Answer:
b
Step-by-step explanation:
Given
3x² - 18x = 21 ( divide all terms by 3 )
x² - 6x = 7
To complete the square
add ( half the coefficient of the x- term )² to both sides
x² + 2(- 3)x + 9 = 7 + 9
(x - 3)² = 16 ( take the square root of both sides )
x - 3 = ± [tex]\sqrt{16}[/tex] = ± 4 ( add 3 to both sides )
x = 3 ± 4, hence
x = 3 - 4 = - 1 and x = 3 + 4 = 7
The correct option is b. [tex]\((x-3)^2=16\); -1 and 7.[/tex]
To solve the given quadratic equation [tex]\(3x^2 - 18x = 21\),[/tex] we first divide the entire equation by 3 to simplify it:
[tex]\[ x^2 - 6x = 7 \]\[ x^2 - 6x + 9 - 9 = 7 \] \[ (x - 3)^2 - 9 = 7 \][/tex]
Now, we isolate the perfect square on one side:
[tex]\[ (x - 3)^2 = 7 + 9 \] \[ (x - 3)^2 = 16 \][/tex]
This is the perfect square equation. To find the solutions, we take the square root of both sides:
[tex]\[ x - 3 = \pm4 \][/tex]
Now, we solve for \(x\) by adding 3 to both sides:
[tex]\[ x = 3 \pm 4 \][/tex]
This gives us two solutions:
[tex]\[ x = 3 + 4 = 7 \] \[ x = 3 - 4 = -1 \][/tex]
Therefore, the solutions to the equation are [tex]\(x = -1\) and \(x = 7\),[/tex] which corresponds to option b. [tex]\((x-3)^2=16\); -1 and 7.[/tex]
Solve the system of equations and choose the correct ordered pair.
2x - 6y = 8
5x - 4y = 31
Answer:
The solution is (7, 1).
Step-by-step explanation:
2x - 6y = 8 Multiply this by -5.
5x - 4y = 31 Multiply this by 2.
-10x + 30y = -40 ...(1)
10x - 8y = 62.........(2)
Adding (1) and (2):
22y = 22
y = 1.
Substitute for y in the first equation:
2x - 6(1) = 8
2x = 14
x = 7.
Final answer:
The system of equations 2x - 6y = 8 and 5x - 4y = 31 can be solved using the elimination method, yielding the solution (7, 1).
Explanation:
To solve the system of equations given by 2x - 6y = 8 and 5x - 4y = 31, we can use the substitution or elimination method.
Let's use the elimination method for efficiency:
Multiply the first equation by 5 and the second equation by 2 to get a common coefficient for x.The solution to the system of equations is the ordered pair (x, y) = (7, 1).
Find all solutions to the equation sin(3x)cosx+sinx cos(3x)=0 on the interval [0,2pi]
a- x=0,pi/4,pi/2,3pi/4,pi,3pi/2,2pi
b- x=0,pi/2,pi,3pi/2,2pi
c- x=0,pi,2pi
d- x=0,pi/2,3pi/2
Answer:
Step-by-step explanation:
Please have in mind that sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
So what we do is:
sin(3x)cos(x) - sin(x)cos(3x) = 0 => sin(3x - x) = 0
sin(2x) = 0
2x = 0, π, 2π, 3π, 4π
x = 0, π/2, π, 3π/2, 2π
Chris wanted to transform the graph of the parent function Y= cot (x) by horizontally compressing it so that it has a period of 2/π units, horizontally Terslating it π/4 units to the right, and vertically translating it 1 unit up. To do so, he graphed the function y= cot (2x-π/4)+1 as shown. What did he do wrong?
Answer:
The answer is C: He graphed the function y=cot(2x-pi/4)+1 correctly but it was not the right function to graph. He should have graphed y=cot(2x-pi/2)+1.
Step-by-step explanation:
The reason why it is C is because we want a period of pi/2, which would mean that b must be equal to 2 (if you use the period equation for tan and cot, pi/b, in order for pi/b to be equal to pi/2, b must be 2). The form for a trigonometric function is: y = acotb(x-h)+k. And if you notice, the equation he uses has the b already distributed inside the parenthesis, which means that both x and h were already multiplied. If we divide 2x and pi/4 by two, we get x, but h becomes pi/8, which is not equal to pi/4 as required by the problem. The correct equation would be: y = cot(2x-pi/2)+1 because when you divide out the two from inside the parenthesis, you get: y = cot2(x-pi/4)+1, which is the equation that he should've graphed.
I hope this helped you out!
If you have any further questions don't be afraid to ask.
Chris made a mistake by multiplying the x variable by 2 instead of π/2 for the horizontal compression and by not correctly adjusting the phase shift for the horizontal translation. The correct transformed function to meet the desired criteria should be y = cot((π/2)x - π/4) + 1.
Explanation:Chris wanted to alter the graph of the parent function Y = cot(x) to achieve a certain transformation: a horizontal compression for a new period of 2/π units, a horizontal translation of π/4 units to the right, and a vertical translation of 1 unit up. He graphed the function y = cot(2x - π/4) + 1. However, there was a mistake in his transformation.
The correct transformation for a horizontal compression to adjust the period to 2/π units would be by multiplying the x variable by π/2. However, Chris multiplied by 2, which would give the transformed function a period of π units, not 2/π units as intended. Moreover, for a horizontal translation of π/4 units to the right, the correct function would include (x - π/4) inside the cotangent function, not (2x - π/4) as Chris graphed . The correct transformation of the parent function thus should have been y = cot((π/2)x - π/4) + 1 .
What is the total number of common tangents that can be drawn to the circles?
A. 0
B. 2
C. 1
D. 3
When a circle is inside of another circle and touch each other as shown there is 1 common tangent ( where they touch).
The answer is C. 1
Answer:
only 1 tangent can drawn to the circle .
Step-by-step explanation:
Given : Two circle with common one point.
To find : What is the total number of common tangents that can be drawn to the circles
Solution : We have given two circle
A tangent to a circle is a straight line which touches the circle at only one point.
We can see both circle are touching at s single point.
By the definition of tangent: A tangent to a circle is a straight line which touches the circle at only one point.
So, only one common point hence only one tangent can be drawn to the circles.
Therefore, only 1 tangent can drawn to the circle .
One liter is approximately equal to 0.26 gallons. Find the volume rounded to the nearest hundredth of a liter of a container that holds approximately 5.5 gallons.
To find the volume of a container in liters, knowing it holds 5.5 gallons, we use the conversion factor that 1 gallon is approximately 3.85 liters. Multiplying this conversion factor with the gallon, we obtain approximately 21.175 liters. Rounded to the nearest hundredth, the volume is 21.18 liters.
Explanation:The subject of this question falls under Mathematics, particularly volume conversions. Given that 1 liter is approximately equal to 0.26 gallons, you want to find out the volume of a container, to the nearest hundredth of a liter, that holds 5.5 gallons.
To help you understand the process, here is a step-by-step explanation:
Firstly, let's use the given conversion factor. Since 1 liter equals 0.26 gallons, we can say that 1 gallon is approximately equal to 1/0.26, equivalent to about 3.85 liters.Now, if a container holds 5.5 gallons, to find out the volume of this container in liters, you simply multiply the number of gallons by the conversion factor: 5.5 gallons * 3.85 liters/gallon. This gives us approximately 21.175 liters.However, the question asks us to round this to the nearest hundredth. So, rounded to the nearest hundredth, the container's volume is roughly 21.18 liters.Remember, all conversions rely on the accuracy of the conversion factor. In this case, the conversion factor of 1 liter being approximately equal to 0.26 gallons was provided, and we took the inverse of it to convert gallons to liters.
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find the missing value
/60 = 85/100
The missing value in your ratio equation /60 = 85/100 is determined by cross-multiplying and solving the resulting equation. By doing this, we find that the missing value is 51.
Explanation:This is a Mathematics question about ratios and finding missing values in ratios. Let’s solve the equation /60 = 85/100. The missing value in your ratio equation represents a fraction that is equivalent to 85/100. To find the missing value, start by cross-multiplying:
missing value * 100 = 60 * 85
missing value * 100 = 5100
Finally, divide both sides by 100 to isolate the missing value:
missing value = 5100 / 100 = 51.
Therefore, the missing value in your ratio equation is 51.
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What is the value of x?
x =
Answer:
x = 58
Step-by-step explanation:
The angle 51° outside the circle whose sides are a tangent and a secant is
equal to half the difference of the intercepted arcs, that is
51 = 0.5 (160 - x) ← multiply both sides by 2
160 - x = 102 ( subtract 160 from both sides )
- x = -58 ( multiply both sides by - 1 )
x = 58
What is the range of the function f(x)=-(x+3)^2+7
Answer:
All real numbers less than or equal to 7
Step-by-step explanation:
we have
f(x)=-(x+3)^2+7
we know that
The function is a vertical parabola open downward
The vertex is the point ( -3,7 )
The vertex is a maximum
The range is the interval-----------> (-∞,7]
That means
All real numbers less than or equal to
The range of the function is all real numbers less than or equal to 7. The correct option is A.
What is a function?A function is defined as the expression that set up the relationship between the dependent variable and independent variable. A function's range is the collection of values it can take as input. After we enter an x value, the function outputs this sequence of values.
The given function is f(x)=-(x+3)² + 7. draw the graph of the function. It is observed that the function is a vertical parabola open downward.
The vertex is the point ( -3,7 ). The vertex is a maximum and the range is the interval is (-∞,7).
Therefore, the range of the function is all real numbers less than or equal to 7. The correct option is A.
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Fill in the blank.if necessary, use the slash marks (/) for a function bar. if sin theta= 3/5, then cos theta=
Answer:
4/5 or -4/5
Step-by-step explanation:
We are going to use the Pythagorean Identity:
[tex]\sin^2(\theta)+\cos^2(\theta)=1[/tex]
We are given the value of [tex]\sin(\theta)[/tex] which is 3/5 so plug that in:
[tex](\frac{3}{5})^2+\cos^2(\theta)=1[/tex]
Simplify:
[tex]\frac{9}{25}+\cos^2(theta)=1[/tex]
Subtract 9/25 on both sides:
[tex]\cos^2(\theta)=1-\frac{9}{25}[/tex]
[tex]\cos^2(\theta)=\frac{16}{25}[/tex]
Take the square root of both sides:
[tex]\cos(\theta)=\pm \sqrt{\frac{16}{25}}[/tex]
[tex]\cos(\theta)=\pm \frac{4}{5}[/tex]
Two trains leave towns 1152km apart at the same time and travel toward each other. One train travels 14km/h slower than the other. If they meet in 4 hours, what is the rate of each train?
Answer:
137 km/h and 151 km/h
Step-by-step explanation:
Let x km/h be the rate of slower train, then (x+14) km/h is the rate of faster train.
In 4 hours:
the slower train covers the distance 4x km;the faster train covers the distance 4(x+14) km.Since they meet in 4 hours, then they cover in total the whole distance between two cities, so
4x+4(x+14)=1152
Solve this equation for x:
4x+4x+56-1152
8x=1152-56
8x=1096
x=1096/8
x=137 km/h
x+14=137+14=151 km/h
if f(x)=3 x and g(x)= 1/x , what is the domain of (g o f)(x)?
Answer:
Domain will be x>0 or x<0 and x≠0
Step-by-step explanation:
f(x) = 3x
g(x) = 1/x
(gof)(x) = ?
(gof)(x) = g(f(x))
(gof)(x) = 1/(3x)
The domain of a function is a set of values for which the function is defined.
Find the points for which the function (gof)(x) = 1/(3x) is undefined.
if x=0 then the function is undefined.
So, domain will be x>0 or x<0 and x≠0
Finding angles outside of a circle.
Solve for x :))
Answer:
x=6
Step-by-step explanation:
So we have the difference of the intercept arcs divided by 2 is the angle formed by the two tangents there.
So we have
[tex]\frac{(37x+5)-(23x-5)}{2}=5x+17[/tex]
Clear the fraction by multiplying both sides by 2:
[tex](37x+5)-(23x-5)=2(5x+17)[/tex]
Distribute:
[tex]37x+5-23x+5=10x+34[/tex]
Combine like terms on the left hand side:
[tex]37x-23x+5+5=10x+34[/tex]
Simplify:
[tex]14x+10=10x+34[/tex]
Subtract 10x on both sides:
[tex]4x+10=34[/tex]
Subtract 10 on both sides:
[tex]4x=24[/tex]
Divide both sides by 4:
[tex]x=6[/tex]
Answer:
23x -5 + 37x + 5= 360
x = 6
Step-by-step explanation:
Which ordered pairs are in the solution set of linear equalities?
Answer: The first option. (2,2)(3,1)(4,2)
Step-by-step explanation:
The lengths of two sides of a right triangle are 5 inches and 8 inches. What is the difference between the two possible
lengths of the third side of the triangle? Round your answer to the nearest tenth.
O
O
O
O
3.1 inches
3.2 inches
10.0 inches
15.7 inches
The difference between the two possible lengths of the third side of the triangle is:
3.2 inches
Step-by-step explanation:The lengths of two sides of a right triangle are 5 inches and 8 inches.
This means that the third side could be the hypotenuse of the triangle or it could be a leg of a triangle with hypotenuse as: 8 inches.
Let the third side be denoted by c.
If the third side is the hypotenuse of the triangle.Then by using the Pythagorean Theorem we have:
[tex]c^2=5^2+8^2\\\\i.e.\\\\c^2=25+64\\\\i.e.\\\\c^2=89\\\\i.e.\\\\c=9.434\ inches[/tex]
and if the third side i.e. c is one of the leg of the triangle with hypotenuse 8 inches then the again by using Pythagorean Theorem we have:[tex]8^2=c^2+5^2\\\\i.e.\\\\64=c^2+25\\\\i.e.\\\\c^2=64-25\\\\i.e.\\\\c^2=39\\\\i.e.\\\\c=\sqrt{39}\\\\i.e.\\\\c=6.245\ inches[/tex]
Hence, the difference between the two possible lengths of the third side is:
[tex]=9.434-6.245\\\\=3.189\ inches[/tex]
which to the nearest tenth is: 3.2 inches
Answer:
B) 3.2 inches
Step-by-step explanation:
did it on edge
if 12.5%of x is 6 ,find the value of x
To solve this you must use a proportion like so...
[tex]\frac{part}{whole} = \frac{part}{whole}[/tex]
12.5 is a percent and percent's are always taken out of the 100. This means that one proportion will have 12.5 as the part and 100 as the whole
We want to know out of what number is 6 12.5% of. This means 6 is the part and the unknown (let's make this x) is the whole.
Here is your proportion:
[tex]\frac{6}{x} =\frac{12.5}{100}[/tex]
Now you must cross multiply
6*100 = 12.5*x
600 = 12.5x
To isolate x divide 12.5 to both sides
600/12.5 = 12.5x/12.5
48 = x
This means that 12.5% of 48 is 6
Hope this helped!
~Just a girl in love with Shawn Mendes
Find the value of EB.
A. 5
B. 11
C. 31
D. 25
Answer:
Option C. [tex]31\ units[/tex]
Step-by-step explanation:
Observing the figure
The point E is the midpoint segment FA and the point B is the midpoint segment CD
therefore
[tex](1/2)(AD+FC)=EB[/tex]
substitute the given values and solve for x
[tex](1/2)(38+6x-6)=7x-4[/tex]
[tex](32+6x)=14x-8[/tex]
[tex]14x-6x=32+8[/tex]
[tex]8x=40[/tex]
[tex]x=5[/tex]
Find the value of EB
[tex]EB=7x-4[/tex]
substitute the value of x
[tex]EB=7(5)-4=31\ units[/tex]
Which of the following is equal to the rational expression below when x does not equal -5 or 3? 7(x+5)/(x-3)(x+5)
Answer:
Step-by-step explanation:
Rewrite 7(x+5)/(x-3)(x+5) for greater clarity:
7(x+5)
--------------
(x-3)(x+5)
Now cancel the x + 5 terms. We get:
7
-----------
x - 3
Please note: the problem statement mentions "which of the following ..."
This implies that there were answer choices. Please, next time, share those answer choices. Thank you.
[tex]7(x+5)/(x-3)(x+5) for greater clarity:7(x+5)--------------(x-3)(x+5)Now cancel the x + 5 terms. We get: 7----------- x - 3[/tex]
What is not rational expression?No. Yes. A rational algebraic expression (or rational expression) is an algebraic expression that can be written as a quotient of polynomials, such as x2 + 4x + 4. An irrational algebraic expression is one that is not rational, such as √x + 4.
How do you solve rational expressions step by step?
The steps to solve a rational equation are:
Find the common denominator.Multiply everything by the common denominator.Simplify.Check the answer(s) to make sure there isn't an extraneous solution.Learn more about rational expression here: https://brainly.com/question/1928496
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How would you do this problem? It gives me the right answer but I need to show my work.
Answer:
x=121
Step-by-step explanation:
The exterior angle is equal to the sum of the two opposite interior angles
x = 74+47
x = 121
Square root of 24336 by prime factorization
Answer:
156
Step-by-step explanation:
The prime factorization of 24336 is 2^4*3^2*13^2. The square root of this is the same as dividing the exponent by 2. so 4/2 is and 2/2 is 1. This gives you 2^2*3*13 which is 4*3*13 or 12*13 which is 156.
-2
-1
1
2
pls help!!!
Step-by-step explanation:
5^(3b−1) = 5^(b−3)
Since the bases are equal, the exponents must also be equal.
3b − 1 = b − 3
2b = -2
b = -1
if a + b = -6 and x + y + z = -2, what is 8a - 7x - 7z - 7y + 8b
Answer:
-34
Step-by-step explanation:
a + b = -6
x + y + z = -2
We want 8a so multiply the first equation by 8
8( a + b) = -6*8
8a+8b = -48
We also want -7x so multiply the second equation by -7
-7(x + y + z) = -2*-7
-7x-7y-7z = 14
Add the two equations together
8a+8b = -48
-7x-7y-7z = 14
-------------------------
8a+8b-7x-7y-7z = -34
Rearranging the order
8a - 7x - 7z - 7y + 8b = -34
By substituting the provided equations into the final equation, we find that 8a-7x-7y-7z+8b equals -34.
Explanation:The given equations are a + b = -6 and x + y + z = -2. The equation that we are asked to solve is 8a - 7x - 7z - 7y + 8b. We can rearrange this as 8(a+b) -7(x+y+z). By substituting the given equations into this we get, 8(-6)-7(-2) which equals -48+14=-34. Thus the answer to the equation 8a-7x-7y-7z+8b is -34.
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The area of a triangular-shaped mat is 18 square feet and the base is 3 feet find the height
Answer:
Area is one half base times height. So base times height is 36 then divide out the 3 to get 12.
The height of a triangular-shaped mat with an area of 18 square feet and a base of 3 feet is 12 feet. This is determined using the formula A = 1/2 × base × height, and ensuring the answer has the correct number of significant figures when using different units or measurements.
Explanation:To find the height of a triangular-shaped mat with an area of 18 square feet and a base of 3 feet, we can use the area formula for a triangle: A = 1/2 × base × height. In this case, we need to solve for the height (h).
Area of a triangle = 1/2 × base × height
18 = 1/2 × 3 × height
18 = 1.5 × height
Height = 18 ÷ 1.5
Height = 12 feet
Therefore, the height of the triangle is 12 feet.
Regarding the given examples, when calculating the area of a triangle with different dimensions, remember to convert all measurements to the same unit, typically meters if you need to express in square meters, and then apply the formula A=1/2 × base × height. Ensure the final answer has the correct number of significant figures based on the precision of the given measurements.
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Any suggestions need help on this question?
Answer:I said B(please don't come for me)Step-by-step explanation:I selected my answer because Canada is apart North America and Chicago, Miami,Tokyo and Mexico are cities and India is a country
HELPPP ASAPPP
The data to represent average test scores for a class of 16 students includes an outlier value of 91. If the outlier is included, then the mean is 80. Which statement is always true about the new data when the outlier is removed?
The median would decrease.
The median would increase.
The mean would decrease.
The mean would increase.
Answer:
Option C (The mean would decrease).
Step-by-step explanation:
In this question, there are 16 observations and their mean is 80. There is an outlier which has the value 91. This means that the outlier is on the greater side of the mean. The formula for mean is:
Mean = Sum of observations/Number of Observations.
Sum of observations can be calculated by substituting the values in the above formula.
80 = Sum/16.
Sum = 80*16 = 1280.
Subtracting 91 from the total sum will give the sum of rest of the 15 non-outlier values. Therefore 1280 - 91 = 1189.
Calculating the mean of the 15 values:
Mean = 1189/15 = 79.267 (correct to 3 decimal places).
It can be seen that removing the outlier decreases the mean. Therefore C is the correct answer. The information regarding the median cannot be determined since actual values are not present, which are required to calculate the median. Therefore, C is the correct choice!!!
Answer: The mean would decrease
Step-by-step explanation:
john invested $12000 in a business on january 1 amd an adutional $2400 on april 1 . he withdraws $1440 in june 1 and invested $2880 on october 1 . what was john's average monthly investment balance of the year?
John's average monthly investment balance for the year is calculated based on the amount invested and the duration of each investment period, resulting in an average balance of $13,680.
Explanation:To calculate John's average monthly investment balance for the year, we need to consider the length of time each investment amount was held during the year:
January 1 to March 31 (3 months): $12,000April 1 to May 31 (2 months, after adding $2,400): $14,400June 1 to September 30 (4 months, after withdrawing $1,440): $12,960October 1 to December 31 (3 months, after adding $2,880): $15,840Now, we calculate the total investment balance for each period and then find the average:
$12,000 * 3 months = $36,000$14,400 * 2 months = $28,800$12,960 * 4 months = $51,840$15,840 * 3 months = $47,520Total for the year = $36,000 + $28,800 + $51,840 + $47,520 = $164,160Average monthly investment balance = Total for the year / 12 months = $164,160 / 12 = $13,680Therefore, John's average monthly investment balance for the year was $13,680.
A reflection of (–4, 5) over the x-axis is located in
Quadrant I
Quadrant II
Quadrant III
Quadrant IV
or no Quadrant
PLEASE HELP!!!!!
Answer:
Quadrant III
Step-by-step explanation:
(-4,5) is in the II quadrant because the x is negative and the y is positive
II I
(-,+) (+,+)
------------------------------------- X axis
III IV
(-,-) (+.-)
reflecting over the x axis means it would be in the third quadrant
iven cos θ=3√3 and sinθ<0 .
What is the value of sinθ
Answer:
177.6683°
Step-by-step explanation:
If Cos ∅=3√3 then,
The angle is therefore the inverse of the cosine.
∅= Cos⁻¹ (3√3)
= 2.3317°
If Sin is less than zero then the angle lies in the second quadrant of the unit circle.
Therefore the angle in question is 180°-2.3317°
=177.6683°
Which numbers are imaginary numbers ?
It would be any number with a - inside the sqrt, since taking the square root of a negative number gives a multiple of i. Using, this, we see that the second one, fourth, fifth, and sixth ones are correct.
Hope this helps!
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he equations9x-10y=6, 8x-10y=-23, 9x+10y=-16 and 8x+10y=13 are shown on the graph below.
Which is the approximate solution for the system of equations 8x-10y=-23 and 9x+10y=-16?
(–2.3, 0.5)
(–2.5, 1)
(–2.3, –0.5)
(–2.5, –1)
Answer:
(-2.3,0.5)
Step-by-step explanation:
Take the second equation we have y= (-16-9X)/10
Then, we will subtitute the value of y on the first equation
8X - 10(-16-9X/10)=-23
Because 10 is denominator, it will delate with the numeber 10 that is multipling the -16-9X. Then the equation is
8X-(-16-9X)=-23
then: 8x+16+9X=-23 So, 17x=-23-16 => X=-2.3
Then we put the value of X in the first equation so
9 (-2.3) -10y=-16 => 10y=-16+20.7
So, y=0.5
Answer:
A (-2.3,0.5)
Step-by-step explanation:
The probability that a train leaves on time is 0.8. The probability that the train arrives on time and leaves on time is 0.24. What is the probability that the train arrives on time, given that it leaves on time?
Answer:
Answer is 0.3
Step-by-step explanation:
Let the probability that the train arrives on time. = p
The probability that the train leaves on time = 0.8
The probability that the train leaves on time and arrives on time = 0.24
Then the equation will be:
0.8 * p = 0.24
Move the constant value to the R.H.S
p = 0.24/0.8
p = 0.3
Thus the probability is 0.3....