Let's begin by listing out the information given to us:
2x³ + x² - 7x- 2 = 0
We will proceed to factorise, we have:
(x + 2)(2x² − 3x - 1) = 0
We will proceed to equate the factors to zero, we have:
x + 2 = 0⇒ x = -2
⇒ x = -2
2x² − 3x - 1 = 0
We will use the quadratic formula, we have:
[tex]\begin{gathered} 2x^{2}-3x-1=0 \\ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\ a=2,b=-3,c=-1 \\ x=\frac{-(-3)\pm\sqrt[]{(-3)^2-4(2)(-1)}}{2(2)} \\ x=\frac{3\pm\sqrt[]{9+8}}{4}=\frac{3\pm\sqrt[]{17}}{4} \\ x=\frac{3\pm\sqrt[]{17}}{4}\Rightarrow x=\frac{3+\sqrt[]{17}}{4},\frac{3-\sqrt[]{17}}{4} \\ x=\frac{3+\sqrt[]{17}}{4},\frac{3-\sqrt[]{17}}{4} \end{gathered}[/tex]
find the slope/rate of the line represented by each table
The correct answer is 1/2 or 0.5
Pick 2 points in the table; (x = -10, y = 1) and ( x = 0, y = 6)
The Rate of Change is given by;
[tex]\text{slope = }\frac{y_2-y_1}{x_2-x_1}=\frac{6-1}{0--10}=\frac{5}{10}=\frac{1}{2}[/tex]The rate of change is 1/2 or 0.5
Hence, the correct answer is 1/2 or 0.5
I need help to do these composition of functions. I have a photo if needed.g(a)=a-1f(a)=3a-1Find (g×f)(1)
According to the definition for composite function,
[tex]g\circ f(x)=g\lbrack f(x)\rbrack[/tex]Substitute the values,
[tex]\begin{gathered} g\circ f(1)=g\lbrack f(1)\rbrack \\ g\circ f(1)=g\lbrack3(1)-1\rbrack \\ g\circ f(1)=g\lbrack2\rbrack \\ g\circ f(1)=(2)-1 \\ g\circ f(1)=1 \end{gathered}[/tex]Thus, the value of the given composite function is 1.
5. Given: Line segment AD bisects
Given information:
AD bisects
We are in a triangle that has been bisected , and therefore, the segment AD cuts the side BC of the triangle in two equal parts.
Also, since AD bisects the angle, <1 equals <2.
Step 1: Angle bisector defines two equal angles, then <1 = <2
They also tell us that angle <3 = <1 (premise)
Then, using transitive property we have:
<1 = <3
and <1 = <2 due to result of a bisection
then using transitive property:
<3 = <1 = <2
<3 = <2
Just lmake sure you use "transitive" property as the reason for the congruency in the last step.
Round each number to the nearest ten. 24=311=107=
Round each number to the nearest ten. 24=
311=
107=
When rounding to the nearest ten, use these rules:
A) Round the number up to the nearest ten if the last digit in the number is 5, 6, 7, 8, or 9.
B) Round the number down to the nearest ten if the last digit in the number is 1, 2, 3, or 4.
C) If the last digit is 0, then we do not have to do any rounding, because it is already to the ten.
Step 1
24, the last digit is 4, so we round to the number down to the nearest ten
[tex]24\Rightarrow20[/tex]311. the last digit is 1, so we round to the number down to the nearest ten
[tex]311\Rightarrow310[/tex]107. the last digit is 7, so we round to the number up to the nearest ten
[tex]107\Rightarrow110[/tex]Triangle A B C has vertices (1,4),(5,6) and (3,10) It is reflected across the y axis forming triangle A’B’C’. What are the vertices of the new triangle?
Given:
The coordinates of triangle ∆ABC are (1,4), (5,6) and (3,10).
The triangle is reflected across y axis forming ∆A'B'C'.
The objective is to find the vertices of the new triangle.
Explanation:
If a triangle with coordinate (a,b) is reflected across y axis, then the change in reflected coordinate will be (-a,b).
If a triangle with coordinate (a,b) is reflected across x axis, then the change in reflected coordinate will be (a,-b).
To find vertices:
Since, the given triangle is reflected across y axis, then the vertices of new triangle will be,
[tex]\begin{gathered} A^{\prime}=(-1,4) \\ B^{\prime}=(-5,6) \\ C^{\prime}=(-3,10) \end{gathered}[/tex]Hence, the vertices of the new triangle are (-1,4), (-5,6) and (-3,10).
If you solved the following system by substitution, which of these could be yourcombined equation?y = 3x - 44x + 3y = 1
To solve the given system, we have to combine the equations
[tex]\begin{gathered} 4x+3y=1 \\ 4x+3(3x-4)=1 \\ 4x+9x-12=1 \\ 13x=12+1 \\ 13x=13 \\ x=\frac{13}{13} \\ x=1 \end{gathered}[/tex]Then, we find y
[tex]y=3x-4=3\cdot1-4=3-4=-1[/tex]Hence, the solutions (1,-1,). C is the answer.4. Find the equationof a line with a slopeof 3 and through(2,9)
to find the equation we need to use the slope-point equation, and we get the following
[tex]\begin{gathered} y-9=3(x-2)=3x-6 \\ y=3x-6+9=3x+3 \end{gathered}[/tex]so the answer is
[tex]y=3x+3[/tex]Dennis invest $4000 into an account that pays at a 3.5% interest rate compounded continuously. How many years will it take until Dennis has $6000 in his account? Round your answer to the nearest year 
Answer:
12 years
Explanation:
For an investment whose interest is compounded continuously, the amount in the account after t years is determined using the formula:
[tex]A(t)=P_oe^{rt}\text{ where }\begin{cases}{P_o=\text{ The amount invested}} \\ r=Interest\text{ }{Rate} \\ {t}=Time\end{cases}[/tex]In our given problem:
• A(t) = $6,000
,• Po = $4000
,• r = 3.5% = 0.035
We want to find the value of t.
Substitute the given values into the formula:
[tex]6000=4000e^{0.035t}[/tex]Then solve for t:
[tex]\begin{gathered} \text{ Divide both sides by 4000} \\ \frac{6000}{4000}=\frac{4000e^{0.035t}}{4000} \\ 1.5=e^{0.035t} \\ \text{ Take the ln of both sides:} \\ \ln(1.5)=\ln(e^{0.035t}) \\ 0.035t=\ln(1.5) \\ \text{ Divide both sides by }0.035 \\ \frac{0.035t}{0.035}=\frac{\operatorname{\ln}(1.5)}{0.035} \\ t=11.58 \\ t\approx12\text{ years} \end{gathered}[/tex]It will take Dennis 12 years (rounded to the nearest year) before he has $6,000 in his account.
Pls help: find the rational expression state any restrictions on the variable
Simplification of Rational Expressions
Given the rational expression:
[tex]\frac{n^4-10n^2+24}{n^4-9n^2+18}[/tex]Simplify and state the restriction for the variable n.
Let's work on the numerator and denominator independently. Factoring the numerator:
[tex]\begin{gathered} n^4-10n^2+24=n^4-4n^2-6n^2+24 \\ n^4-10n^2+24=n^2(n^2-4)-6(n^2-4) \\ n^4-10n^2+24=(n^2-6)\mleft(n^2-4\mright) \end{gathered}[/tex]The denominator can be factored in a similar way:
[tex]\begin{gathered} n^4-9n^2+18=n^4-3n^2-6n^2+18 \\ n^4-9n^2+18=n^2(n^2-3)-6(n^2-3) \\ n^4-9n^2+18=\mleft(n^2-3\mright)(n^2-6) \end{gathered}[/tex]Thus, rewriting the expression:
[tex]\frac{n^4-10n^2+24}{n^4-9n^2+18}=\frac{(n^2-4)(n^2-6)}{(n^2-3)(n^2-6)}[/tex]Before simplifying, we must state the restrictions for the variable. The denominator cannot be 0, thus:
[tex]\begin{gathered} n^2-3\ne0\Rightarrow n\ne\pm\sqrt[]{3} \\ n^2-6\ne0\Rightarrow n\ne\pm\sqrt[]{6} \end{gathered}[/tex]Now simplify:
[tex]\frac{n^4-10n^2+24}{n^4-9n^2+18}=\frac{(n^2-4)}{(n^2-3)}[/tex]Combining the final expression with the restrictions, we stick with choice a.
write the equation of a circle given the center (-4, 4) and radius r = 5
Given : the center of the circle = (-4 , 4)
And the radius of the circle = r = 5
The general equation of the circle is :
[tex](x-h)^2+(y-k)^2=r^2[/tex]Where (h,k) is the center of the circle and r is the radius of the circle
So, ( h , k ) = ( -4 , 4 ) and r = 5
so, the equation of the circle will be :
[tex]\begin{gathered} (x-(-4))^2+(y-4)^2=5^2 \\ \\ (x+4)^2+(y-4)^2=25 \end{gathered}[/tex]can someone please help me find the valu of X?
We are asked to find the value of x.
As you can see, the two sides are parallel and when these parallel sides intersect sides of overlapping triangles then the intercepted segments are proportional.
So, we can set up the following proportion.
[tex]\frac{15}{6}=\frac{(3x+10)-8}{8}[/tex]Let us solve the above equation for x.
[tex]\begin{gathered} \frac{15}{6}=\frac{3x+10-8}{8} \\ \frac{15}{6}=\frac{3x+2}{8} \\ 15\cdot8=6\cdot(3x+2) \\ 120=18x+12 \\ 120-12=18x \\ 108=18x \\ \frac{108}{18}=x \\ 6=x \\ x=6 \end{gathered}[/tex]Therefore, the value of x is 6
Figure 1 and figure 2 below are similar. which po8nt corresponds to point U.
SOLUTION
Step 1 :
In this question, we have that Figure 1 and Figure 2 are similar.
The point that corresponds to point U is Point E.
In a city 20% of the cars are electric, 16% of the cars are red, and 14% of the cars are red electriccars. If a car is randomly picked and found to be red, what is the probability that this car is electric?Enter your answer as a decimal number rounded to TWO digits after the decimal point, like 0.12.DO NOT enter it like 12% or 12.
Identify relative maximum and the relative minimum points on the graph, if any y=x^3-2x^2-3x
The relative maximum on the graph is approximately
The relative minimum on the graph is approximately
STEP - BY - STEP EXPLANATION
The width of a rectangular slab of concrete is 16 m less than the length. The area is 80m^2Part 1 of 3(a) What are the dimensions of the rectangle?The length of the slab is?
width = w
length = l
w = l - 16
area = w*l
(l - 16)*l = 80
l^2 - 16l = 80
l^2 - 16l - 80 = 0
(l + 4)(l - 20) = 0
then length = 20
w = 20 - 16 = 4
the width is 4
I need to use the formula for a trapezoid and find the area and perimeter
Hello there. To solve this question, we'll have to remember some properties about trapezoids and right triangles.
Given the following trapezoid:
We have to determine its area and perimeter.
For this, remember that:
The area of a trapezoid with bases B and b (larger and smaller, respectively) and height h can be found by the formula
[tex]A=\dfrac{(B+b)\cdot h}{2}[/tex]The perimeter is the sum of the measures of all sides of the figure.
For the perimeter, we'll use the pythagorean theorem to determine the measure of the legs of the trapezoid.
Okay. Notice that in the trapezoid, the larger base B measures 41, the smaller base measures 21 and the height is 18.
By the formula for area, we get
[tex]A=\dfrac{(41+21)\cdot18}{2}=62\cdot9=558[/tex]Now, notice we can determine a right triangle on the left:
To determine the legs of the triangle, we make
[tex]\dfrac{41-21}{2}=\dfrac{20}{2}=10[/tex]Now we have a right triangle with legs 10 and 18.
Using the Pythagorean theorem:
[tex]a^2+b^2=c^2[/tex]For a triangle with legs a and b and hypotenuse c, the sum of the squares of the legs is equal to the square of the hypotenuse.
Using a = 10 and b = 18, we get
[tex]\begin{gathered} 10^2+18^2=c^2 \\ 100+324=c^2 \\ 424=c^2 \\ 4\cdot106=c^2 \\ c=2\sqrt{106} \end{gathered}[/tex]Since the other right triangle is the same, the other leg have the same measure, hence we add
[tex]\text{ Perimeter }=21+41+2\cdot2\sqrt{106}=62+4\sqrt{106}[/tex]We can approximate this value using a calculator
[tex]\text{ Perimeter }\approx103.18[/tex]Status
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Convert the following equation
into slope intercept form.
x-13y = 26
y = x -
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10
Enter
Answer:
y=x+2
Step-by-step explanation:
x-13y=26
Bring the x to the other side
-13y=x+26
To get y by itself divide both sides by -13
-13y = x-26
----------- -----------
-13 -13
y=x+2
Find an equation of a parabola that satisfies the given conditions.
Focus at (8,0), directrix x = -8
The equation of a parabola for the focus at (8,0), directrix x = -8 is found as y² = 32x.
What is meant by the term parabola?A parabola is an open plane symmetrical curve created by the intersection of the a cone and a plane parallel towards its side. A projectile's path under the effect of gravity ideally continues to follow a curve of the this shape.The standard equation of parabola,
(y−n)² = 4p(x−m),
In which,
Vertex is (m,n)Axis of symmetry is y = mFocus is (p+m,n)Directrix is x =m−p.For the given value in question;
p+m = 8
n = 0
m−p = -8
m = 0
p = 8
Put the obtained values in general equation;
y² = 4×8(x+0)
y² = 32x
Thus, the equation of a parabola for the focus at (8,0), directrix x = -8 is found as y² = 32x.
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10. Write the slope-intercept form of the equation of the line through the given points. Write answer as y=mx+b. 1 po through: (0, 2) and (-5, -5) Your answer
First of all, remember what the equation of a line is:
y = mx+b
Where:
m is the slope, and
b is the y-intercept
First, let's find what m is, the slope of the line...
[tex]\begin{gathered} m=\frac{y2-y1}{x2-x1} \\ m=\frac{-5-2}{-5-0} \\ m=\frac{-7}{-5} \\ m=\frac{7}{5} \\ \end{gathered}[/tex]Now, what about b, the y-intercept?
[tex]\begin{gathered} b=y-mx \\ b=2-\frac{7}{5}(0) \\ b=2 \end{gathered}[/tex]The equation of the line that passes through the points
[tex]y=\frac{7}{5}x+2[/tex]A meteorologist collected data about wind speed in a city, in miles per hour, on consecutive days of a month. Her data is shown using the dot plot. Create a box plot to represent the data. (1 point)
dot plot titled Monthly Wind Speed and number line from 9 to 10 in increments of 1 tenth labeled Wind Speed (in miles per hour) with zero dots over 9, 1 dot over 9 and 1 tenth, 2 dots over 9 and 2 tenths, 1 dot over 9 and 3 tenths, 3 dots over 9 and 4 tenths, zero dots over 9 and 5 tenths, 1 dot over 9 and 6 tenths, 2 dots over 9 and 7 tenths, 1 dot over 9 and 8 tenths, zero dots over 9 and 9 tenths, and zero dots over
box plot with minimum value 9 and 2 tenths, lower quartile 9 and 3 tenths, median 9 and 5 tenths, upper quartile 9 and 8 tenths, and maximum value 9 and 9 tenths
box plot with minimum value 9 and 1 tenth, lower quartile 9 and 2 tenths, median 9 and 4 tenths, upper quartile 9 and 7 tenths, and maximum value 9 and 8 tenths
box plot with minimum value 9 and 1 tenth, lower quartile 9 and 3 tenths, median 9 and 4 tenths, upper quartile 9 and 6 tenths, and maximum value 9 and 8 tenths
box plot with minimum value 9 and 1 tenth, lower quartile 9 and 2 tenths, median 9 and 5 tenths, upper quartile 9 and 7 tenths, and maximum value 9 and 8 tenths
The correct box plot to represent the data on wind speed is: minimum at 9.1, first quartile at 9.2, median at 9.4, 3rd quartile at 9.7, maximum at 9.8.
What is a boxplot?
A boxplot refers to a type of chart that can be used to graphically represent and show the five-number summary of a data set with respect to locality, skewness, and spread. Thus, the five-number summary include the following:
Minimum
First quartile
Median
Third quartile
Maximum
Based on the data on wind speed in a city, the minimum should be at 9.1, first quartile at 9.2, median at 9.4, 3rd quartile at 9.7, and maximum at 9.8.
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Answer:
I believe the answer is C.
Step-by-step explanation:
The point starts at 91 then goes up all the way to 94 making that the line thing (I cant remember the terms) then the end of the box would be 97 because you want the line to end at the last dot thats on 98
I hope this isnt confusing thats just how I got C
Find the direction angle of vector v to the nearest tenth of a degree.Equation editor does not include the grouping symbols "<" and ">" that are necessary for writing avector in component form. For this question, use braces to write a vector in component form. Forexample, the vector < 2,3> should be written as {2,3}.
The direction angle is approximately 9.5 degrees
Explanation:The vectors are {-5, 0} and {7, 2}
Direction vector is {7 - (-5), 2 - 0 } = {12, 2}
Direction angle is:
[tex]\tan^{-1}(\frac{2}{12})\approx9.5^o[/tex]PLEASE ANSWER Given: a = 7 and b = 2 Then the m∠A=_?_ . ROund to the nearest degree. Enter a number answer only.
Answer:
A = 74 degrees
Step-by-step explanation:
a = b tan A
tan A = a/b
tan A = 7/2
A = arctan 7/2
A = 74 degrees
15 1/3 ÷ 3 5/6 A. 45 6/9 B. 5 1/4 c . 4 d. 4 1/3
To compute 15 1/3 ÷ 3 5/6, first transform the mixed numbers into fractions, as follows:
[tex]15\frac{1}{3}=\frac{15\cdot3+1}{3}=\frac{46}{3}[/tex][tex]3\frac{5}{6}=\frac{3\cdot6+5}{6}=\frac{23}{6}[/tex]Then, 15 1/3 ÷ 3 5/6 = 46/3 ÷ 23/6. Dividing by a fraction is equivalent to multiply by its inverse, then 46/3 ÷ 23/6 = 46/3 x 6/23:
[tex]\frac{46}{3}\cdot\frac{6}{23}=\frac{46}{23}\cdot\frac{6}{3}=2\cdot2=4[/tex]43 pointThe length of a rectangular box is 5 inches longer than twice the width (x).The height is 6 inches.Which is the volume (y) when the width (x) is 9 inches
L = 2*9+5=23
In ABCD, the measure of ZD=90°, CB = 53, BD = 45, and DC = 28. What ratiorepresents the sine of ZB?|
SOLUTION
Step 1 :
In this question, we asked to find the value of
[tex]\sin \text{ B}[/tex][tex]\begin{gathered} \text{where }\angle D=90^0 \\ BD\text{ = 45} \\ DC\text{ = 28} \\ BC\text{ = 53} \end{gathered}[/tex]Step 2 :
We can are see clearly that 28 , 45, 53 ) iPythagoras' Triple, since:
[tex]28^2+45^2=53^2[/tex]Step 3 :
[tex]\begin{gathered} \sin \text{ B = }\frac{28}{53} \\ =\text{ 0.5283} \end{gathered}[/tex]CONCLUSION :
[tex]\sin \text{ B = 0.5283}[/tex]Solve the given equation over the interval [0, 2.2): 2 cos2 x + cos x + 15 = 0.X = 0 and x = 2.0T57x= - and x=66There are no real valued solutions for the equation.T371x= and x =2
The given function is
[tex]2\cos ^2x+\cos x+15=0[/tex]Solve the equation to get:
[tex]\cos x=\frac{-1\pm\sqrt[]{1-4\ast2\ast15}}{4}=\frac{-1\pm\sqrt[]{-119}}{4}[/tex]The square root of a negative number is not real hence there are no real valued solutions
Option C is correct.
Which picture below represents ?
5 2
10
Pls help
Answer:
b
Step-by-step explanation:
Last month the mail carrier delivered mail on the morning route 16 times and on the afternoon route 12 times for a total distance of 141 miles. This month the mail carrier delivered mail on the morning route 10 times and on the afternoon route 15 times, for a total distance traveled of 123.75 miles. What is the distance of the morning route in miles?
The distance of the morning route is 5.25 miles.
What is the distance of the morning route?The first step is to form a system of equations that would represent the information in the question:
16m + 12a = 141 equation 1
10m + 15a = 123.75 equation 2
The equations would be solved using the elimination method:
Multiply equation 1 by 5 and equation 2 by 4
80m + 60a = 705 equation 3
40m + 60a = 495 equation 4
Subtract equation 4 from equation 3:
40m = 210
Divide both sides of the equation by 40
m = 210 / 40
m = 5.25 miles
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Which equation represents the vertical line passing through (1,-9)?
A. x = -9
B.x = 1
C. y=-9
D. y = 1
Answer:
B is the correct equation.
While your family is visiting Deep Creek Lake, you and your mother decide to go to boating. The rangers charge $6.50 per hour in addition to a $25 deposit to rent a canoe. If the total cost to rent the canoe from 12:30 pm to 3:30pm, write and solve a linear equation to find the total cost to rent the canoe.
Given: The cost of renting a canoe is $6.50 per hour in addition to a $25 deposit.
Required: If the family rented the canoe from 12:30 PM-3:30 PM, write and solve a linear equation to find the total cost to rent the canoe.
Explanation: Let x denote the number of hours the family rented the canoe. Then the linear equation representing the total cost of renting the canoe is given by-.
[tex]Cost,\text{ }C=6.50x+25[/tex]Now, since the family rented the canoe for 3 hours. Putting x=3 in the above equation gives,
[tex]\begin{gathered} Cost=6.50\times3+25 \\ Cost=\text{\$}44.50\text{ } \end{gathered}[/tex]Final Answer: The equation representing the total cost of renting the canoe is-
[tex]C=6.50x+25[/tex]And the total cost of renting the canoe for 3 hours is $44.50.