Answer:
A.16b^4/81
step-by-step explanation:
(2b/3)^4
= (2b)^4/3^4
= (2^4×b^4)/3^4
=16b^4/81
(as 16 and 81 cant simplify each other)
Answer:
A. 16b^4/81
Step-by-step explanation:
(2b/3)^4
We know (a/b)^c = a^c / b^c
(2b)^4 / 3^4
We also know (ab)^c = a^c * b^c
2^4 * b^4 / 3^4
16 b^4/ 81
Solve the system of linear equations by graphing.
y = –x – 7
x + 2y = 4
What is the solution to the system of linear equations?
(–4.5, 4.25)
(–1.7, –2.8)
(0, –7)
(3, 0.5)
The solution to the system of linear equations is (3, 0.5).
To solve the system of linear equations by graphing, we'll plot the two equations on the same set of axes and find the point of intersection, which represents the solution.
Start with the first equation, y = -x - 7. Choose a few x-values, calculate the corresponding y-values, and plot the points on the graph.For the second equation, x + 2y = 4, rewrite it in slope-intercept form (y = mx + b) by solving for y: 2y = -x + 4 → y = -0.5x + 2. Plot points using this equation.The point where the two graphs intersect is the solution. Confirm that the coordinates of this point satisfy both equations.The solution is (3, 0.5). Confirming for the first equation: y = -3 - 7 = -10 and for the second equation: 3 + 2(0.5) = 4. The coordinates match both equations.
The coordinates of △ABC are A(−11,7), B(−5,−3), C(−2,3). After a dilation, the coordinates are A'(22,−14), B'(10,6), C'(4,−6). Find the scale factor.
Answer:
-2
Step-by-step explanation:
Before a dilation you have the point (x,y).
After a dilation of a scale factor of r you have (r*x,r*y).
Let's look at one pair of corresponding points.
A(-11,7) and A'(22,-14)
We need to figure out what we can multiply to -11 to get 22.
We need to figure out what we can multiply to 7 to get -14.
Hopefully it is the same number or this isn't a dilation.
So to get from pre-image to image you need to multiply by -2 because -11*-2=22 and 7*-2=-14.
The scale factor is -2.
The dilation is this: (x,y)->(-2x,-2y)
The required scale factor is -2
What is scale factor?The scale factor is a measure for similar figures, who look the same but have different scales or measures.
How to find the scale factor? We know that if we have the point (x, y) before a dilation and (ax, ay) after a dilation, then a is the scale factorLet's consider one pair of corresponding points.
A(-11,7) and A'(22,-14)To get the scale factor we need to find out what we can multiply to -11 and 7 to get 22 and -14Clearly we need to multiply by -2.
This is applicable for all the points
So, the scale factor is -2.
Find more about "Scale factor" here: https://brainly.com/question/10253650
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In each family of functions, the _____function is the most basic function in
the family.
Answer:
parent funtion is most basic
Step-by-step explanation:i just took the test
Plz help me! Find the perimeter of the shape
Answers and the shape are in the picture
I believe the answer would be 12.4 the first choice.
Answer:
=12.4 units
Step-by-step explanation:
We can use the Pythagoras theorem to find the lengths of EF, FG, and HE
(GF)²=(ΔX)²+(Δy)²
(GF)²=(2--1)²+(4-3)²
=3²+1²
=10
GF=√10=3.16 units
(EF)²=(Δx)²+(Δy)²
=(2--1)²+(6-4)²
=3²+2²
=9+4
=13
EF=√13=3.61 units
(HE)²=(Δx)²+(Δy)²
=(-1--3)²+(6-3)²
=2²+3²
=4+9
=13
HE=√13=3.61 units
GH=(-1--3)=2 units
Perimeter =GF+EF+EH+HG
=3.16+3.61+3.61+2
=12.38 units
=12.4 units to 1 decimal place.
Select the equivalent forms.
50% =
10% =
75% =
Answer:
The answers would be:
1) 1/2
2) 1/10
3) 3/4
Step-by-step explanation:
Percentages are always calculated by multiplying the percentages with 100. Like in this example, 50% means 0.5. This 0.5 was multiplied before by 100 to get this percentage. So to find out the equivalent forms, we would simplify the percentage like:
50%-------> 0.5
10%--------> 0.1
75%--------> 0.75
So
0.5 means 1/2 of the total value
0.1 means 1/10 of the total value
0.75 means 3/4 of the total value.
Answer:
Took Quiz 100%.... 50%= 0.50 and 10%= 0.10 and 75% =0.75
How many multiples of three are between 10 and 787 and show work pls
As per the concept of arithmetic sequence, there are 259 multiples of three between 10 and 787
To solve this problem, we need to find the first and last multiples of three within the given range and then count how many integers are there in between
Step 1: Find the first multiple of three greater than or equal to 10.
The first multiple of three greater than or equal to 10 is 12.
Step 2: Find the last multiple of three that is less than or equal to 787.
The largest multiple of three less than or equal to 787 can be calculated as follows:
Divide 787 by 3:
787 ÷ 3 ≈ 262.33
The largest multiple of three less than or equal to 787 is:
262 × 3 = 786
Step 3: Count the multiples of three within the range.
Now that we have found the first and last multiples of three within the range, we need to count the number of multiples of three between them, including the endpoints.
The multiples of three between 12 and 786 are:
12, 15, 18, ..., 783, 786.
Step 4: Calculate the total count of multiples.
To calculate the total count, we can use the formula for finding the number of terms in an arithmetic sequence:
Number of terms = (last term - first term) / common difference + 1.
In this case, the first term is 12, the last term is 786, and the common difference (the difference between consecutive terms) is 3.
Number of terms = (786 - 12) / 3 + 1
Number of terms = 774 / 3 + 1
Number of terms = 258 + 1
Number of terms = 259.
To know more about multiples here
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The point (-3,-2) is rotated 180 degrees about the origin. The coordinates of its image are:
Answer:
(3,2)
Explanation:
The rotation of a point 180 degrees about the origin follows the rule:
(x,y) → (-x, -y)That means that both the x-coordinate and the y-coordinate transform into their negative values.
So, - 3 transforms into - (-3) = 3, and - 2 transforms into - (-2) = 2.
The result is (-3, -2) → (3,2).
Solve the inequality: –3(x + 2) > 4x + 5(x – 7)
Answer:
29/12 > x
Step-by-step explanation:
–3(x + 2) > 4x + 5(x – 7)
Distribute
-3x -6 > 4x +5x-35
Combine like terms
-3x-6 > 9x -35
Add 3x to each side
-3x+3x-6 > 9x+3x -35
-6 > 12x-35
Add 35 to each side
-6+35 > 12x -35+35
29 > 12x
Divide each side by 12
29/12 > 12x/12
29/12 > x
What is the solution to the system of equations graphed below?
y = --3/2x+2
y = 5x + 28
Answer:
(-4, 8) → x = -4 and y = 8Step-by-step explanation:
We only need two points to plot the graph of each equation.
[tex]y=-\dfrac{3}{2}x+2\\\\for\ x=0\to y=-\dfrac{3}{2}(0)+2=0+2=2\to(0,\ 2)\\\\for\ x=2\to y=-\dfrac{3}{2}(2)+2=-3+2=-1\to(2,\ -1)\\\\y=5x+28\\\\for\ x=-4\to y=5(-4)+28=-20+28=8\to(-4,\ 8)\\\\for\ x=-6\to y=5(-6)+28=-30+28=-2\to(-6,\ -2)[/tex]
Look at the picture.
Read the coordinates of the intersection of the line (solution).
Answer:
(-4,8)
Step-by-step explanation:
Given system of equations,
[tex]y=-\frac{3}{2}x+2-----(1)[/tex]
[tex]y=5x+28------(2)[/tex]
In equation (1), If x = 0, y = 2,
If y = 0,
[tex]-\frac{3}{2}x+2=0\implies -\frac{3}{2}x=-2\implies -3x=-4\implies x=\frac{4}{3}[/tex]
Join the points (0,2) and (4/3,0) in the graph we get the line (1),
In equation (2), if x = 0, y = 28,
If y = 0,
[tex]5x+28=0\implies 5x=-28\implies x=-5.6[/tex]
Join the points (0, 28) and (-5.6,0) in the graph we get the line (2),
Hence, by graph,
The intersection point of line (1) and (2) is (-4,8)
Which is the required solution.
PLEASE HELP! WILL MARK BRAINLIEST!!
A surveyor is “shooting a line” to a point on a tree 70 m from his current position. After rotating his surveying instrument 25° to the left, he “shoots” another line to a point on a fence post 35 m away. Determine the distance between the point on the tree to the point on the fence post. (Do not assume that the triangle shown is a right triangle) Show all work. Round answer to the nearest hundredths.
Use the Law of Sines to find the measure of
Find the measure of
Answer:
41.04 meters
Step-by-step explanation:
The questions which involve calculating the angles and the sides of a triangle either require the sine rule or the cosine rule. In this question, the two sides that are given are adjacent to each other the given angle is the included angle. The initial position is given by A. The tree is denoted as C and the fence post is denoted as B. Since the use of sine rule will complicate the question, it will be easier to solve this question using the cosine rule. Therefore, cosine rule will be used to calculate the length of BC. The cosine rule is:
BC^2 = AB^2 + AC^2 - 2*AB*AC*cos(BAC).
The question specifies that AC = 70 meters, BAC = 25°, and AB = 35 meters. Plugging in the values:
BC^2 = 35^2 + 70^2 - 2(35)(70)*cos(25°).
Simplifying gives:
BC^2 = 1684.091844.
Taking square root on the both sides gives BC = 41.04 meters (rounded to two decimal places).
Therefore, the distance between the point on the tree to the point on the fence post is 41.04 meters!!!
Given that h=10, b=5, and r=3.8, find the area of the unshaded region. Use 3.14 for π as necessary. All answers are expressed in square units.
A. 70.34
B. 36.93
C. 25
D. 20.34
The area of the circle is found using the formula Area = PI x r^2
The area of a triangle is found using the formula Area = 1/2 x base x height.
Using the given dimensions:
Area of the circle = 3.14 x 3.8^2 = 45.3416 square units.
The area of the triangle is 1/2 x 5 x 10 = 25 square units.
To find the area of the unshaded part, subtract the area of the shaded triangle from the circle:
Area = 45.3416 - 25 = 20.3416
Round to two decimal places = 20.34
The answer is D. 20.34
D. 20.34
In this case, we must calculate first the Areas of the Triangle ([tex]A_{t}[/tex]), in square units, and the Circle ([tex]A_{c}[/tex]), in square units, later we subtract the Area of the former from the Area of the latter to determine the Area of unshaded region ([tex]A_{u}[/tex]), in square units. The Area formulas for each figure are, respectively:
Triangle
[tex]A_{t} = \frac{1}{2}\cdot b\cdot h[/tex] (1)
Circle
[tex]A_{c} = \pi\cdot r^{2}[/tex] (2)
Unshaded Area
[tex]A_{u} = A_{c} - A_{t}[/tex] (3)
Where:
[tex]h[/tex] - Height of the triangle.
[tex]b[/tex] - Base of the triangle.
[tex]r[/tex] - Radius of the circle.
If we know that [tex]h = 10[/tex], [tex]b = 5[/tex] and [tex]r = 3.8[/tex], then the area of the unshaded region is:
[tex]A_{t} = \frac{1}{2}\cdot (5)\cdot (10)[/tex]
[tex]A_{t} = 25[/tex]
[tex]A_{c} = \pi\cdot 3.8^{2}[/tex]
[tex]A_{c} \approx 45.365[/tex]
[tex]A_{u} = 45.365-25[/tex]
[tex]A_{u} = 20.365[/tex]
The correct answer is D.
Please see this question related to Area problems: https://brainly.com/question/16151549
Which shows x^2 + 2x = 3 as a perfect square equation? What are the solution(s)?
a. x^2+2x-3=0; -3 and 1
b. x^2+2x+1=0; -1
c. (x+1)^2=4; -3 and 1
d. (x+1)^2=0; -1
First we can rewrite the equation to,
[tex]x^2+2x-3=0[/tex]
Which factors to,
[tex](x+3)(x-1)=0[/tex]
And this leads towards two solutions,
[tex]x_1\Longleftrightarrow x+3=0\Longrightarrow x_1=-3[/tex]
and,
[tex]x_2\Longleftrightarrow x-1=0\Longrightarrow x_2=1[/tex]
The answer is A.
Hope this helps.
r3t40
Answer:
c
Step-by-step explanation:
Given
x² + 2x = 3
To complete the square
add ( half the coefficient of the x- term )² to both sides
x² + 2(1)x + 1 = 3 + 1
(x + 1)² = 4 ( take the square root of both sides )
x + 1 = ± [tex]\sqrt{4}[/tex] = ± 2 ( subtract 1 from both sides )
x = - 1 ± 2, hence
x = - 1 - 2 = - 3 and x = - 1 + 2 = 1
Situation 1: In Lakeville, Melinda’s television cable bill is a flat rate of $50 per month, plus $1.50 for every movie she rents. After one month of service, her bill is $66.25.
Situation 2: In Oceanside, Kimberly’s television cable bill is a flat rate of $60 dollars per month, plus $1.25 for every movie she rents.
If you were to model these two situations with equations to solve for all the unknowns, what similarities and differences would the equations have? Can you solve for all the unknowns in both situations?
Answer:
situation 1 : flat rate of 50....1.50 per movie.....after 1 month her bill is 66.25
66.25 = 1.50x + 50
situation 2 : flat rate of 60....1.25 for every movie...
y = 1.25x + 60
both situations have a flat rate fee plus a charge for every movie. In situation 1, u can solve for x (the number of movies rented) because u know how much she was charged. However, in situation 2, without more information, u cannot solve this equation...what u would need to solve it would either be the number of movies rented or the total bill showing what she was charged. So basically, in situation 1, u are given more information then in situation 2.....enough information to be able to solve for the variable x...whereas, in situation 2, u are not given enough info to solve for a variable and get a numerical answer
BRAINLIEST PLEASE!!!!!
Answer: Situation 1 involves one independent variable that stands for the number of movies Melinda rented. Situation 1 gives a total for the monthly bill, but situation 2 does not give a total for the monthly bill. Because situation 2 leaves the total value out, it needs an additional variable. Therefore, situation 2 involves two variables—the number of movies Kimberly rented (the independent variable) and the monthly cable bill (the dependent variable). In this situation, we can’t solve for one variable without knowing the other.
Step-by-step explanation: ed.men.tum answer. DO NOT copy and paste!
There are 22 animals in the barn. Some are geese and some are goats. There are 82 legs in all. How many of each animal are there? Please answer
Let the number of geese be x.
Then number of goats is 22-x.
A goose has 2 legs, and a goat has 4.
There are a total of 82 legs.
So,
2x + 4(22-x) = 82
2x + 88 - 4x = 82
-2x = -6
x = 3
There are 3 geese and (22-3=) 19 goats.
Please mark Brainliest if this helps and feel free to ask doubts!
What is the solution to the equation -4(2x+3) = 2x+6-(8x+2)?
0
x=-10
0
L
0
|
0
Answer:
x = -8
Step-by-step explanation:
-4(2x+3) = 2x+6-(8x+2)
Distribute
-8x-12= 2x+6-8x-2
Combine like terms
-8x-12 = -6x+4
Add 8x to each side
-8x-12 +8x = -6x+4+8x
-12 = 2x+4
Subtract 4 from each side
-12-4 = 2x+4-4
-16 = 2x
Divide each side by 2
-16/2 = 2x/2
-8 =x
the graph shows a system of equations
What is the solution to the system of equations?
A.(-1,3)
B.(1, -3)
C.(1,3)
D.(-1,-3)
Answer:
B. (1, -3)Step-by-step explanation:
The solution of the system of equations is the coordinates of the intersection of the lines.
(1, -3)
Answer:
B.(1, -3)
Wherever the lines intersect is the solution. Hope this helps!
Step-by-step explanation:
40 POINTS
The sides of an isosceles triangle are 5, 5, and 7. Find the measure of the vertex angle to the nearest degree.
Answer:
Answer:
89
°
to the nearest degree.
Explanation:
The base of the triangle 7 can be divided in half by a line of symmetry of the isosceles triangle, which will bisect the vertex angle. This creates two right triangles:
Each with a base of 3.5 and a hypotenuse of 5.
The side opposite the half of the vertex angle is 3.5, the hypotenuse is 5.
The sine function can be used to find the angle.
sin
θ
=
o
p
p
h
y
p
sin
θ
=
3.5
5
=
0.7
Use the inverse sin function or a table of trig functions to find the corresponding angle . (Arcsin)
arcsin
0.7
=
44.4
°
Remember that this is the value of half of the vertex angle so double the value to find the vertex angle.
2
×
44.4
=
88.8
°
rounded off to the nearest whole degree =
89
°
What is the sqaure root of 40?
Answer:
6.32455532034 or just 6
Step-by-step explanation:
Answer:
2√10
Step-by-step explanation:
Find two numbers that multiply to forty, where one of them is a NON-PERFECT square. Those numbers would be 10 and 4. Take the square root of both and you will see that 2 comes from the 4, so that moves to the outside, and √10 stays the way it is because there is no perfect square to factor from this. With that being said, you have your answer.
I am joyous to assist you anytime.
An item is regularly priced at $39. Ashley bought it at a discount of 55% off the regular price.
55% off the regular price means she paid 45% of the regular price. ( 100% - 55% = 45%
Multiply the original price by 45%:
39 x 0.45 = 17.55
She paid $17.55
(If you need to know how much she saved: 39 - 17.55 = $21.45)
Answer:
Ashley bought the item for $17.55
you can calculate the probabilityof a random event, such as the rolling of dice or dealing of cards, by _____
Answer:
Making a fraction.
Step-by-step explanation:
Put the favored outcome as the numerator. Then, put the total number of outcomes as the denominator and boom, you have calculated probability. *Thumbs Up*
find sec theta if theta is in quadrant 4 and sin theta= -1/5
Answer:
[tex]\frac{5}{2\sqrt{6} }[/tex]
Step-by-step explanation:
Since Θ is in fourth quadrant then cosΘ > 0, as is secΘ
Given
sinΘ = - [tex]\frac{1}{5}[/tex], then
cosΘ = [tex]\sqrt{1-(-1/5)^2}[/tex]
= [tex]\sqrt{1-\frac{1}{25} }[/tex] = [tex]\sqrt{\frac{24}{25} }[/tex] = [tex]\frac{2\sqrt{6} }{5}[/tex]
Hence
secΘ = [tex]\frac{1}{\frac{2\sqrt{6} }{5} }[/tex] = [tex]\frac{5}{2\sqrt{6} }[/tex]
Answer:
[tex]\sec(\theta)=\frac{5\sqrt{6}}{12}[/tex]
The answer is the last one.
Step-by-step explanation:
If we are in quadrant 4, then x (cosine) is positive and y (sine) is negative.
Since cosine is positive, secant is positive because secant is the reciprocal of cosine.
So we already know the answer is not the 1st one or the 3rd one.
I'm going to use a Pythagorean Identity to find cosine value of theta.
[tex]\cos^2(\theta)+\sin^2(\theta)=1[/tex]
Enter in -1/5 for [tex]\sin(\theta)[/tex]:
[tex]\cos^2(\theta)+(\frac{-1}{5})^2=1[/tex]
Simplify a bit:
[tex]\cos^2(\theta)+\frac{1}{25}=1[/tex]
Subtract 1/25 on both sides:
[tex]\cos^2(\theta)=1-\frac{1}{25}[/tex]
Write 1 as 25/25 so you have a common denominator on the right hand side:
[tex]\cos^2(\theta)=\frac{25}{25}-\frac{1}{25}[/tex]
[tex]\cos^2(\theta)=\frac{24}{25}[/tex]
Take the square root of both sides:
[tex]\cos(\theta)=\pm \sqrt{\frac{24}{25}}[/tex]
[tex]\cos(\theta)=\pm \frac{\sqrt{24}}{\sqrt{25}}[/tex]
I will worry about simplifying the square root part when finding secant.
We said that cosine was positive because we were in the fourth quadrant.
[tex]\cos(\theta)=\frac{\sqrt{24}}{\sqrt{25}}[/tex]
Now recall that cosine and secant are reciprocals of each other:
[tex]\sec(\theta)=\frac{\sqrt{25}}{\sqrt{24}}[/tex]
Let's simplify the square part not.
Usually people hate the square root on both and also if you look at your choices none of the choice have square root on bottom.
So we are going to multiply top and bottom by [tex]\sqrt{24}[/tex]. I'm going to also write 5 instead of [tex]\sqrt{25}[/tex].
[tex]\sec(\theta)=\frac{5}{\sqrt{24}} \cdot \frac{\sqrt{24}}{\sqrt{24}}[/tex]
[tex]\sec(\theta)=\frac{5\sqrt{24}}{24}[/tex]
Now let's simplify the square root of 24.
We know 24 is not a perfect square, but 24 does contain a factor that is a perfect square. That factor is 4.
[tex]\sec(\theta)=\frac{5\sqrt{4}\sqrt{6}}{24}[/tex].
[tex]\sec(\theta)=\frac{5(2)\sqrt{6}}{24}[/tex]
[tex]\sec(\theta)=\frac{10\sqrt{6}}{24}[/tex]
Now both 10 and 24 share a common factor of 2 so let's divide top and bottom by 2:
[tex]\sec(\theta)=\frac{5\sqrt{6}}{12}[/tex]
The answer is the last one.
Determine whether the relation represents y as a function of x.
1.) x^2+y^2=9
2.) 2xy=1
Answer:
1 is not a function
2 is a function because you can write it (AS) f(x)=1/(2x).
Step-by-step explanation:
1) x^2+y^2=9 is a circle with center (0,0) and radius 3.
To get this all I did was compare to (x-h)^2+(y-k)^2=r^2 where (h,k) is the center and r is the radius.
A circle is not a function.
You can solve solve for and see that you will get two values for y which is no go for a function.
Let's do that:
[tex]x^2+y^2=9[/tex]
Subtract x^2 on both sides:
[tex]y^2=9-x^2[/tex]
Square root both sides:
[tex]y=\pm \sqrt{9-x^2}[/tex].
2) 2xy=1
Divide both sides by 2x:
y=1/(2x).
This is a function only one y there.
which of the following is a factor of x^6 + 1000?
Well,
[tex]x^6+1000=(x^2)^3+10^3[/tex]
From here we use [tex]a^3+b^3=(a+b)(a^2-ab+b^2)[/tex],
[tex](x^2+10)\boxed{(x^4-10x^2+100)}[/tex]
And the found our factor.
Hope this helps.
r3t40
Answer:
Option B
Step-by-step explanation:
solve the system by graphing.
y=x+2
y=-2x+2
A.(0,2)
B.(2,0)
C.(0,-2)
I need help solving
[tex]3D-2E=3\left[\begin{array}{cc}5&5\\-3&7\\1&3\end{array}\right] -2\left[\begin{array}{cc}6&7\\2&-7\\0&5\end{array}\right] \\\\3D-2E=\left[\begin{array}{cc}15&15\\-9&21\\3&9\end{array}\right] -\left[\begin{array}{cc}12&14\\4&-14\\0&10\end{array}\right] \\\\3D-2E=\left[\begin{array}{cc}3&1\\-13&35\\3&-1\end{array}\right][/tex]
Why is this the graph of the function f(x)=4x^2-8x+7?
Answer:
I'll be referring to this form: ax^2-bx+c
The 4x^2 is the rate that it goes up. If a is greater than 1, then that means the graph gets narrower from the parent function x^2. You can clearly see that in that graph.
The c is always the y-intercept. In this case, you can see that it's 7. The graph clearly shows 7 as it's y-intercept as well, so that matches well.
Sadly, I do not know how to compare the 8x into this situation without a calculator, but that information should be quite enough to see how that function is in that graph.
if A= (-1,3) and B=(11,-8) what is the length of ab
Answer:
16.3 units ( to 1 dec. place )
Step-by-step explanation:
Using the distance formula
d = √ (x₂ - x₁ )² + (y₂ - y₁ )²
with (x₁, y₁ ) = A(- 1, 3) and (x₂, y₂ ) = B(11, - 8)
AB = [tex]\sqrt{(11+1)^2+(-8-3)^2}[/tex]
= [tex]\sqrt{12^2+(-11)^2}[/tex]
= [tex]\sqrt{144+121}[/tex]
= [tex]\sqrt{265}[/tex] ≈ 16.3 ( to 1 dec. place )
Given that (X+3) is a factor of the expression x^3 + 4x^2 + px + 3 , find the value of p. Hence, solve the equation x^3 + 4x^2 + px + 3=0, expressing the complex number in the form a + bi
Answer:
[tex]p=4[/tex]
[tex]x=\frac{-1}{2} \pm \frac{\sqrt{3}}{2}i[/tex]
Step-by-step explanation:
We are given (x+3) is a factor of [tex]x^3+4x^2+px+3[/tex], which means if were to plug in -3, the result is 0.
Let's write that down:
[tex](-3)^3+4(-3)^2+p(-3)+3=0[/tex]
[tex]-27+36-3p+3=0[/tex]
[tex]9-3p+3=0[/tex]
[tex]9+3-3p=0[/tex]
[tex]12-3p=0[/tex]
[tex]12=3p[/tex]
[tex]p=4[/tex]
So the cubic equation is actually [tex]x^3+4x^2+4x+3=0[/tex] that they wish we solve for [tex]x[/tex].
To find another factor of the given cubic expression on the left, I'm going to use synthetic division with that polynomial and (x+3) where (x+3) is divisor. Since (x+3) is the divisor, -3 will be on the outside like so:
-3 | 1 4 4 3
| -3 -3 -3
---------------------
1 1 1 0
So the other factor of [tex]x^3+4x^2+4x+3[/tex] is [tex](x^2+x+1)[/tex].
We must solve [tex]x^2+x+1=0[/tex].
Compare this to [tex]ax^2+bx+c=0[/tex].
We have [tex]a=1,b=1, \text{ and } c=1[/tex].
The quadratic formula is
[tex]x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}[/tex].
Plug in the numbers we have for [tex]a,b, \text{ and } c[/tex].
[tex]x=\frac{-1 \pm \sqrt{1^2-4(1)(1)}}{2(1)}[/tex].
Simplify inside the square root while also performing the one operation on bottom:
[tex]x=\frac{-1 \pm \sqrt{1-4}}{2}[/tex]
[tex]x=\frac{-1 \pm \sqrt{-3}}{2}[/tex]
Now our answer will include an imaginary part because of that sqrt(negative number).
The imaginary unit is [tex]i=\sqrt{-1}[/tex].
So our final answer is:
[tex]x=\frac{-1}{2} \pm \frac{\sqrt{3}}{2}i[/tex]
Final answer:
To find the value of p, substitute -3 into the polynomial since (X+3) is a factor, thus yielding p=3. With p known, the polynomial becomes [tex]x^3 + 4x^2 + 3x + 3[/tex] = 0, and can now be solved for x.
Explanation:
Finding the Value of p
Given the polynomial [tex]x^3 + 4x^2 + px + 3[/tex] and the fact that (X+3) is a factor, we can use polynomial division or synthetic division to find the value of p. Since (X+3) is a factor, when we substitute -3 for x in the polynomial, the result should be zero.
Substituting -3 into the polynomial yields:
[tex](-3)^3 + 4(-3)^2 + p(-3) + 3[/tex] = 0
-27 + 36 - 3p + 3 = 0
9 - 3p = 0.
Solving for p gives us:
3p = 9
p = 3.
Solving the Equation
Now that we know p, we rewrite the polynomial as [tex]x^3 + 4x^2 + 3x + 3 = 0[/tex] and use the fact that (X+3) is a factor to perform the division. The remainder of the division gives us a quadratic polynomial which we can solve using the quadratic formula or factoring.
Drag steps in the given order to evaluate this expression. -3(-3+2)-6
Answer:
The value of the expression is -3.
Step-by-step explanation:
Consider the provided expression.
[tex]-3(-3+2)-6[/tex]
Step 1: Solve the parentheses.
[tex]-3(-1)-6[/tex]
Step 2: Open the parentheses.
[tex]3-6[/tex]
Step 3: Subtract the like terms.
[tex]-3[/tex]
Hence, the value of the expression is -3. The required steps is shown above.
The function f(x) 32(1.5) represents the chipmunk population of a forest x years after it was first studied. What was the original population of chipmunks?
A) 32
B) 21
C) 48
D) 72
Answer:
A) 32
Step-by-step explanation:
I will assume the function is
f(x) = 32 (1.5)^x
This is in the form
y =a b^x
in which a is the initial value, b is the growth rate, x is the time
a =32, which is the initial population
b= 1.5, which means 1.5-1 = .5. which means it grows at a 50% increase
x = number of years