[tex]\frac{2 \tan x}{1-\tan ^{2} x}+\frac{1}{2 \cos ^{2} x-1}=\frac{\cos x+\sin x}{\cos x-\sin x}[/tex] is proved
Solution:
Given that,
[tex]\frac{2 \tan x}{1-\tan ^{2} x}+\frac{1}{2 \cos ^{2} x-1}=\frac{\cos x+\sin x}{\cos x-\sin x}[/tex]
Let us first solve the L.H.S
[tex]\text { L. H.S }=\frac{2 \tan x}{1-\tan ^{2} x}+\frac{1}{2 \cos ^{2} x-1}[/tex] --- (1)
By trignometric identities,
[tex]\tan 2 x=\frac{2 \tan x}{1-\tan ^{2} x}[/tex]
[tex]\cos 2 x=2 \cos ^{2} x-1=\cos ^{2} x-\sin ^{2} x[/tex]
By using these in (1) we get,
[tex]\text { L. H.S }=\tan 2 x+\frac{1}{\cos 2 x}[/tex]
By definition of tan,
[tex]tan x = \frac{sinx}{cosx}[/tex]
Therefore,
[tex]L. H.S =\frac{\sin 2 x}{\cos 2 x}+\frac{1}{\cos 2 x}\\\\L. H . S=\frac{1+\sin 2 x}{\cos 2 x}$[/tex] --- (ii)
By trignometric identities,
[tex]\cos 2 x=2 \cos ^{2} x-1=\cos ^{2} x-\sin ^{2} x[/tex]
[tex]\begin{aligned}&\sin 2 x=2 \sin x \cos x\\\\&\cos ^{2} x+\sin ^{2} x=1\end{aligned}[/tex]
By using these in (ii)
[tex]\text { L. H.S }=\frac{\cos ^{2} x+\sin ^{2} x+2 \sin x \cos x}{\cos ^{2} x-\sin ^{2} x}[/tex] ----- (iii)
We know that,
[tex](a+b)^2 = a^2 + 2ab + b^2[/tex]
Similarly,
[tex](cosx + sinx)^2 = cos^2x + 2cosxsinx + sin^2x[/tex]
Also,
[tex]a^2 - b^2 = (a+b)(a-b)[/tex]
Similarly,
[tex]cos^2x - sin^2x = (cosx+sinx)(cosx-sinx)[/tex]
Therefore apply these in (iii)
[tex]\begin{aligned}&\text { L. } H . S=\frac{(\cos x+\sin x)^{2}}{\cos ^{2} x-\sin ^{2} x}\\\\&\text { L. H.S }=\frac{(\cos x+\sin x)(\cos x+\sin x)}{(\cos x+\sin x)(\cos x-\sin x)}\end{aligned}[/tex]
Cancel out (cos x + sin x) on numerator and denominator
[tex]\text { L. H.S }=\frac{\cos x+\sin x}{\cos x-\sin x}[/tex]
[tex]\frac{2 \tan x}{1-\tan ^{2} x}+\frac{1}{2 \cos ^{2} x-1}=\frac{\cos x+\sin x}{\cos x-\sin x}[/tex]
Thus L.H.S = R.H.S
Thus proved
12z-7z-2=13 solve for z
Answer: z = 3
Step-by-step explanation: To solve this equation for z, we can first combine our like terms on the left side of the equation. Since 12 and 7 both have z after their coefficient, we can subtract 12z - 7z to get 5z.
Now we have 5z - 2 = 13.
To solve from here, we add 2 to the left side of the equation in order to isolate 5z. If we add 2 to the left side, we must also add 2 to the right side. On the left side, the -2 and +2 cancel out. On the right, 13 + 2 simplifies to 15.
Now we have 5z = 15.
Solving from here, we divide both sides of the equation by 5 to get z alone. On the left side, the 5's cancel out and we are simply left with z. On the right side, 15 divided by 5 simplifies to 3 so we have z = 3.
What is D N E? Given sets in photo
Answer:
OPTION B: {5, 7, 9}
Step-by-step explanation:
D [tex]$ \cap $[/tex] E is the collection of elements that are present in both D and E.
Given the set D = {5, 6, 7, 8, 9, 10}
and E = {1, 3, 5, 7, 9, 11}
So, the elements present in both D and E are: {5, 7, 9}.
Hence, D [tex]$ \cap $[/tex] E = {5, 7, 9}.
OPTION B is the answer.
What type of triangle is created by the three cities?
The distance between city A and city B is 22 miles. The
distance between city B and city C is 54 miles. The distance
between city A and city C is 51 miles.
an acute triangle, because 222 + 542 >512
an acute triangle, because 222 + 512 >542
an obtuse triangle, because 222 + 542 >512
an obtuse triangle, because 222 + 512 >542
Answer:
An acute triangle, because [tex]22^2 + 51^2 >54^2[/tex]
Step-by-step explanation:
Given:
The distance between city A and city B is 22 miles.
The distance between city B and city C is 54 miles.
The distance between city A and city C is 51 miles.
thus we have a triangle ABC with side lengths :
AB = 22 miles (shortest side)
BC= 54 miles (longest side)
AC = 51 miles (shorter side)
For an obtuse triangle :
[tex](Shorter\ side)^2+(Shortest\ side)^2<(Longest\ side)^2[/tex]
For acute triangle :
[tex](Shorter\ side)^2+(Shortest\ side)^2>(Longest\ side)^2[/tex]
Comparing sum of squares of shorter sides with the square of longest side:
[tex]AB^2+AC^2=22^2+51^2=484+2601=3085[/tex]
[tex]BC^2=54^2=2916[/tex]
We see that:
[tex]22^2+51^2>54^2[/tex]
Thus, the condition [tex](Shorter\ side)^2+(Shortest\ side)^2>(Longest\ side)^2[/tex] fulfills proving this triangle to be an acute triangle.
Answer:
The answer is B on Edge 2020 Thank me Later!
Step-by-step explanation:
I did the Exam
Determine if the triangles in the figure are congruent
Answer:
The triangles are congruent by Angle Side Angle congruency statement and the reason is below.
Step-by-step explanation:
Given:
∠ VUW ≅ ∠ XYW
VW ≅ YW
To Prove:
Δ VUW ≅ Δ XYW
Proof:
In Δ VUW and Δ XYW
∠ VUW ≅ ∠ XYW ……….{Given}
VW ≅ YW ............{Given}
∠ VWU ≅ ∠ XWY …………..{Vertically Opposite Angles are equal}
Δ VUW ≅ Δ XYW .….{ By Angle-Side-Angle congruence test}
......Proved
caprice adds equations a and b to solve this system of equations. What makes this approach a valid method in general for solving a system of equations?
Answer:
Its the Adding a quantity one or the bottom right
Adding equations is a legitimate method in solving systems because it utilizes the elimination process to simplify the system. It also respects the fundamental properties of equations and provides a precise analytical technique for finding solutions, which can be more accurate than graphical methods.
Explanation:Adding equations, known as the method of elimination, is a valid method for solving a system of equations. This approach works because it leverages the property that if two equations both equal the same variable, they can be added together to eliminate one of the unknowns, simplifying the system. For instance, suppose we have equation a: x + 2y = 6, and equation b: 3x - y = 4. Adding them gets 4x + y = 10, which is a new equation that can help solve for one of the variables more directly. This takes advantage of the fact that equations describe relationships between variables that hold true under various mathematical operations, including addition.
Another key aspect is the understanding that solutions to equations are often not unique when they include an unknown squared, leading to two potential solutions. However, contextual knowledge can determine which solution is reasonable, such as in physics problems where time or velocity must make sense in the real world.
Moreover, while analytical methods of solving systems are precise, solving equations graphically can offer a visual understanding but may lack accuracy due to potential scaling and drawing inaccuracies. Therefore, analytical techniques, like adding equations, are generally more accurate but can be complemented with graphical methods for a better conceptual grasp.
Rafael bought n packs of pencils. Each pack has 15 pencils. Write an equation to represent the total number of pencils p that Rafael
Answer:
The equation to represent total number of pencils p that Rafael bought is 15 n .
Step-by-step explanation:
Given as :
The total numbers of pencils packs bought by Rafael = n
The number of pencils in each pack = 15 pencils
Let The total number of pencil does Rafael bought = p
Now, According to question
The total number of pencil does Rafael bought = total numbers of pencils packs bought by Rafael × number of pencils in each pack
i.e p = n × 15
So, The total number of pencil does Rafael bought = p = 15 n
Hence, The equation to represent total number of pencils p that Rafael bought is 15 n . Answer
Final answer:
The equation to represent the total number of pencils Rafael bought is p = 15 × n, where p is the total number of pencils and n is the number of packs bought.
Explanation:
To represent the total number of pencils p that Rafael bought, we create an equation that multiplies the number of packs n by the number of pencils in each pack.
Since each pack contains 15 pencils, the equation is:
p = 15 × n
Using this equation, if you know the number of packs Rafael bought, you can calculate the total number of pencils by substituting n with the number of packs.
Solve the equations:
Answer:X=5/3
Step-by-step explanation:
(2∧(3*x-4))2=(-2)²
2∧2*(3*x-4)=2²
-------------------------
2*(3*x-4)=2
3*x-4=1
3*x=5
x=5/3
The surface are of a cylinder is given by the formula SA=2pir^2+2pirh,where r is the radius of the base of the cylinder and h is the height of the cylinder. Solve the formula for h in the space given below.show all the steps.
Answer:
[tex]h=\frac{SA}{2\pi r}-r[/tex]
Step-by-step explanation:
The surface area of a cylinder is equal to
[tex]SA=2\pi r^{2} +2\pi rh[/tex]
Solve for h
That means ----> isolate the variable h
subtract 2πr² both sides
[tex](SA-2\pi r^{2})=2\pi rh[/tex]
Divide by 2πr both sides
[tex]\frac{SA-2\pi r^{2}}{2\pi r}=h[/tex]
Rewrite
[tex]h=\frac{SA-2\pi r^{2}}{2\pi r}[/tex]
Simplify
[tex]h=\frac{SA}{2\pi r}-r[/tex]
Help with steps pls!!!
Step-by-step explanation:
you can convert both inequalities to slope intercept form and then graph it.
So 4x+y>-1 would convert to y>-4x-1
And x+y ≥ 2 would be y≥-x+2
Then graph it
Working with the first one
4x+y > -1
Treat as an equation and find the x and y intercept
4x+y= -1
when x=0
4(0)+y = -1
y= -1
(0,-1)
when y=0
4x+0=-1
4x= -1
divide both sides by 4
x= -1/4 or -0.25
(-0.25, 0)
working with the second one
x+y≥2
same process
when x=0
0+y=2
y=2
(0,2)
when y=0
x+0=2
x=2
(2,0)
Now on your graph you have to plot the points (0, -1) ,( -0.25,0), (0,2) ,(2,0) using an appropriate scale
The boiling point of water (measured in degrees), at altitude (measured in feet), is modeled by the function () = −0.0018 + 212. In terms of altitude and temperature, which statement describes the meaning of the slope?
A. The boiling point increases by 18 degrees as the altitude increases by 1,000 feet.
B. The boiling point increases by 1.8 degrees as the altitude increases by 1,000 feet.
C. The boiling point decreases by 18 degrees as the altitude increases by 1,000 feet.
D. The boiling point decreases by 1.8 degrees as the altitude increases by 1,000 feet.
The statement which describes the meaning of the slope is " The boiling point decreases by 1.8 degrees as the altitude increases by 1,000 feet " ⇒ D
Step-by-step explanation:
The boiling point of water (measured in degrees), at altitude (measured in feet), is modeled by the function T(a) = −0.0018 a + 212, where
a is the altitude measured in feetT(a) is the boiling point of water in degreeThe form of the linear function is f(x) = mx + b, where m is slope (the rate of change), and b is the y-intercept (the value of f(x) at x = 0)
By comparing between the form of the function and the given function
∴ m = -0.0018 ⇒ degrees per foot
∵ The sign of the slope is negative
∵ The slope = Δf(x)/Δx
- That means T(a) is decreasing function
∴ The boiling point is decreases when the altitude increasing
∵ -0.0018 = [tex]-\frac{18}{10,000}[/tex] degrees per feet
∴ The boiling point decreases by 18 degrees as the altitude
increases by 10,000 feet or the boiling point decreases by 1.8
degrees as the altitude increases by 1,000 feet
The statement which describes the meaning of the slope is " The boiling point decreases by 1.8 degrees as the altitude increases by 1,000 feet "
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The meaning of slope is best described by the boiling point decreases by 1.8 degrees as the altitude increases by 1,000 feet. The option (D) is correct.
To understand the meaning of the slope in the given function[tex]\( T(a) = -0.0018a + 212 \)[/tex], where [tex]\( T \)[/tex] represents the boiling point of water in degrees and [tex]\( a \)[/tex] represents the altitude in feet, we need to analyze the slope of the function.
The slope of the function is given by the coefficient of [tex]\( a \)[/tex], which is [tex]\( -0.0018 \)[/tex]. This slope tells us how the boiling point of water changes with altitude. To interpret the slope in terms of degrees per foot:
A slope of [tex]\( -0.0018 \)[/tex] means that for every 1 foot increase in altitude, the boiling point decreases by 0.0018 degrees.
To find out how much the boiling point changes over an increase of 1,000 feet in altitude, we multiply the slope by 1,000:
[tex]\( \text{Change in boiling point} = \text{slope} \times \text{change in altitude} \)[/tex]
[tex]\( \text{Change in boiling point} = -0.0018 \times 1000 \)[/tex]
[tex]\( \text{Change in boiling point} = -1.8 \)[/tex]
Therefore, the correct interpretation of the slope is that the boiling point decreases by 1.8 degrees as the altitude increases by 1,000 feet, which corresponds to option D.
The complete question is:
The boiling point of water, T (measured in degrees), at altitude a (measured in feet) is modeled by the function T(a)=-0.0018a+212. In terms of altitude and temperature, which statement describes the meaning of the slope?
A. The boiling point increases by 18 degrees as the altitude increases by 1,000 feet.
B. The boiling point increases by 1.8 degrees as the altitude increases by 1,000 feet.
C. The boiling point decreases by 18 degrees as the altitude increases by 1,000 feet.
D. The boiling point decreases by 1.8 degrees as the altitude increases by 1,000 feet.
Rhonda deposits $5,600 in a savings account that pays 1.5% interest, compounded semi-annually.
Round to the nearest cent.
Answer:
Part a) [tex]\$42[/tex]
Part b) [tex]\$5,642[/tex]
Part c) [tex]\$42.32[/tex]
Part d) [tex]\$5,684.32[/tex]
Part e) [tex]\$84.32[/tex]
Step-by-step explanation:
The complete question is
Rhonda deposits $5,600 in a savings account that pays 1.5% interest, compounded semi-annually. Round to the nearest cent.
a. How much interest does the account earn in the first 6 months?
b. What is the ending balance after 6 months?
c. How much interest does the account earn in the second 6 months?
d. What is the balance after 1 year?
e. How much interest does the account earn the first year?
we know that
The compound interest formula is equal to
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]
where
A is the Final Investment Value
P is the Principal amount of money to be invested
r is the rate of interest in decimal
t is Number of Time Periods
n is the number of times interest is compounded per year
Part a) How much interest does the account earn in the first 6 months?
in this part we have
[tex]t=6/12=0.5\ years\\ P=\$5,600\\ r=1.5\%=1.5/100=0.015\\n=2[/tex]
substitute in the formula above
[tex]A=5,600(1+\frac{0.015}{2})^{2*0.5}[/tex]
[tex]A=5,600(1.0075)^{1}[/tex]
[tex]A=\$5,642[/tex]
Find out the interest
[tex]I=A-P[/tex]
[tex]I=\$5,642-\$5,600=\$42[/tex]
Part b) What is the ending balance after 6 months?
we know that
The ending balance after 6 months is the same that the final investment value of A after 6 months
so
[tex]A=\$5,642[/tex] ----> see part a)
Part c) How much interest does the account earn in the second 6 months?
in this part we have
[tex]t=6/12=0.5\ years\\ P=\$5,642\\ r=1.5\%=1.5/100=0.015\\n=2[/tex]
substitute in the formula above
[tex]A=5,642(1+\frac{0.015}{2})^{2*0.5}[/tex]
[tex]A=5,642(1.0075)^{1}[/tex]
[tex]A=\$5,684.32[/tex]
Find out the interest
[tex]I=A-P[/tex]
[tex]I=\$5,684.32-\$5,642=\$42.32[/tex]
Part d) What is the balance after 1 year?
we know that
The balance after 1 year is equal to the initial deposit of $5,600 plus the interest earned in the first 6 months plus the interest earned in the second 6 months
so
[tex]\$5,600+\$42+\$42.32=\$5,684.32[/tex]
Part e) How much interest does the account earn the first year?
The total interest the first year is equal to the interest earned in the first 6 months plus the interest earned in the second 6 months
so
[tex]\$42+\$42.32=\$84.32[/tex]
The owner of the Outdoor Furniture Center decides to use the 120% markup. At the end on
the season, he wants to sell all the benches that are in stock. He sells the benches for 20% off
What is the total price of a bench with this discount plus a 5% sales tax.
Answer: The total price of a bench is 1.008x.
Step-by-step explanation:
Since we have given that
Let the cost price be 'x'.
Mark up % = 120%
So, Mark up value would be
[tex]\dfrac{120}{100}x\\\\=1.20x[/tex]
Discount % = 20%
Amount of discount is given by
[tex]\dfrac{20}{100}\times 1.2x\\\\=0.2\times 1.2x\\\\=0.24x[/tex]
So, it becomes,
Amount after discount is given by
[tex]1.2x-0.24x\\\\=0.96x[/tex]
Sales tax = 5%
Amount of sales tax would be
[tex]\dfrac{100+5}{100}\times 0.96x\\\\=\dfrac{105}{100}\times 0.96x\\\\=1.05\times 0.96x\\\\=1.008x[/tex]
Hence, the total price of a bench is 1.008x.
A rectangular field has a perimeter of (10a - 6 ) meters and a width of 2a meters. write a polynomial to represent the length
Answer:
The length of the rectangular field is (3 a - 3) meters
Step-by-step explanation:
Given as :
The Perimeter of rectangular field = p = ( 10 a - 6 ) meters
The width of the rectangular field = w = 2 a meters
Let The length of the rectangular field = L meters
Now From The perimeter formula
Perimeter of rectangular field = 2 × Length + 2 × width
Or, p = 2 × L + 2 × w
Or, ( 10 a - 6 ) meters = 2 × L meters + 2 × 2 a meters
Or, 10 a - 6 = 2 × L + 4 a
Or, 10 a - 4 a - 6 = 2 L
Or, 6 a - 6 = 2 L
∴ L = [tex]\dfrac{6 a - 6}{2}[/tex]
i,e L = 3 a - 3
So, The length of the rectangular field = L = (3 a - 3) meters
Hence,The length of the rectangular field is (3 a - 3) meters Answer
Caroline rolls a fair dice 90 times. How many times would she expect to roll a number greater then 3
Of the possible numbers (1,2,3,4,5,6), 3 are greater than 3. The probability to roll a number greater than 3 is 3 over the 6 total. That can be simplified to 1/2 for the probability. If she rolls the dice 90 times 90 x 1/2 or 45 rolls will receive a number greater than 3.
answer: 45
What is the solution to 3/4 a > -16 Hellp Plz
Answer:
Step-by-step explanation:
a>−64/3
Hope this helps mark me as brainiest pls
The solution of the a is greater than negative 21 and one-third. Then the correct option is A.
What is inequality?Inequality is simply a type of equation that does not have an equal sign in it. Inequality is defined as a statement about the relative size as well as is used to compare two statements.
The inequality equation is given below.
[tex]\dfrac{3}{4} \times a > -16\\[/tex]
On solving the inequality, the value of the a will be
[tex]\rm a > -16 \times \dfrac{4}{3}\\\\a > - \dfrac{64}{3}\\\\a > - 21 \dfrac{1}{3}[/tex]
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On a coordinate plane, a line goes through (negative 12, negative 2) and (0, negative 4). A point is at (0, 6).
Which point is on the line that passes through (0, 6) and is parallel to the given line?
(–12, 8)
(–6, 6)
(2, 8)
(6, 0)
Point (-12 , 8) is on the line that passes through (0, 6) and is parallel to the given line ⇒ 1st
Step-by-step explanation:
Parallel lines have:
Same slopesDifferent y-interceptsThe formula of the slope of a line which passes through points [tex](x_{1},y_{1})[/tex] and [tex](x_{1},y_{1})[/tex] is [tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]
∵ The given line passes through points (-12 , -2) and (0 , -4)
∴ [tex]x_{1}[/tex] = -12 , [tex]x_{2}[/tex] = 0
∴ [tex]y_{1}[/tex] = -2 , [tex]y_{2}[/tex] = -4
- Use the formula of the slope above to find the slope of the given line
∵ [tex]m=\frac{-4-(-2)}{0-(-12)}=\frac{-4+2}{12}=\frac{-2}{12}=\frac{-1}{6}[/tex]
∴ The slope of the given line is [tex]\frac{-1}{6}[/tex]
∵ The two lines are parallel
∴ Their slopes are equal
∴ The slope of the parallel line = [tex]\frac{-1}{6}[/tex]
∵ The parallel line passes through point (0 , 6)
- The form of the linear equation is y = mx + b, where m is the slope
and b is the y-intercept (y when x = 0)
∵ m = [tex]\frac{-1}{6}[/tex] and b = 6
∴ The equation of the parallel line is y = [tex]\frac{-1}{6}[/tex] x + 6
Let us check which point is on the line by substitute the x in the equation by the x-coordinate of each point to find y, if y is equal the y-coordinate of the point, then the point is on the line
Point (-12 , 8)
∵ x = -12 and y = 8
∵ y = [tex]\frac{-1}{6}[/tex] (-12) + 6
∴ y = 2 + 6 = 8
- The value of y is equal the y-coordinate of the point
∴ Point (-12 , 8) is on the line
Point (-12 , 8) is on the line that passes through (0, 6) and is parallel to the given line
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The point on the line that passes through (0, 6) and is parallel to the given line is (2, 8).
To find a point on a line parallel to the given line, we can use the slope of the given line, which is equal to the slope of the parallel line.
The slope of the given line can be calculated using the coordinates (-12, -2) and (0, -4):
Slope (m) = (change in y) / (change in x) = (-4 - (-2)) / (0 - (-12)) = (-2) / (12) = -1/6.
Now that we know the slope of the parallel line is -1/6, we can use the point-slope form of a linear equation to find the equation of the parallel line:
y - y1 = m(x - x1),
where (x1, y1) is the point (0, 6) and m is the slope (-1/6). Plugging in these values:
y - 6 = (-1/6)(x - 0),
y - 6 = (-1/6)x,
y = (-1/6)x + 6.
Now, we can choose any x-value to find the corresponding y-value. If we plug in x = 2 into the equation, we get:
y = (-1/6)(2) + 6 = -1/3 + 6 = 8.
So, the point (2, 8) is on the line that passes through (0, 6) and is parallel to the given line.
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Unit 3, Lesson 7
practice Problems
A car travels 55 miles per hour for 2 hours. Complete the table.
time (hours)
distance (miles)
1
55
110
Answer:
Find the complete table in attached file.
Step-by-step explanation:
Find the Incomplete table in attached file
Given:
A car travels 55 miles per hour for 2 hours.
So the speed of the car 55 miles per hour is constant for 2 hours.
Using speed formula.
[tex]Speed=\frac{distance}{time}[/tex]
Second row.
From the second row of the table [tex]time = \frac{1}{2}=0.5\ sec[/tex] and speed is constant [tex]speed = 55\ mi/hr[/tex], so we need to find only distance.
[tex]Speed=\frac{distance}{time}[/tex]
[tex]55=\frac{distance}{0.5}[/tex]
[tex]distance = 55\times 0.5[/tex]
[tex]distance = 27.5\ mi[/tex]
Third row.
From the third row of the table [tex]time =1\frac{1}{2}=\frac{3}{2} = 1.5\ sec[/tex] and speed is constant [tex]speed = 55\ mi/hr[/tex], so we need to find only distance.
[tex]Speed=\frac{distance}{time}[/tex]
[tex]55=\frac{distance}{1.5}[/tex]
[tex]distance = 55\times 1.5[/tex]
[tex]distance = 82.5\ mi[/tex]
Fourth row.
From the fourth row of the table [tex]distance =110\ miles[/tex] and speed is constant [tex]speed = 55\ mi/hr[/tex], so we need to find time.
[tex]Speed=\frac{distance}{time}[/tex]
[tex]time=\frac{distance}{speed}[/tex]
[tex]time=\frac{110}{55}[/tex]
[tex]time = 2\ sec[/tex]
Therefore we complete the table from the above calculation.
Write the equation in vertex form that has the root of -7 and has a vertex of (-1,-9)
Answer:
[tex]y=\frac{1}{4}(x+1)^2-9[/tex]
Step-by-step explanation:
Method 1
we know that
The equation of a vertical parabola in vertex form is equal to
[tex]y=a(x-h)^2+k[/tex]
where
a is the leading coefficient
(h,k) is the vertex
we have
(h,k)=(-1,-9)
substitute
[tex]y=a(x+1)^2-9[/tex]
Remember that
one root is (-7,0)
substitute and solve for a
[tex]0=a(-7+1)^2-9[/tex]
[tex]0=a(-6)^2-9[/tex]
[tex]0=36a-9[/tex]
[tex]a=\frac{1}{4}[/tex]
therefore
[tex]y=\frac{1}{4}(x+1)^2-9[/tex]
Method 2
I use the fact that the roots are the same distance from the vertex
the distance from the given root to the vertex is equal to
6 units
so
If one root is x=-7
then the other root is
x=-1+6=5
The general equation of the quadratic equation is equal to
[tex]y=a(x+7)(x-5)[/tex]
we have the vertex (-1,-9)
substitute the value of x and the value of y and solve for a
[tex]-9=a(-1+7)(-1-5)[/tex]
[tex]-9=a(6)(-6)[/tex]
[tex]-9=-36a[/tex]
[tex]a=\frac{1}{4}[/tex]
[tex]y=\frac{1}{4}(x+7)(x-5)[/tex]
so
Expanded the equation, complete the square and rewrite as vertex form
express each of the following as product of its prime factors, giving your answer in expanded form. a) 24. b) 63. c) 84. d) 150
1
2
3
4
5
6
7
8
9
10
You have a spinner, shown below. You wish to test its fairness, so you conduct several trials in which you repeatedly spin the spinner and record how many times you come up with a result of violet. Your results are shown in the table next to the spinner.

Trial
1
2
3
4
Times Spun
85
156
34
127
Times Violet
10
18
2
14
Calculate the experimental probability of your spinner showing a result of violet, written as a percentage to two decimal places.
a.
10.05%
b.
10.95%
c.
11.02%
d.
11.11%
Answer:
B
Step-by-step explanation:
85+156+34+127=402
10+18+2+14=44
44/402=10.95%
Answer is B. 10.95%
Need help breaking this down step by step
Distribute and combine
1/2(n-8)+7-n
Answer:
[tex]3-\frac{1}{2}n[/tex]
Step-by-step explanation:
We will use the distributive property shown below to break it down first.
Distributive Property: [tex]a(b-c)=ab-ac[/tex]
The expression is:
[tex]\frac{1}{2}(n-8)+7-n[/tex]
Let's distribute:
[tex]\frac{1}{2}(n-8)+7-n\\=\frac{1}{2}(n)-\frac{1}{2}(8)+7-n[/tex]
Now, we multiply:
[tex]\frac{1}{2}(n)-\frac{1}{2}(8)+7-n\\=\frac{1}{2}n-4+7-n[/tex]
Now we combine like terms (variables and numbers separately):
[tex]\frac{1}{2}n-4+7-n\\=-4+7-n+\frac{1}{2}n\\=3-\frac{1}{2}n[/tex]
This is the simplified expression.
help a girl out thank you
Answer: number 2
Step-by-step explanation:
Answer:
2
Step-by-step explanation:
77.86 divided by 0.85 show work
Answer:
91.6
Step-by-step explanation:
7786/85
Model and partial quotient solve:
783/4
Answer:
195.75
Step-by-step explanation:
There are 6 forks in the silverware drawer there are twice as many spoons as knives How many picecs of silverware are there in all? Only the answer
Answer:
30 or 52
I'm not sure. Sorry. I tried my best. Don't let the "Helping Hand" fool you. You see you said 6 forks and twice as many spoons which equals 12+6. Then "as" knives. I got confused there. I think there it's either 12 or twice as many knives as spoons. In other words, 12×2.
6 + (6 × 2) + (6 × 2) = 30
6 + (6 ×2) + (12 × 2) = 52
Given the equation square root of quantity 2x minus 1 end quantity equals 7, solve for x and identify if it is an extraneous solution.
one half; solution is extraneous
one half; solution is not extraneous
25; solution is extraneous
25; solution is not extraneous
Answer:
x=25 solution is not extraneous
Step-by-step explanation:
given, [tex]\sqrt{2x-1}[/tex]=7
squaring both side we get,
2x-1=7^2
2x-1=49
2x=50
x=25 solution is not extraneous answer
Select the correct answer.
If the point (4,-2) is included in a direct variation relationship, which point also belongs in this direct variation?
Answer:
(-4,2)
Step-by-step explanation:
Answer:
(-4,2)
Step-by-step explanation:
what is the lcm and gcm of 12:56
Step-by-step explanation:
[tex]\begin{array}{c|cc}12&2\\6&2\\3&3\\1\end{array}\qquad\qquad\begin{array}{c|cc}56&2\\28&2\\14&2\\7&7\\1\end{array}\\\\12=\boxed{2}\cdot\boxed{2}\cdot3\\\\56=\boxed{2}\cdot\boxed{2}\cdot2\cdot7\\\\GCF(12,\ 56)=\boxed{2}\cdot\boxed{2}=4\\\\LCM(12,\ 56)=\boxed{2}\cdot\boxed{2}\cdot3\cdot2\cdot7=168[/tex]
if Jerry can type an average of 35 words per minute about how long will it take him to type a four-page documentary that has 125 words on each page
Answer:
About 14.29 minutes
Step-by-step explanation:
Each page has 125 words
There are 4 pages
SO, total number of words:
125 * 4 = 500 words
The rate of typing is 35 wpm (words per minute)
So, to find time it will take to type 500 words is found by dividing the number of words (500) by the rate (35), so we have:
500/35 = 14.2857 minutes
So, it will take about 14.29 minutes to type up the whole documentary
Ben's family went on a car trip. They
stopped for gas 4 times. Each time
they stopped they spent an average
of 15 minutes at the gas station. On their
trip, about how much time did Ben's
family spend stopping for gas?
A 45 minutes
B 60 minutes
C 75 minutes
Answer:
B. 60 Minutes
Step-by-step explanation: