The required answers are:
a. The linear approximating polynomial at a = 1: P₁(x) = x - 2
b. The quadratic approximating polynomial at a = 1: P₂(x) = -3 + 3x - x²
c. Appromating the given quantity -1/1.06 for P₁(x) and P₂(x): -3.06 and -7.30 respectively.
Using Taylor polynomial series, the required approximating polynomials are calculated.
What is the Taylor polynomial series?The Taylor polynomial series is
Pₙ(x) = ∑ [tex]\frac{f^n(a)}{n!}[/tex](x - a)ⁿ; limits from 0 to n
Where fⁿ(a) is the nth derivation of f(x) at a
The first order(linear) Taylor polynomial series is
P₁(x) = f(a) + [tex]\frac{f'(a)}{1!}[/tex](x - a)¹
The second order(quadratic) Taylor polynomial series is
P₂(x) = f(a) + [tex]\frac{f'(a)}{1!}[/tex](x - a) + [tex]\frac{f''(a)}{2!}[/tex](x - a)²
Calculation:The given function is f(x) = -1/x at a = 1
f(a) = f(1) = -1
f'(x) = -(-1/x²) = 1/x²; f'(a) = f'(1) = 1
f''(x) = -2/x³; f''(a) = f''(1) = -2
a. Linear approximating polynomial:
From the Taylor series, we have
P₁(x) = f(a) + [tex]\frac{f'(a)}{1!}[/tex](x - a)¹
On substituting,
P₁(x) = -1 + (1/1!)(x - 1)¹
= -1 + (x - 1)
= x - 2
b. Quadratic approximating polynomial:
From the Taylor series, we have
P₂(x) = f(a) + [tex]\frac{f'(a)}{1!}[/tex](x - a) + [tex]\frac{f''(a)}{2!}[/tex](x - a)²
On substituting,
P₂(x) = f(1) + (1/1!)(x - 1) + (-2/2!)(x - 1)²
= -1 + (x - 1) - (x - 1)²
= -1 + x - 1 - (x² - 2x + 1)
= x - 2 - x² + 2x - 1
= -3 + 3x - x²
c. Approximating the given quantity -1/1.06:
Substituting x = -1.06 in P₁(x) and P₂(x),
P₁(x) = x - 2
⇒ P₁(-1.06) = (-1.06) - 2 = -3.06
P₂(x) = -3 + 3x - x²
⇒ P₂(-1.06) = -3 + 3(-1.06) - (-1.06)² = -7.30
Therefore, the approximating polynomials are calculated by using Taylor polynomial series.
Disclaimer: The given question is incomplete. Here is the complete question.
Question:
a. Find the linear approximating polynomial for the following function centered at the given point a.
b. Find the quadratic approximating polynomial for the following function centered at the given point a.
c. Use the polynomials obtained in parts a. and b. to approximate the given quantity.
f(x) = -1/x , a = 1; approximate -1/1.06
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Question 13 (5 points
if the current mortgage rates are high and expected to drop in the near future, which of the following loans do you not want to get?
fixed rate
15-year
variable rate
30-year
When the current mortgage rates are high and expected to drop in the near future, the loans that will be gotten is one that has C. variable rate.
What is mortgage?It should be noted that a mortgage simply means a loan that's made by a bank to purchase a house.
In this case, the current mortgage rates are high and expected to drop in the near future, the loans that will be gotten is one that has a variable rate. This is because once the mortgage rate decreases, one will pay a lesser amount.
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1.an examination in three subjects algebra,biology ,chemistry was taken by 41 students. the following table shows how many students failed in each single subject and their various combinations: subject a b c ab ac bc abc failed 12 5 8 2 6 3 1 how many students failed at least in one subject?
HELP NEEDED ASAP WILL GIVE BRAINLIEST!!!!!!!!!!!!!!!!!!!
Consider function h.
What is the range of function h?
Answer: (-[tex]\infty[/tex], [tex]\infty[/tex])
Step-by-step explanation:
The following graph is a graph with a slant asymptote, which is an asymptote that isn't horizontal or vertical (i.e. doesn't have a slope of 0 or [tex]\infty[/tex]).
To review, an asymptote is a line that the graph of a function approaches, but will never touch. Normally, a horizontal asymptote would limit the range as the graph can never cross a certain y-value.
However, there is no such y-value that must not be crossed in a slant asymptote; therefore, there is not limitation to the range. The range would be all real numbers.
Hence, the range of function h is (-[tex]\infty[/tex], [tex]\infty[/tex]).
Help me solve this image please!!!
Based on the information, Christian would have $5525.5 of an annuity.
How to calculate the annuity?According to the given information, the number of coffees per week is 3 then, per month is 3x4 = 12
Each coffee is $4.5. Then monthly expenditure for coffees is 12 x 4.5 = $54
Rate of interest r = 1.6% = 1.6/100 = 0.016 and for monthly compounding r = 0.016/12 = 0.00133
n = number of payments = 8 x 12 = 96
We can use the formula for finding the future value as below
FV = C x [ ( 1 + r )n-1 ] / ( r )
FV = 54 x [ ( 1 + 0.00133 )96 – 1 ] / (0.00133)
= 54 x [ (1.13609 - 1)] / (0.00133)
= 54 x 0.13609 / (0.00133)
= 54 x 102.3233
= 5525.5
Therefore Christian would have $5525.5 of the annuity.
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Rectangle 1 has length x and width y. Rectangle 2 is made by multiplying each dimension of Rectangle 1 by a factor of k, where k > 0. Are Rectangle 1 and Rectangle 2 similar? Why or why not? Write a paragraph proof to show that the perimeter of Rectangle 2 is k times the perimeter of Rectangle 1. Write a paragraph proof to show that the area of Rectangle 2 is times the area of Rectangle 1. Answer:
Yes, the two rectangles are similar, because rectangle 2 is a dilation of rectangle 1.
Are the two rectangles similar?
We know that rectangle 1 has dimensions L and W.
And rectangle 2 is made by multiplying the dimensions of rectangle 1 by a factor k > 0.
Then, rectangle 2 is just a dilation of rectangle 1, this means that in fact, the two rectangles are similar by definition.
Then:
Dimensions of rectangle 1:
Length = LWidth = W.Perimeter = 2*(W + L)Area = W*LFor rectangle 2:
Length = k*LWidth = k*WPerimeter = 2*(k*L + k*W) = k*(2*(L + W))Area = (k*L)*(k*W) = k²(L*W)Above we can see that the perimeter of rectangle 2 is k times the perimeter of rectangle 1, and the area of rectangle 2 is k squared times the area of the rectangle 1.
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Given the following vectors in component form:
r = ⟨2, 3⟩
s = ⟨5, –3⟩
t = ⟨–8, 6⟩
Check all expressions whose sum represents the same vector as (r + s) + t.
LeftAngleBracket 0, 7 RightAngleBracket + LeftAngleBracket negative 8, 6
RightAngleBracket LeftAngleBracket 7, 0 RightAngleBracket + LeftAngleBracket negative 8, 6 RightAngleBracket LeftAngleBracket 2, 3 RightAngleBracket + LeftAngleBracket negative 3, 3 RightAngleBracket LeftAngleBracket 2, 3 RightAngleBracket + LeftAngleBracket 3, negative 3 RightAngleBracket LeftAngleBracket negative 6, 9 RightAngleBracket + LeftAngleBracket 5, negative 3 RightAngleBracket
All expressions whose sum represents the same vector as (r + s) + t. is Option b,c,e
What are the expressions that equals (r + s) + t. ?Where
r= (2,3)
s= (5,-3)
t= (-8,6)
Generally, the equation for statement is mathematically given as
(7,0)+(-8,6) ,(2,3)+(-3,3) and (-6,9)+(5,-3) are equal to (r+s)+t
Therefore
x=(2,3)+(5,-3)
x=(7,0)
Now we can calculate (r+s)+t as
(7,0)+(-8,6)
(7-8,0+6)
(-1,6)
For (b)
x=(7,0)+(-8,6)
x=(-1,6)
in this scenario x expressions is equal to (r+s)+t
For (c)
x=(2,3)+(-3,3,)
x=(-1,6)
in this scenario x expressions is equal to (r+s)+t
For (e)
[tex]x=(-6,9)+(5,-3)[/tex]
x=(-1,6)
in this scenario x expressions is equal to (r+s)+t
In conclusion, all expressions whose sum represents the same vector as (r + s) + t. is b,c,e
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If the base angles of a trapezoid are congruent, what can be said about the diagonals of the trapezoid?
They intersect at the median.
They split the trapezoid into 4 congruent parts.
They are congruent.
They bisect each other.
If the base angles of a trapezoid are congruent, the diagonals are congruent.
Properties of a trapezium?A trapezium is a quadrilateral and the sum of the interior angles is 360 degrees. Therefore, the properties of a trapezium includes the following:
The bases are parallel by definition.Each lower base angle is supplementary to the upper base angle on the same side.If the base angles are congruent, therefore, the trapezium is an isosceles trapezium. This means the diagonals are also congruent.learn more on trapezium here: https://brainly.com/question/9422487
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30men can do a piece of work in 40 days how long will it take to complete the work by 25 men
Step-by-step explanation:
men × days=men1×days1
30×40=40×x
x=30days
The ages of all employees at a small convenience store are 36, 32, 38, and 28. What is standard deviation of ages for this population
The standard deviation is approximately 3.84
Step 1: Find the mean
Find the sum of the values, and divide by number of terms.
Mean (average) = (36+32+38+28)/4 = 134/4 = 33.5
We must use Standard Deviation formula which is attached as an image below.
X is the mean here, so lets plug in 33.5
Step 2. Distance from the mean squared |x-mean∣²
For each data point, we will plug in the numbers.
|36-33.5|²=2.5²=6.25
|32-33.5|²=1.5²=2.25
|38-33.5|²=4.5²=20.25
|28-33.5|²=5.5²=30.25
Step 3: Find the sum (∑)
We find the sum here by adding all the distances from the mean squared
∑|x-mean|²=59
Step 4: Divide the sum by the amount of points, 4
59/4=14.75
Step 5: Take the square root of step 4
[tex]\sqrt{14.75}[/tex] is about 3.84
The standard deviation is approximately 3.84
i really need hep with this question
(5x-7)-5(7x-12)+7=0
Answer:
x = 2
Step-by-step explanation:
Simplifying expressions:5x - 7 - 5(7x - 12) + 7 = 0
Distribute (-5) to each term of (7x - 12)
5x - 7 + 7x*(-5) + (-12)*(-5) + 7 = 0
5x - 7 - 35x + 60 + 7 = 0
5x - 35x - 7 + 60 +7 = 0
Combine like terms.
-30x + 60 = 0
Subtract 60 from both sides
-30x = -60
Divide both sides by (-30)
x = (-60)/(-30)
x = 2
Answer:
2
Step-by-step explanation:
Simplifying the Algebraic Equation.
( 5x - 7 ) -5 (7x - 12 ) + 7 = 0
Removing brackets.
5x - 7 - 35x + 60 + 7 = 0
Combining like terms.
5x - 35x + 60 + 7 - 7 = 0
- 30x + 60 = 0
- 30x = - 60
dividing bothsides by - 30
[tex] \frac{ - 30x}{ - 30} = \frac{ - 60}{ - 30} \\ \\ x \: = 2[/tex]
Find the magnitude of the
resultant vector.
(8, 10)
W
(7, -1)
[?]
Round to the nearest tenth.
17.49 is the magnitude of the resultant vector. This can be obtained by adding the vectors and finding magnitude.
Calculate the magnitude of the resultant vector:Resultant vector: vector that gives the combined effect of all the vectors. When we add two or more vectors, the outcome is the resultant vector.
For example,
Sum of vector (3,4) and vector (5,7)(3,4) + (5,7) = (3+5,4+7) = (8,11)
(8,11) is the resultant vector of vector (3,4) and vector (5,7)
Magnitude of a vector: the length of the vector. The magnitude of the vector a is denoted as ∥a∥.
Magnitude of a = (a₁, a₂)
∥a∥=[tex]\sqrt{a_{1}^{2} +a_{2}^{2} }[/tex]
In the given question,
v (8,10) and w (7, -1)
Adding the vectors,
resultant vector = (8,10) + (7, -1) = (15, 9)
Magnitude of a vector (a,b) = √a²+b²
Here, magnitude = √15²+9² = √225+81 = √306 ≈ 17.49
Hence 17.49 is the magnitude of the resultant vector.
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4z-10=10 what do i solve for z
Answer:
z = 5
Step-by-step explanation:
To solve for z, means to get z all by itself on one side of the equation. In your equation z has had two things done to it. First, z has been times by 4, then 10 is subtracted. So, you "un-do" that in reverse order.
FIRST, add 10. To reverse or "un-do" subtracting.
4z - 10 = 10
4z -10+10 = 10+10
4z = 20
Then, divide by 4 to reverse the multiplying.
4z/4 = 20/4
z = 5
see image.
Also, you can check you answer.
put in 5 in place of z and be sure that you get a true statement.
4(5) - 10 = 10
20 - 10 = 10
10 = 10 Checked!
Answer:
[tex]z=5[/tex]
Step-by-step explanation:
First, substitute in 5 for z: [tex]4(5) -10=10[/tex]
Since [tex]4*5=20[/tex], we can now simplify the equation to [tex]20-10=10[/tex].
Now, check the equation to make sure that the value of z is correct. Do this by subtracting [tex]20-10[/tex] to see if it results in a solution of 10.
Since the equation [tex]20-10=10[/tex] is equal/true, we can conclude that [tex]z=5[/tex].
Suppose that a rectangular piece of glass has a perimeter of 20 meters. if we multiply both its length and its width by 4, what will be the perimeter of the new rectangular piece of glass?
The perimeter of the new rectangular piece of glass is 80m.
What is the perimeter of a rectangle?The sum of all sides of a rectangle is the perimeter of the rectangle.A rectangle has two equal lengths and two equal widths.Therefore, the perimeter 'P' of a rectangle is the sum of two equal lengths and two equal widths. That is, 'P=2l+2w' where 'l' is the length and 'w' is the width of the rectangle.
Here, a rectangular piece of glass has a perimeter of 20 meters.
As we know the perimeter P of a rectangle is given by the formula,
P = 2l+2w...(1)
Therefore,
20 = 2l + 2w
20 = 2(l + w)
20/2 = l + w
10 = l + w ....(2)
Now, both the length and width of the rectangle are multiplied by 4,
That is,
The new length of the rectangle is, l = 4l
The new width of the rectangle is, w = 4w
Therefore by substituting this in the equation(1) we get,
P = 2(4l) + 2(4w)
P = 8l + 8w
P = 8(l+w)....(3)
Substitute equation (2) in (3),
That is,
P = 8*10
P = 80m
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PLEASE HELP CORRECT ANSWER ONLY PLS
Answer:
y = -1x + 5
Step-by-step explanation:
y = mx + c
y = gradient(x intercept) + y intercept
y = -1x + 5
what is the surface area of the pyramid?
Answer: SA = 445.05
Step-by-step explanation:
10 points please help me and i will mark brainlyist
Answer:
Min: 32
Lower Quartile: 36
Median: 42
Upper Quartile: 47
Max: 53
Interquartile: 11
Step-by-step explanation:
The min. is just the lowest number. The lower quartile is the number in the middle of the numbers below 42, the median is the number in the very middle, the upper quartile is the middle number above 41 and the interquartile is the upper quartile minus the lower quartile.
Find the measure of each interior angle of a regular octagon. (A regular polygon is one that has all the same sides and angles). [1]
The interior angle of a regular octagon is 135 degrees.
What is an octagon?An octagon is a polygon that has 8 sides. Therefore, a regular octagon is a octagon with all it sides and angles equal to each other.
Therefore, the sum of angles in an octagon is each interior angle multiply by the number of sides.
Therefore,
n = number of sides = 8
Hence,
interior angle = (n - 2)180 / n
interior angle = (8 - 2)180 / 8
interior angle = 6 × 180 / 8
interior angle = 1080 / 8 = 135°
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A 10-foot board is to be cut into 3 pieces. Two of the pieces will be the same length and one piece will be 2 feet longer
than the other two.
Which equation could be used to solve for the lengths?
x+x+x=10+2
x+x+ 2 x= 10
x+x+x+2=10
x+x+x-2=10
Answer:
the third : x + x + x + 2 = 10
Step-by-step explanation:
x = the length of the first 2 pieces.
the third piece will be x+2 ft long.
so, the total length of the original board is the sum of all 3 pieces and must be all in all 10 ft.
10 = x + x + (x + 2) = x + x + x + 2
so, it is the third answer option.
Yellowstone National Park is a popular field trip destination. This year the senior class at High School A and the senior class at High School B both planned trips there. The senior class at High School A rented and filled 5 vans and 2 buses y with 115 students. High School B rented and filled 1 van () and 6 buses () with 163 students. Each van carried the same number of students and each bus carried the same number of students. Write equations to represent the scenario. Then, graph to find the number of students in each van and in each bus.
Each van carries 13 students and each bus carries 25 students.
DistributionGiven that the senior class at High School A and the senior class at High School B both planned trips to Yellowstone, and the senior class at High School A rented and filled 5 vans and 2 buses with 115 students, while High School B rented and filled 1 van and 6 buses with 163 students, and each van carried the same number of students and each bus carried the same number of students, to determine the number of students in each van and in each bus, the following calculation must be made:
5X + 2Y = 1151X + 6Y = 1635X + 2Y = 1155X + 30Y = 815815 - 115 = 28Y700 = 28YY = 700 / 28y = 251X + 6x25 = 1631X = 163 - 150X = 13Therefore, each van carries 13 students and each bus carries 25 students.
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Answer:
Each van carries 13 students and each bus carries 25 students.
Step-by-step explanation:
Given A(2, 3) B(1, 4) C(1, 3) D(-2, 2). How are AB and CD related?
The vector AB is not related with the vector CD as k is not the same for each pair of components.
Are two vectors similar?In this question we must prove if the vector AB is a multiple of the vector CD, that is:
[tex]\overrightarrow{AB} = k \cdot \overrightarrow {CD}[/tex]
[tex]\vec B - \vec A = k \cdot [\vec D - \vec C][/tex]
(1, 4) - (2, 3) = k · [(- 2, 2) - (1, 3)]
(- 1, 1) = k · (- 3, - 1)
Hence, the vector AB is not related with the vector CD as k is not the same for each pair of components.
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Which equation shows an example of the associative property of addition? (–7 i) 7i = –7 (i 7i) (–7 i) 7i = 7i (–7i i) 7i × (–7i i) = (7i – 7i) (7i × i) (–7i i) 0 = (–7i i)
The equation which shows the associative property of addition is; (–7 + i) +7i = –7 (i +7i).
What is the associative property of addition?Associative property of addition postulates that Changing the grouping of addends does not change the sum of the numbers. As an instance, ( 5 + 3 ) + 4 = 5 + ( 3 + 4 )
Consequently, the equation which shows the associative property of addition is; (–7 + i) +7i = –7 (i +7i)
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A car-making factory has 480 wheels. how many cars can be completed with the wheels?
Answer:
See below
Step-by-step explanation:
If the car has 4 wheels 480/4 = 120 cars
If the car has 3 wheels 480/3 = 160 cars
If the car has 6 wheels ( like a dual truck) 480 / 6 = 80
urgent!!!! please help just dont know if it is the use of words or what
Answer:
The translation rule is: (x,y) → (x-3, y+3)
Step-by-step explanation:
A translation rule in geometry is basically asking you what was done to move the shape to its new location on a graph.
In this particular problem, visualizing the graph isn't actually necessary. You only need to figure out what was added or subtracted to get the new numbers from the old ones.
In the beginning, point D was at the x coordinate 1. To make 1 into -2, you subtract three. The points E and F were also changed from their old numbers to the new ones by subtracting 3.
So, we know that the first half of our rule is x-3.
In the beginning, point D was at the y coordinate 3, but it changed to a 6. To get from 3 to 6, you add three more. Points E and F can also be made into their new numbers by adding 3.
So, we know that the second half of the rule is y+3.
I hope this helped. Please let me know if you have any questions :)
Solve for x.
A. 3
B. 10
C. 6
D. 1
The values of x is 1 and 6.
According to the statement
we have two tangents which are represented by x+5 and x+1 then
we have to find the value of x to get the perfect value of tangent
so, due to interior values of tangent they are equal to each other but in opposite sign.
But here we have to find the value of x one by one.
It means
Here we take interior value of tangent and exterior value of tangent equal to each other
So, the equations become are written below
X+5= 6 then x = 1
X+1= 7 then x = 6
So, The values of x is 1 and 6.
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A line passes through the points (-1,-1) and (5,8) which points lie on the same line?
Answer: all points [tex](x,y)[/tex] that satisfy [tex]y=\frac{3}{2}x+\frac{1}{2}[/tex]
Step-by-step explanation:
The slope of the line is
[tex]\frac{-1-8}{-1-5}=\frac{-9}{-6}=\frac{3}{2}[/tex]
So, substituting into point-slope form, the equation is
[tex]y+1=\frac{3}{2}(x+1)\\\\y+1=\frac{3}{2}x+\frac{3}{2}\\\\y=\frac{3}{2}x+\frac{1}{2}[/tex]
Therefore, all points [tex](x,y)[/tex] that satisfy [tex]y=\frac{3}{2}x+\frac{1}{2}[/tex] will lie on the same line.
Volume=
Help me please!! Asap! Thanks so much
The volume of the oblique prism is 16261 cubic units
How to determine the volume?The given parameters are:
Height = 23
Base area = 707
The volume is calculated as:
V = base area * height
So, we have:
V = 707 * 23
Evaluate
V = 16261
Hence, the volume of the oblique prism is 16261 cubic units
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30. Isla swims 4 laps in 2 minutes, and 1
lap = 25 meters. Which of the following
calculations would give her average
speed in meters per second?
The average speed is 50 meters per second
How to determine the average speed?The given parameters are:
Laps = 4
Time = 2 minutes
1 lap = 25 meters
Start by calculating the total distance
Distance = Laps * Meters
Distance = 4 * 25 meters
Distance = 100 meters
The average speed is then calculated as:
Speed = Distance/Time
This gives
Speed = 100 meters/2 minutes
Evaluate
Speed = 50 meters per second
Hence, the average speed is 50 meters per second
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PLEASE HELP IM SO STUCK PLS
Answer:
x-intercept: (-22, 0)y-intercept: (0, 6)Step-by-step explanation:
Each intercept is found by solving for the variable after the other variable is set to zero.
X-interceptSet y = 0 and solve for x:
-3x = 66
x = 66/(-3) = -22 . . . . . divide by the coefficient of x
The x-intercept is (-22, 0).
Y-interceptSet x = 0 and solve for y:
11y = 66
y = 66/11 = 6 . . . . . . . . divide by the coefficien tof y
The y-intercept is (0, 6).
__
Additional comment
The given equation is in standard form (almost). This form makes it easy to find the intercepts by dividing the constant by the variable's coefficient, as we did above.
(We say "almost" standard form because the leading coefficient is negative. It would be positive in standard form: 3x -11y = -66.)
Answer: (-22,0) and (0,6) in order
Step-by-step explanation:
x-intercept means point of graph that intersects the x-axis and y=0
y-intercept means point of graph that intersects the y-axis and x=0
So the x-intercept is -3x + 0 = 66 as y = 0
x = -22 so the coordinate is (-22,0)
The y-intercept is 0 + 11y = 66 as x = 0
y = 6 so the y-intercept is (0,6)
Which equations are true for x = –2 and x = 2? Select two options
x2 – 4 = 0
x2 = –4
3x2 + 12 = 0
4x2 = 16
2(x – 2)2 = 0
The equations that are true for x = –2 and x = 2 are x^2 - 4 = 0 and 4x^2 = 16
Equations and expressionsGiven the solution to a quadratic equation to be -2 and 2, the equation is expressed as:
(x-(-2))(x -2) = 0
(x+2)(x - 2) = 0
Expand
x^2 - 2x + 2x - 4 = 0
x^2 - 4 = 0
Hence the equations that are true for x = –2 and x = 2 are x^2 - 4 = 0 and 4x^2 = 16
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I need help with cancelling units
the trick being, you flip the ratio as needed, hmmm in this case we want 750 oz converted to lbs, so hmmm we'd want to cancel out the "oz" part and only leave the "lbs" part,
[tex]750~~~\begin{matrix} oz \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~\cfrac{1lbs}{16~~\begin{matrix} oz \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}\implies \cfrac{750}{16}lbs\implies 46.875lbs[/tex]
notice, to cancel out the "oz", we placed the 16 at the bottom.