Use the method of equating coefficients to find the values of a, b, and c: (x + 4) (ar²+bx+c) = 2x³ + 9x² + 3x - 4.A. a = -2; b= 1; c= -1OB. a=2; b= 1; c= 1OC. a=2; b= -1; c= -1OD. a=2; b= 1; c= -1
Answers
To find the coefficients we first need to make the multipliation on the left expression:
[tex]\begin{gathered} (x+4)(ax^2+bx+c)=ax^3+bx^2+cx+4ax^2+4bx+4c \\ =ax^3+(4a+b)x^2+(4b+c)x+4c \end{gathered}[/tex]Then we have:
[tex]ax^3+(4a+b)x^2+(4b+c)x+4c=2x^3+9x^2+3x-4[/tex]Two polynomials are equal if and only if their coefficients are equal, this leads to the following equations:
[tex]\begin{gathered} a=2 \\ 4a+b=9 \\ 4b+c=3 \\ 4c=-4 \end{gathered}[/tex]From the first one it is clear that the value of a is 2, from the last one we have:
[tex]\begin{gathered} 4c=-4 \\ c=-\frac{4}{4} \\ c=-1 \end{gathered}[/tex]Plugging the value of a in the second one we have:
[tex]\begin{gathered} 4(2)+b=9 \\ 8+b=9 \\ b=9-8 \\ b=1 \end{gathered}[/tex]Therefore, we conclude that a=2, b=1 and c=-1 and the correct choice is D.
Related Questions
The quadratic equation y= -16t^2 +4t+2 represents a moving objects trajectory where y is the objects height in feet above the ground after t seconds . At what time will the objects hit the ground ?
Answers
Since y is the object's height, it will be on the ground when y = 0. So let's do that:
[tex]0=-16t^2+4t+2[/tex]Here, we can use Bhaskara's Formula to find the roots of the equation:
[tex]\begin{gathered} t=\frac{-4\pm\sqrt[]{4^2-4\cdot(-16)\cdot2}}{2\cdot(-16)} \\ t=\frac{-4\pm\sqrt[]{16+128}}{-32}=\frac{-4\pm\sqrt[]{144}}{-32}=\frac{-4\pm12}{-32} \\ t_1=\frac{-4+12}{-32}=\frac{8}{-32}=-0.25 \\ t_2=\frac{-4-12}{-32}=\frac{-16}{-32}=0.5 \end{gathered}[/tex]Since the time at start is 0, we can't have a negative sign, it would be like saying what happened before the object was in the air. The it will hit the ground at t = 0.5 s.
Let v be the vector from initial point P1=(−4,−9) to terminal point P2=(6,2). Write v in terms of i and j.
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Step 1;
P1 = ( - 4 , -9 )
P2 = ( 6 , 2 )
Step 2:
[tex]\begin{gathered} \text{Let P}_1=(x_1,y_1)_{} \\ P_2=(x_2,y_2\text{ ) } \end{gathered}[/tex]Step 3:
[tex]\text{v = (x}_2-x_1)i+_{}(y_2-y_1\text{ ) j}[/tex]Step 4:
[tex]\begin{gathered} \text{v = (x}_2-x_1)i+_{}(y_2-y_1\text{ ) j} \\ \text{v = (6}-(-4))i+_{}(2-(-9)\text{) j} \\ v\text{ = (6+4)i + (2 + 9)j} \\ v\text{ = 10i + 11 j} \end{gathered}[/tex]Y.11 Multi-step problems with customary uni You have prizes to reveal! Go to your Tracy decides to take her puppy for a walk. After 90 feet, they stop to smell some roses. Then, Tracy runs into a friend 200 yards up the road. They start talking, and soon it's time for Tracy to go home. So, she and her puppy head back to her house. How many feet long was Tracy's walk? feet Submit
Answers
Given:
The distance travelled by Tracy till she stopped to smell roses, x=90 feet.
The distance from roses to the friend, y=200 yards.
The distance travelled by Tracy one side,
[tex]\begin{gathered} D=x+y \\ =90\text{ f}eet+200\times3feet \\ =90\text{ f}eet+600\text{ f}eet \\ =690\text{ f}eet \end{gathered}[/tex](1 yard=3 feet).
Now, the total distance travelled byTracy both sides is,
[tex]\begin{gathered} d=2D \\ =2\times690\text{ f}eet \\ =1380\text{ f}eet \end{gathered}[/tex]Therefore, Tracy walk was 1380 feet long.
What does slope mean?
Answers
Slope is a measure of its steepness
Mathematically,
Slope = Rise / Run
Rise = y2 - y1
Run = x2 - x1
Slope = y2 - y1 / x2 - x1
Answer:
Suppose a linear equation describes something (say, population growth). The slope is the rate (say, of growth) and the y-intercept gives the starting value.
Step-by-step explanation:
IN Date OUT IN OUT Employee Time Card: 7:30 10/1 11:30 4:15 12:00 John Apple 10/2 8:15 11:00 5:15 11:45 10/3 11:15 3:55 7:00 12:10 Dept: Cust. Serv. 4:30 10:55 12:00 6:25 10/4 NOTE: NO OVERTIME 1:30 5:00 12:45 10/5 6:00 TOTAL HOURS RATE per hour: $13.75 What is John's total pay for the week? deneaker notes
Answers
I can see it now
thank you
11:30-7:30= 4h
11:00-8.15=2:45h
11:15-7:00=4:15h
10:55-6:25=4:30h
10:45-6:00=4:45h
Total = 4+2.75+4.25+4.5+4.75=20.25
4:15-12:00=4:15h
5:15-11:45=5:30h
3:55-12:10=3:45h
4:30-12:00=4:30h
5:00-1:30=3:30h
Total = 4.25+5.5+3.75+4.5+3.5=21.5
Total hours = 21.5+20.25=41.75
ok, the total pay would be:
Rate per hour * total hours:
[tex]13.75\times41.75=574.0625[/tex]Did you get the same value? hello? are you still with me? ok
do you have any question? oh, remember: After our session, the answer is saved in your profile . My pleasure
What is an equation of the line that passes through the points (-3,-5) and (-5, -3)? Put your answer in fully reduced form.
Answers
Express the general equation of a line passing through two points (x1,y1) and (x2,y2).
[tex]y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]Put (-3,-5) for (x1,y1) and (-5,-3) for (x2,y2) implies,
[tex]\begin{gathered} y+5=\frac{-3+5}{-5+3}(x+3) \\ y+5=\frac{2}{-2}(x+3) \\ y+5=-x-3 \end{gathered}[/tex]Further simplifying gives,
[tex]y=-x-8[/tex]Therefore, the equation of the line is y=-x-8.
Parallel and Perpendicular LinesDetermine whether the following lines are parallel, perpendicular, orneither. Write the corresponding letter on the line next to the question.A = parallel, B = perpendicular, or C = neither1. y = }x+6 and y =- *x + 4
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One of the criteria for lines being perpendicular is the fact that the slope of the function in a perpendicular line is the inverse of the slope of the first times -1.
And as you can see m (being the slope of the first equation) is the inverse of the second equiation:
[tex]m=\frac{7}{3},m_1=-\frac{1}{m}[/tex][tex]-\frac{1}{m}=-\frac{1}{\frac{7}{3}}=-\frac{3}{7}[/tex]Therefore line 1 is perpendicular to line 2.
Four times a number decreased by three is between -15 and 41?
Answers
Answer:
The number will lie between -3 and 11
Step-by-step explanation:
Let the number be 'x'
According to the question,
-15 < 4x - 3 < 41
-12 < 4x < 44 (Adding 3)
-3 < x < 11 (Dividing by 4)
The area of a semicircle is 0.5652 square inches. What is the semicircle's diameter? Use 3.14 for a inches Submit can you explain
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Given :
The area of semicircle is given as 0.5652 sq.inches.
To find:
The diameter of semicircle which is denoted as d.
Explanation:
The area of semicircle is given as
[tex]A=\frac{\pi r^2}{2}[/tex]The relation between radius and diameter is
[tex]d=2r[/tex]Now substitute the given area in the area of semicircle formula.
[tex]0.5652=\frac{3.14\times r^2}{2}[/tex][tex]r=\sqrt[]{\frac{2\times0.5652}{3.14}}=\sqrt[]{0.36}[/tex][tex]r=0.6in[/tex]The semicircle diameter is determined as
[tex]d=2r\Rightarrow2\times0.6=1.2in[/tex]Answer:
Hence the diameter of semicircle is determined as 1.2 in.
A homeowner estimates that it will take 9 days to roof his house. A professional roofer estimates that he could roof the house in 5 days. How long ( in days ) will it take if the homeowner helps the roofer?
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Solution:
If x denote the days, the rate unit being Jobs per day is:
[tex]\frac{1}{x}=\frac{1}{9}+\frac{1}{5}[/tex]this is equivalent to
[tex]\frac{1}{x}=\frac{5+9}{45}=\frac{14}{45}[/tex]solving for x, we get:
[tex]x\text{ = }\frac{45}{14}=3.2\text{ days}[/tex]that is just a little more than 3 days.
Solve for x in the parallelogram below.
Answers
Answer:
3
Step-by-step explanation:
In parallelogram, opposite sides are equal.
Here,
5x + 2 and 17 are opposite sides.
5x + 2 = 17
5x = 17 - 2
5x = 15
x = 15 / 5
x = 3
The equation should look like this:
5x + 2 = 17
Why 17? Because if you look at the opposite side of the parallelogram, you can see a parallel line that is the exact same length. Since it also isn’t an equation, we can tell this would be the true length of it.
Okay, to solve it, first subtract 2 on both sides of the equation. Why? Because on the left side of the equation (the numbers before the equal sign) we can see that there is one coefficient (number with a variable) and one constant (just a plain number). In equations like this, you want only coefficients or constants per side. To remove the constant from the left side, do the inverse of the number. Always remember that whatever you do on one side, has to be done to the other.
5x + 2 - 2 = 17 - 12
This equals:
5x = 15
Now, just divide by the coefficient, in this case 5, on both sides. This removes the 5.
5x/5 = 15/5
This leaves:
x = 3
To mail the envelope first class do US post office charges $.39 for the 1st ounce and $.22 for each additional ounce . use inequality to find the maximum number of whole ounce of that can be mailed for $7.24
Answers
Let N be the total amount of whole ounces that are mailed.
Since mailing the first ounce has a cost of $0.39, then there will be N-1 ounces charged for $0.22 each.
The total cost of mailing N ounces will be:
[tex]0.39+0.22\times(N-1)[/tex]If that cost cannot exceed $7.24, then:
[tex]0.39+0.22\times(N-1)\le7.24[/tex]Solve the inequality for N:
[tex]\begin{gathered} \Rightarrow0.22\times(N-1)\le7.24-0.39 \\ \Rightarrow0.22N-0.22\le6.85 \\ \Rightarrow0.22N\le6.85+0.22 \\ \Rightarrow0.22N\le7.07 \\ \Rightarrow N\le\frac{7.07}{0.22} \\ \Rightarrow N\le32.136\ldots \end{gathered}[/tex]Since N must be a whole number, the maximum value of N that satisfies the inequality is 32.
Therefore, the maximum number of whole ounces that can be mailed for $7.24 is:
[tex]32[/tex]h(x) = 3a + 410-8-h612-X10-8-668-2-2-10Select the correct answer from each drop-down menu.Function h is alwaysThe function'sis located at (0,5), and there is noThe function isfor all values of x.
Answers
We are given the following exponential function.
[tex]h(x)=3^x+4_{}[/tex]The function h(x) is always increasing as can be seen in the given graph.
The y-intercept of a graph is the point where the function intersects/crosses the y-axis.
As you can see from the graph, the graph intersects the y-axis at the point (0, 5)
Therefore, the function's y-intercept is located at (0, 5)
The x-intercept of a graph is the point where the function intersects/crosses the x-axis.
As you can see from the graph, the graph does not intersect the x-axis at any point.
Therefore, there is no x-intercept.
Notice that the graph of the function h(x) is always positive for all values of x.
Answer:
Step-by-step explanation:
A triangle has squares on its three sides as shown below. What is the value of x? 4 centimeters 7 centimeters 5 centimeters 3 centimeters
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45. (09.01) Let A = {1, 2, 3, 4, 5} and B = {2,4}. What is A n B? O {2,4) O {1, 2, 3) O {1, 2, 3, 4 } O {1, 2, 3, 4,5)
Answers
Answer:
{2,4}
Explanation:
Given sets A and B defined below:
[tex]\begin{gathered} A=\mleft\{1,2,3,4,5\mright\} \\ B=\mleft\{2,4\mright\} \end{gathered}[/tex]The set A Π B is the set of elements common to sets A and B.
[tex]A\cap B=\{2,4\}[/tex]Suppose you deposit $600 into an account that pays 5% annual interest, compounded continuously. How much will you have in the account in 4 years? ƒ(t) = ae^rt
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To determine the amount that will be on the account after 4 years you have to apply the given exponential function that models the amount of money on the account with respect to the time.
[tex]f(t)=ae^{rt}[/tex]Where
a represents the initial amount
r represents the interest rate expressed as a decimal value
t is the time period in years
The initial amount on the account is a= $600
The time period is t= 4 years
The interest rate is r=5%, divide it by 100 to express it as a decimal value:
[tex]r=\frac{5}{100}=0.05[/tex]Using this information, you can calculate the final amount:
[tex]\begin{gathered} f(t)=ae^{rt} \\ f(4)=600e^{0.05\cdot4} \\ f(4)=600e^{0.2} \\ f(4)=732.84 \end{gathered}[/tex]After 4 years there will be $732.84 on the account. The correct option is B.
what is 9.77 with 8% tax
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it will be 9.77+0.08(9.77)=10.5516
Solve the following compound inequality:0< x+7< 9
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you need to subtract 7 in each section of the inequality is
-7< x<2
-7< x and x<2
what is the answer to a negative 4 divided by a positive 6?
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The expression given as negative 4 divided by a positive 6 has a value of -2/3
How to evaluate the expression?From the question, the expression is given as
negative 4 divided by a positive 6
Rewrite the expression properly
This is rewritten as follows
-4 divided by +6
This can be represented as
-4/6
There are no like terms in the above expression
So, we have the following equation
-4/6 = -4/6
Divide 4 and 6 by a common factor
The common factor is 2
So, we have
-4/6 = -2/3
The expression cannot be further simplified
So, we have the following equation
-4/6 = -2/3
Hence, the value of the expression is -2/3
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The expression given as "negative 4 divided by a positive 6" has a value of that is -2/3
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
From the problem, the expression is given as;
"negative 4 divided by a positive 6"
Rewrite the expression properly then;
-4 divided by +6
This can be express as;
-4/6
There are no like terms in the expression
So, we have the equation;
-4/6 = -4/6
Divide 4 and 6 by a common factor;
The common factor is 2
-4/6 = -2/3
So, we have the equation;
-4/6 = -2/3
Hence, the value of the expression will be; -2/3
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The volume of the rectangular prism is 105 cubic yards. What is the surface area of the prism in square feet?
Answers
Answer:
198.18 is the answer
Step-by-step explanation:
the answer is 198.18
hope it helps
The graph of y=(x + 2)^2 – 1 is reflected across the x axis and then translated up 3 units and right 4 units. What is the equation for the transformed graph?
Answers
ANSWER
[tex]y=-(x-2)^2\text{ + 4}[/tex]EXPLANATION
We have that the graph of y is:
[tex]y=(x+2)^2\text{ - 1}[/tex]It is first reflected about the x axis.
A reflection about the x axis is represented as:
y = -f(x)
which means that we find the negative of the function:
[tex]\begin{gathered} \Rightarrow y=-\lbrack(x+2)^2\text{ - 1\rbrack} \\ y=-(x+2)^2\text{ + 1} \end{gathered}[/tex]Then, it is translated 3 units up (vertical shift) and 4 units right (horizontal shift).
A translation is represented as:
y = f(x - a) + b
where a = horizontal shift; b = vertical shift
So, we have to find:
y = f(x - 4) + 3
That is:
[tex]\begin{gathered} y\text{ = }-\lbrack(x-4)+2\rbrack^2\text{ + 1 + 3} \\ y=-(x-4+2)^2\text{ + 4} \\ y=-(x-2)^2\text{ + 4} \end{gathered}[/tex]Therefore, that is the equation of the transformed graph.
what is an equation of the line that passes through the point -6 and -7 and is perpendicular to the line 6x+5y=30I got y=5/6-2 but apparently its wrong
Answers
First we can find the slope. The standard form of the equation of a line is:
[tex]y=ax+b[/tex]Where a is the slope and b is the intercept.
When 2 lines are perpendicular, the slopes are reciprocal and opposite to each other. If we write the given equation of the perpendicular line in the standard form we have:
[tex]6x+5y=30\rightarrow y=-\frac{6}{5}x+\frac{30}{5}\rightarrow y=-\frac{6}{5}x+6[/tex]So you got the slope right, it's 5/6.
Now, with the given point we find the intercept. The point is x = -6 and y = -7, so we replace these values into the expression we have until now:
[tex]y=\frac{5}{6}x+b[/tex][tex]-7=\frac{5}{6}(-6)+b[/tex]And solve for b
[tex]-7=-5+b\rightarrow b=-7+5=-2[/tex]So the equation of the line is:
[tex]y=\frac{5}{6}x-2[/tex]Which parabola corresponds to the quadratic function y = 2x2 + 4x - 16? D. A. B. C. 10:13 1618 10- 12 =10 10 28 -20
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We can see that the y-intercept would be (0,-16) since this is the result of replacing x=0 in the function.
We can also find the x-intercepts solving the equation 0=2x^2+4x-16. Doing so, we have:
[tex]\begin{gathered} 0=2x^2+4x-16 \\ 0=x^2+2x-8\text{ (Dividing by 2 on both sides of the equation)} \\ 0=(x+4)(x-2)\text{ (Factoring)} \\ \text{ We can see that the solutions of the equation are x=-4 and x=2} \\ \text{Therefore the x-intercepts are (-4,0) and (2,0)} \end{gathered}[/tex]The graph that satisfies the conditions we have found previously is the option A.
how do I graph the line with the given slope m and y-intercept b.
m=5/3,b=-4
Answers
y=(5/3)x+4
I am aware that the slope is "big," m = - 5 /3, and that the yy-intercept is "left(0, 4), right" (0,4). The final graph of the line should be declining when viewed from left to right because the slope is negative.
y = mx+c
how to draw this graph?
step 1: Plot the given equation's yy-intercept, which is left(0,4right), first (0,4).
On the xy axis, the position (0,4) .
step2: Use the slope largem = -5 /3
m= 5/3
to locate a different point using the y-intercept b as a guide. The slope instructs us to move 3 units to the right after dropping down 5 units.
To find the opposite spot, start at (0,4) and go 5 units down and 3 units to the right.
Step 3: Make a line that goes through all of the points.
Create a line that joins the coordinates (0,4) and (3,5)
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a triangular pyramid has four faces h = b = 1. What is the pryimands surface area?(There's no image)(
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Let's find the area of one face
[tex]A=\frac{bh}{2}[/tex]Where h = b = 1.
[tex]A=\frac{1\cdot1}{2}=\frac{1}{2}[/tex]Given that there are four faces, we have to multiply the area above by 4
[tex]S=4\cdot\frac{1}{2}=2[/tex]Hence, the answer is 2 square units.If a,b ,and c represent the set of all values of x that satisly the equation below, what is the value(A+ b+ c) + (abc)?X^3-20x = x^2(A) -1(B) 0(C) 1(D) 9
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First, we need to find the solutions a, b, and c of the equation:
[tex]x^3-20x=x^2[/tex]We can rewrite it as:
[tex]\begin{gathered} x^3-x^{2}-20x=0 \\ \\ x(x^{2}-x-20)=0 \\ \\ x=0\text{ or }x^{2}-x-20=0 \end{gathered}[/tex]Thus, one of the solutions is a = 0.
To find the other solutions, we can use the quadratic formula. We obtain:
[tex]\begin{gathered} x=\frac{-(-1)\pm\sqrt[]{(-1)^{2}-4(1)(-20)}}{2(1)} \\ \\ x=\frac{1\pm\sqrt[]{1+80}}{2} \\ \\ x=\frac{1\pm\sqrt[]{81}}{2} \\ \\ x=\frac{1\pm9}{2} \\ \\ b=\frac{1-9}{2}=-4 \\ \\ c=\frac{1+9}{2}=5 \end{gathered}[/tex]Now, we need to find the value of the expression:
[tex]\mleft(a+b+c\mright)+abc[/tex]Using the previous solutions, we obtain:
[tex]\mleft(0-4+5\mright)+0(-4)(5)=1+0=1[/tex]Therefore, the answer is 1.
Position Value of Term 1 1 2 3 -18 1-24 5 -30 What expression shows the relationship between the value of any term and n, its position in the sequence?
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basically they are the negative multiples of 6, so:
[tex]a_n=-6n[/tex]What is the equation of this line?
A. y=4/3x−5
B. y=3/4x−5
C. y=−43/x−5
D. y=4/3x+5
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the y intercept is -5
therefore the true answer is A
Drag "Yes" if the lengths could create a triangle, or "No" if the lengths could not create a triangle.
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Find the missing number so that the equation has infinitely many solutions.
Answers
we have the equation
-2x-9=-2x-?
Remember that
If in a system of two linear equations, we have two identical lines
then
The system has infinite solutions
therefore
the missing number is 9
-2x-9=-2x-9