In this problem we find the case of a rational function that must be evaluated at two different x-values: x = 11, x = - 2. The procedure is summarized below:
Write the entire expression.Substitute on x with the given value.Simplify the expression by algebra properties.Now we proceed to present the complete procedure:
f(11) = 11 / (11 + 1) - 1
f(11) = 11 / 12 - 1
f(11) = 11 / 12 - 12 / 12
f(11) = (11 - 12) / 12
f(11) = - 1 / 12
f(- 2) = (- 2) / (- 2 + 1) - 1
f(- 2) = - 2 / (- 1) - 1
f(- 2) = 2 - 1
f(- 2) = 1
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50 Points! Multiple choice algebra question. Photo attached. Thank you!
Answers
The exact value of the cosine of θ is given as follows:
a) 3/5.
What are the trigonometric ratios?The three trigonometric ratios are the sine, the cosine and the tangent, and they are obtained according to the rules presented as follows:
Sine of angle = opposite side/hypotenuse.Cosine of angle = adjacent side/hypotenuse.Tangent of angle = opposite side/adjacent side = sine/cosine.Applying the Pythagorean Theorem, the hypotenuse length is given as follows:
x² = 6² + (-8)²
x² = 100
x = 10.
As the x-coordinate is used for the cosine, the cosine is given as follows:
cos(θ) = 6/10
cos(θ) = 3/5.
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A rectangular prism measures 8 inches in width, 12 inches in length, and 4 inches in height. What is the surface area of the prism?
Answers
Step-by-step explanation:
It has
two sides 8x12
two sides 12x4
two sides 4x8 total 352 in^2
Which graph has the same end behavior as f (x) = StartFraction 10 x cubed + x squared + 3 Over 2 x squared + 1 EndFraction?
On a coordinate plane, a line goes through (negative 2, negative 10), (0, 0), and (2, 10).
On a coordinate plane, a graph approaches the x-axis in quadrant 2, increases up to (0, 10), and then curves down and approaches the x-axis in quadrant 1.
On a coordinate plane, a parabola opens up. It goes through (negative 2, 8), has vertex (0, 1), and goes through (2, 8).
Answers
The function with the same end behavior as f(x) = (10x³ + x² + 3)/(2x² + 1) is given as follows:
On a coordinate plane, a line goes through (-2, -10), (0, 0), and (2, 10).
What is the end behavior of a function?The end behavior of a function refers to how the function behaves as the input variable approaches positive or negative infinity.
The function for this problem is given as follows:
f(x) = (10x³ + x² + 3)/(2x² + 1)
As we are looking for the end behavior, we look at the terms with the highest exponent, hence:
f(x) = 10x³.
Then the end behavior is given as follows:
As x -> - ∞, 10(-∞)³ -> -∞.As x -> ∞, 10(∞)³ -> ∞.Hence the function with the same end behavior is the increasing line, which is given by the first option.
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Tanya flew 2,299 miles the first year on the job. She flew 3,831 miles the second year. Tanya flew 1,510 more miles the third year than the first and second years combined. How many miles did Tanya fly the third year?
Please help me!!
Answers
Answer: 7,640
Step-by-step explanation: Because we know that she flew 2,299 the first year and 3,831 the second, we can add them together to get: 6,130. It says that she flew 1,510 more miles the third year than the first and second combined. Therefore: 6,130 + 1,510 = 7,640
To solve it, find the sum of the first 2 years,
2,299+3,831= 6,130
and then add the additional 1,510 to it,
6,130+1,510=7,640
A dentist was making note of her upcoming appointments with different aged patients and
the reasons for their visits.
Patients under 18
Patients 19-60
Patients over 60
Regular cleaning Broken tooth
Submit
5
3
3
2
6
2
What is the probability that a randomly selected appointment is not with a patient 19-60 or is
not for a broken tooth?
Simplify any fractions.
Answers
Probability that a randomly selected appointment is with patients over 60 years old and is for a regular cleaning = (5/21)
The probability of an event is calculated as the number of elements in that event divided by the total number of elements in the sample space.
For this question, we are told to find the probability that a randomly selected appointment is with patients over 60 years old and is for a regular cleaning.
Number of patients that are over 60 years and are for regular cleaning = 10
Total number of patients = 2 + 8 + 6 + 4 + 10 + 12 = 42
Probability that a randomly selected appointment is with patients over 60 years old and is for a regular cleaning = (10/42)
= (5/21)
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Help me solve this problem please
Answers
Answer:
450°
Step-by-step explanation:
Stuck on this and cant move along until I get it correct. Please help.
Fill in the missing value.
The measure of angle F is ___°
Answers
The solution is: the measure of angles 2, 3, and 4 are:
m∠2 = 95°
m∠3 = 85°
m∠4 = 95°
Vertical Angle Theorem:
When two straight lines intersect, the opposite vertical angles formed are always equal (congruent) to each other.
Therefore,
m∠3 = 85°
m∠2 = m∠4
Linear pair:
Two adjacent angles which, when combined, form a line, so the sum of a linear pair is 180°.
Therefore,
85° + m∠2 = 180°
m∠3 + m∠4 = 180°
If 85° + m∠2 = 180°
⇒ m∠2 = 180° - 85°
⇒ m∠2 = 95°
As m∠2 = m∠4 then m∠4 = 95°
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complete question:
Find the measure of angles 2, 3, and 4. input the measures then click done. i-ready
please help i missed a day of class i cant figure this out
how do I change 1/4 to a percentage .
Answers
Answer: You divide it, get your answer, move the decimal point to the right twice, and that's you're percent.
Step-by-step explanation:
1/4=.25
.25 > 25%
Answer:
The answer is 25%
Step-by-step explanation:
To change decimal to percentage multiply by 100
1/4×100%
=25%
Solve for y
Finally, solve the proportion for y.
y/8 = 10/y
y = √ [?]
M
Enter the square root NOT simplified.
=
Answers
The square root of y is not simplified and can be represented as:
y = √([tex]2^4 \times[/tex] 5) or y = 4√5.
To solve the proportion y/8 = 10/y for y, we can cross-multiply:
y [tex]\times[/tex] y = 8 [tex]\times[/tex]10
This simplifies to:
[tex]y^2[/tex] = 80
To solve for y, we can take the square root of both sides of the equation:
[tex]\sqrt{y^2[/tex] = √80
Since the square root of [tex]y^2[/tex] is simply y, we have:
y = √80
The square root of 80 can be further simplified by breaking it down into its prime factors. Since 80 can be expressed as 2 [tex]\times[/tex]2 [tex]\times[/tex]2 [tex]\times[/tex]2 [tex]\times[/tex]5, we can simplify the square root:
y = [tex]\sqrt(2 \times 2 \times 2 \times 2 \times 5)[/tex]
Therefore, the square root of y is not simplified and can be represented as:
y = [tex]\sqrt{(2^4 \times 5)[/tex] or y = 4√5.
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Answer:
[tex]y = \sqrt{80\\} =4\sqrt{5}[/tex]
Step-by-step explanation:
A jewlary shop sells 240 necklaces in a month 180 of the necklaces were sold in a via the shops website the rest were sold in a high street shop .
Work out the ratio of online sales to shop sales
Give ur answers in its simplest form
Answers
180 were sold online and 60 were sold in shops
the ratio in its simplest form would be 3:1
With a short time remaining in the day, a delivery driver has time to make deliveries at seven locations among the 8 locations remaining. How many different routes are possible? Question content area bottom Part 1 There are enter your response here possible different routes. (Simplify your answer.)
Answers
The number of different routes possible for the delivery driver, given the time to make deliveries at 7 out of the remaining 8 locations, is 8.
The number of different routes possible for the delivery driver can be calculated using combinatorics. Specifically, we need to determine the number of ways to choose 7 locations out of the remaining 8 locations. This can be found using the combination formula.
The combination formula, also known as "n choose k," is given by the expression C(n, k) = n! / (k! * (n-k)!), where n is the total number of items to choose from and k is the number of items to choose.
In this case, there are 8 locations remaining and the driver needs to choose 7 of them. Applying the combination formula, we have C(8, 7) = 8! / (7! * (8-7)!) = 8! / (7! * 1!) = 8.
Therefore, there are 8 possible different routes for the delivery driver to make deliveries at the remaining 8 locations, given that the driver has time to make deliveries at 7 of them.
To simplify the answer, we can state that there are 8 possible different routes for the driver.
In summary, the number of different routes possible for the delivery driver, given the time to make deliveries at 7 out of the remaining 8 locations, is 8.
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if x+y+z=27 and x+y=17 then Z=
Answers
Answer:
Step-by-step explanation:
x+y+z=27
x+y=17
z=?
since x+y=17
then
17+z=27
z=27-17
z=10
5. (6 pts pts) The displacement of a spring vibrating in damped harmonic motion is given by
y = 4e-3t sin(2πt)
Find the times when the spring is at its equilibrium position (y = 0). '
Answers
The times when the spring is at its equilibrium position (y = 0) are t = 0, 1/2, 1, 3/2, ... and so on.
To find the times when the spring is at its equilibrium position (y = 0), we can set the displacement equation equal to zero and solve for t:
[tex]4e^{(-3t)[/tex] sin(2πt) = 0
Since the product of two factors is zero if and only if at least one of the factors is zero, we have two cases to consider:
4[tex]e^{(-3t)[/tex] = 0
This equation has no solution since [tex]e^{(-3t)[/tex] is always positive and nonzero.
sin(2πt) = 0
The sine function is zero at integer multiples of π.
So, we can set 2πt equal to integer multiples of π:
2πt = 0, π, 2π, 3π, ...
Solving for t in each case, we get:
t = 0, 1/2, 1, 3/2, ...
Therefore, the times when the spring is at its equilibrium position (y = 0) are t = 0, 1/2, 1, 3/2, ... and so on.
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who can help me solve this pls need help
Answers
Answer:
Step-by-step explanation:
1+3=4
4x4=16
10/2=5
16-5=11
answer is 11
What is the apparent solution set for the system of equations:
x ^ 2 + 3x - 2;
- x ^ 2 + 2x + 4
A) {(- 3.5, 0) , (-1.2, 0), (0.5, 0), (3.3, 0)}
B) {(0, 2), (1.5, - 4.25)}
C) {(- 2, - 4), (1.5, 4.75)}
D) {(0, 4), (1, 5)}
Answers
There are no apparent solutions that satisfy both equations.
We have,
The system of equations is:
x² + 3x - 2 = 0
-x² + 2x + 4 = 0
To find the solutions, we can solve each equation separately:
x² + 3x - 2 = 0
This equation can be factored as:
(x + 2)(x - 1) = 0
Setting each factor equal to zero gives:
x + 2 = 0
x = -2
And,
x - 1 = 0
x = 1
-x² + 2x + 4 = 0
This equation does not factor easily, so we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
For this equation, a = -1, b = 2, and c = 4.
Plugging in the values into the quadratic formula:
x = (-2 ± √(2² - 4(-1)(4))) / (2(-1))
x = (-2 ± √(4 + 16)) / (-2)
x = (-2 ± √20) / (-2)
Simplifying further:
x = (-2 ± 2√5) / (-2)
x = 1 ± √5
Now,
let's check which of these solutions satisfies both equations:
For x = -2:
(-2)² + 3(-2) - 2 = 4 - 6 - 2 = -4 ≠ 0
-(-2)² + 2(-2) + 4 = -4 - 4 + 4 = -4 ≠ 0
x = -2 does not satisfy both equations.
For x = 1:
1² + 3(1) - 2 = 1 + 3 - 2 = 2 ≠ 0
-(1)² + 2(1) + 4 = -1 + 2 + 4 = 5 ≠ 0
x = 1 does not satisfy both equations.
For x = 1 ± √5:
We can substitute these values into the equations, but it can be seen that they will not satisfy both equations either.
Therefore,
There are no apparent solutions that satisfy both equations.
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