Answer:
[tex]-\sqrt{2} \le m \le \sqrt{2}[/tex] would ensure that at least one real root exists for this equation when solving for [tex]x[/tex].
Step-by-step explanation:
Apply the Pythagorean identity [tex]1 - \sin^{2}(x) = \cos^{2}(x)[/tex] to replace the cosine this equation with sine:
[tex](1 - \sin^{2}(x)) - (m^2 - 3)\, \sin(x) + 2\, m^2 - 3 = 0[/tex].
Multiply both sides by [tex](-1)[/tex] to obtain:
[tex]-1 + \sin^{2}(x) + (m^2 - 3)\, \sin(x) - 2\, m^2 + 3 = 0[/tex].
[tex]\sin^{2}(x) + (m^2 - 3)\, \sin(x) - 2\, m^2 + 2 = 0[/tex].
If [tex]y = \sin(x)[/tex], then this equation would become a quadratic equation about [tex]y[/tex]:
[tex]y^{2} + (m^2 - 3)\, y + (- 2\, m^2 + 2) = 0[/tex].
[tex]a = 1[/tex].[tex]b = m^{2} - 3[/tex].[tex]c = -2\, m^{2} + 2[/tex].However, [tex]-1 \le \sin(x) \le 1[/tex] for all real [tex]x[/tex].
Hence, the value of [tex]y[/tex] must be between [tex](-1)[/tex] and [tex]1[/tex] (inclusive) for the original equation to have a real root when solving for [tex]x[/tex].
Determinant of this quadratic equation about [tex]y[/tex]:
[tex]\begin{aligned} & b^{2} - 4\, a\, c \\ =\; & (m^{2} - 3)^{2} - 4 \cdot (-2\, m^{2} + 2) \\ =\; & m^{4} - 6\, m^{2} + 9 - (-8\, m^{2} + 8) \\ =\; & m^{4} - 6\, m^{2} + 9 + 8\, m^{2} - 8 \\ =\; & m^{4} + 2\, m^{2} + 1 \\ =\; &(m^2 + 1)^{2} \end{aligned}[/tex].
Hence, when solving for [tex]y[/tex], the roots of [tex]y^{2} + (m^2 - 3)\, y + (- 2\, m^2 + 2) = 0[/tex] in terms of [tex]m[/tex] would be:
[tex]\begin{aligned}y_1 &= \frac{-b + \sqrt{b^{2} - 4\, a\, c}}{2\, a} \\ &= \frac{-(m^{2} - 3) + \sqrt{(m^{2} + 1)^{2}}}{2} \\ &= \frac{-(m^{2} - 3) + (m^{2} + 1)}{2} = 2\end{aligned}[/tex].
[tex]\begin{aligned}y_2 &= \frac{-b - \sqrt{b^{2} - 4\, a\, c}}{2\, a} \\ &= \frac{-(m^{2} - 3) - \sqrt{(m^{2} + 1)^{2}}}{2} \\ &= \frac{-(m^{2} - 3) - (m^{2} + 1)}{2} \\ &= \frac{-2\, m^{2} + 2}{2} = -m^{2} + 1\end{aligned}[/tex].
Since [tex]y = \sin(x)[/tex], it is necessary that [tex]-1 \le y \le 1[/tex] for the original solution to have a real root when solved for [tex]x[/tex].
The first solution, [tex]y_1[/tex], does not meet the requirements. On the other hand, simplifying [tex]-1 \le y_2 \le 1[/tex], [tex]-1 \le -m^{2} + 1 \le 1[/tex] gives:
[tex]-2 \le -m^{2} \le 0[/tex].
[tex]0 \le m^{2} \le 2[/tex].
[tex]-\sqrt{2} \le m \le \sqrt{2}[/tex].
In other words, solving [tex]y^{2} + (m^2 - 3)\, y + (- 2\, m^2 + 2) = 0[/tex] for [tex]y[/tex] would give a real root between [tex]-1 \le y \le 1[/tex] if and only if [tex]-\sqrt{2} \le m \le \sqrt{2}[/tex].
On the other hand, given that [tex]y = \sin(x)[/tex] for the [tex]x[/tex] in the original equation, solving that equation for [tex]x\![/tex] would give a real root if and only if [tex]-1 \le y \le 1[/tex].
Therefore, the original equation with [tex]x[/tex] as the unknown has a real root if and only if [tex]-\sqrt{2} \le m \le \sqrt{2}[/tex].
I need HELP ASAP!! Please explain how to solve the problem
Answer:
[tex](x+1)^2+(y+4)^2=9\\[/tex]
Step-by-step explanation:
The general format for the equation of a circle is the following:
[tex](x-h)^2+(y-k)^2=a^2\\[/tex]
Where [tex](h,k)[/tex] is the center of the circle and ([tex]a[/tex]) is the circle's radius. Please note, that the circle ([tex](x-h)^2+(y-k)^2=a^2\\[/tex]) has a center that is (h) units to the right of the origin, and (k) units above the origin.
The given circle has a center at [tex](-1,-4)[/tex], moreover, its radius is (3) units. Therefore, one must substitute these points into the equation of a circle and simplify to find its equation:
[tex](x-h)^2+(y-k)^2=a^2\\[/tex]
[tex](x-(-1))^2+(y-(-4))^2=(3)^2\\[/tex]
[tex](x+1)^2+(y+4)^2=9\\[/tex]
Answer:
Step-by-step explanation: Let's first determine the center of the circle
which is represented by the red dot and it has the coordinates (-1, -4).
The radius of the circle is a segment that joins the center of the
circle to a point on the circle and all radii of a circle are congruent.
The radius of the circle shown here is 3.
Now, the equation of a circle is (x - h)² + (y - k)² = r² where
(h, k) is the center of the circle and r is the radius.
Now we plug all our given information into the formula.
So we have [x - (-1)]² + [y - (-4)]² = (3)².
Notice that I changed the parentheses in the formula to brackets
so that we wouldn't be dealing with too many sets of parentheses.
Changing the brackets back to parentheses,
our equation is (x + 1)² + (y + 4)² = 9.
Find the first five terms. Please solve
Answer:
a1=3, a2=6, a3=12, a4=24, a5=48
Step-by-step explanation:
a7=a*r^6=192
a10=a*r^9=1536, r^3=8, r=2 and a=3
a1=3, a2=6, a3=12, a4=24, a5=48
pls help, and explain. I will give brainliest
Answer:
Nicole should take 13 1/8 cups of snack mix.
Step-by-step explanation:
If a serving size is 7/8 and Nicole wants to take 15 serving sizes (15 7/8's), then we must multiply 7/8 by 15:
15 × 7/8
Write 15 over 1 (15 = 15/1) to make the calculations easier:
15/1 × 7/8
Multiply the numerators and denominators separately:
15 × 7 / 1 × 8
105 / 8
Instructions don't say you have to do this but I will convert this improper fraction into a mixed number:
105/8 = 13 1/8
I'LL GIVE BRAINLIEST !!! FASTER
please explain how do you get the answer
Answer:
84°
Step-by-step explanation:
angles in a quadrilateral add to 360°. 360-(114+76)=5x =170°. 170°/5 = 32°. x=32°
angles on a straight line add to 180°.
2x = 64°. 180-64=116°. y=116°.
y-x = 116-32 = 84°
Answer:
[tex]78[/tex]
Step-by-step explanation:
The inner angles of a quadrilateral all add up to 360. This means we can write the following
[tex]114 + 76 + 3x + 2x = 360\\190 + 5x = 360\\5x = 170\\x = 34[/tex]
Now that we have x we can find y. Notice that y and 2x are on the same line. Any line cutting another straight line will create two angles that add up to 180.
Therefore we can write
[tex]2x + y = 180\\2(34) + y = 180\\y = 112[/tex]
Finally computing y - x
[tex]y - x = 112 - 34 = 78[/tex]
Scott and Ashley each improved their yards by planting daylilies and ivy. They bought their
supplies from the same store. Scott spent $170 on 12 daylilies and 13 pots of ivy. Ashley spent
$172 on 14 daylilies and 2 pots of ivy. What is the cost of one daylily and the cost of one pot of
ivy?
Answer:
x = cost of daylily = $12
y = cost of ivy = $2
Step-by-step explanation:
Let
x = cost of daylily
y = cost of ivy
Scott:
12x + 13y = 170
Ashley:
14x + 2y = 172
12x + 13y = 170 (1)
14x + 2y = 172 (2)
Multiply (1) by 14 and (2) by 12
168x + 182y = 2380 (3)
168x + 24y = 2064 (4)
Subtract (4) from (3) to eliminate x
182y - 24y = 2380 - 2064
158y = 316
y = 316/158
y = 2
Substitute y = 2 into (1)
12x + 13y = 170 (1)
12x + 13(2) = 170
12x + 26 = 170
12x = 170 - 26
12x = 144
x = 144/12
x = 12
x = cost of daylily = $12
y = cost of ivy = $2
the third term and the fifth term of a geometric progression are 2 and 1/8 respectively. If all terms are positive, find the sum to the infinity of the progression
Answer:
42 + 2/3
Step-by-step explanation:
First, to calculate the sum of an infinite geometric series, our formula is
a₁/(1-r), with a₁ being the first term of the series and r being the common ratio. Therefore, we want to find both a₁ and r.
To find r, we can first determine that 2 * r = a₄ and a₄ * r = a₅, as the ratio separates one number from the next in a geometric series. Therefore, we have
2 * r * r = a₅
2 * r² = 1/8
divide both sides by 2 to isolate the r²
r² = 1/16
square root both sides to isolate r
r =± 1/4. Note the ± because r²=1/16 regardless of whether r = 1/4 or -1/4. However, because all terms are positive, r must be positive as well, or a₄, for example, would be 2 * (-1/4) = -0.5
Therefore, r = 1/4 .
To find the first term, we know that a₁ * r = a₂, and a₂ * r = a₃. Therefore, a₁ * r² = a₃ = 2
a₁ * 1/16 = 2
divide both sides by 1/16 to isolate a₁
a₁ = 2 * 1/ (1/16)
= 2 * 16
= 32
Plugging a₁ and r into our infinite geometric series formula, we have
a₁/(1-r)
= 32 / (1-1/4)
= 32/ (3/4)
= 32/ 0.75
= 42 + 2/3
Please someone tell me the answer of these questions
Answer:
VERTICALLY OPP ANGLES
Step-by-step explanation:
U
w
R
Which angles
are adjacent?
T
S
A. RWS and UWT
B. UWT and SWR
C. UWT and TWU
D. RWS and SWT
Answer:
D. RWS and SWT
Step-by-step explanation:
Adjacent angles have a common side and a common vertex but they don't overlaps each other
The two angles that has these requirements are :
RWS and SWT
The two angles are said to be adjacent angles when they share the common vertex and side.
So , adjacent angles are :
RWS and SWTOption D is the correct answer.
Give an example of when you would encounter different types of fractions in the
real world.
Step-by-step explanation:
when I had to share a bar of chocolate with 3 of my cousins....... the bar had a size of 18 cm and there were 3 of us so we divided it by 3 (18÷3) and ended up with 6, so 6cm/cousin..... ..
create a problem using radios of for your guardian/parent or friends solve. write both of your problems and Solutions in your Journal
Answer:
The following,... and uh I don't have a journal ;->
Step-by-step explanation:
You and your 10 friends each have 121 books that you will be bringing to a Best Book Barrage book exchange. Once everyone arrives you count a total of books. Assuming everyone brings the same amount of books how many people came to the exchange?
Answer: 8,228
please help me guys im desprate at this point
Answer:
Does the answer help you?
A student simplified the rational expression
using the steps shown.
(x^2/5 • x^4/5 / x^2/5)^1/2 = (x^6/5/x^2/5)^1/2=(x^3)^1/2=x^3/2
Is the answer correct? Explain.
Answer:
Does the answer help you?
Answer:
[tex]\textbf{No, the answer is not correct }[/tex].
Step-by-step explanation:
The student didn't use the quotient of powers property correctly. Instead of subtracting, the student divided the exponents within the parenthesis. So, x to the two-fifths power is the correct simplified form.
[tex](\frac{x^{2/5}\times x^{4/5} }{x^{2/5} } )[/tex]
[tex]=(\frac{x^{6/5} }{x^{2/5} } )^{1/2}[/tex]
[tex]=x^{6/5-2/5} )^{1/2}[/tex]
[tex]=(x^{4/5} )^{1/2} =x^{2/5}[/tex]
OAmalOHopeO
The perimeter of a rectangle is 18cm . if the length is (x+2), find it's width.
Answer:
W = 7 - x
Step-by-step explanation:
The perimeter is P= 2L + 2×W , where L is the length and W is the width.
If L = (x+2) , replacing L with the expression x+2 we have
P= 2×(X+2) + 2W ⇔ 18 = 2x + 4 + 2W ⇔ 2W =18 - 2x - 4 ⇔ 2W = 14 - 2x
⇔ W = 7 - x
What is the surface area of a cube measure 8 c/w?
Answer:
512
Step-by-step explanation:
if 8 is the edge using the formula
V=a³=8³
V=512
the tub started with gallons of water
Answer:
huh?
Step-by-step explanation:
Explain the steps necessary to convert a quadratic function in standard form to vertex form.
Step-by-step explanation:
A quadratic function in standard form is converted to vertex form by completing the square. The first two terms are used to create a perfect square trinomial after a zero pair is added. The zero pair is found by taking half of the x-term coefficient and squaring it.
reduce the following rational expression to the lowest form
[tex]\frac{64x^{5} - 64x}{( 8x^{2} +8) (2x +2) }[/tex]
please answer this. But no spam answers please
hurry
Answer:
4x(x - 1)
Step-by-step explanation:
Factor the numerator and denominator
64[tex]x^{5}[/tex] - 64x ← factor out 64x from both terms
= 64x([tex]x^{4}[/tex] - 1) ← difference of squares
= 64x(x² - 1)(x² + 1) ← x² - 1 is also a difference of squares
= 64x(x - 1)(x + 1)(x² + 1)
---------------------------------
(8x² + 8)(2x + 2) ← factor out 8 and 2 from each factor
= 8(x² + 1) × 2(x + 1)
= 16(x² + 1)(x + 1)
Then expression can be written as
[tex]\frac{64x(x-1)(x+1)(x^2+1)}{16(x^2+1)(x+1)}[/tex] ← cancel (x² + 1) and (x + 1) on numerator/ denominator
= [tex]\frac{64x(x-1)}{16}[/tex] ← cancel common factor 16 on numerator/ denominator
= 4x(x - 1)
what is the volume of a cube with and edge length of 4 cm
Answer:
V = 64 cm^3
Step-by-step explanation:
Volume of a cube is given by
V = s^3 where s is the side length
V = 4^3
V = 64 cm^3
Answer:
64
Step-by-step explanation:
i guess so as the volume is 4×4×4
đồ thị hàm số có bao nhiêu tiệm cận
Answer:
c
Step-by-step explanation:
Grace bought a property valued at $200,00.00 and 20% down and a mortgage amortized over 10 years. She makes equal payments due at the end of every months. Interest on the mortgage is 4% compounded semi-annually and the mortgage is renewable after five years. a) What is the size of each monthly payment? b) What is the outstanding principal at the end of the five-year term? c) What is the cost of the mortgage for the first five years?
Answer:
Size of each monthly payment = $161.69 per month
Step-by-step explanation:
Given:
Value of property = $20,000
Downpayment = 20%
Number of payment = 12 x 10 = 120
Interest rate = 4% = 4% / 12 = 0.33 %
Computation:
Loan balance = 20,000 - 20%
Loan balance = $16,000
A] Size of each monthly payment [In Excel]
Size of each monthly payment = PMT(0.33%,120,16000,0)
Size of each monthly payment = $161.69 per month
Match function with its corresponding graph
Answer:
Step-by-step explanation:
We can see that there are roots at (-2,0) and (-1,0)
also, the root at (-2,0) should bounce right off
and the root at (-1,0) should go through
With all that being said it has to be B
The point B (-2, 1) has been transformed to B' (-5, -3). The transformation is described as
A T (-3,-4)
B R x=2
C T (-3,-2)
D D3
Answer:
Option A: T(-3, -4)
Step-by-step explanation:
For a general point (x, y), if we apply the transformation:
T(a, b)
at that point, the new point that we will get is:
T(a,b)[ (x, y) ] = (x + a, y + b)
Notice that if w take the difference between the new point (x + a, y + b) and the original point (x, y)
we get:
(x + a, y + b) - (x, y) = (x + a - x, y + b y) = (a, b)
These are x-value and y-value that describe our transformation T(a, b).
Then if we know that the original point is B (-2, 1)
and the transformed point is B'( -5, -3)
We can just take the difference to get:
(-5, -3) - (-2, 1) = (-5 - (-2), -3 - 1) = (-5 + 2, -4) = (-3, -4)
Then the transformation applied is:
T(-3, -4)
The correct option is A.
I have gymnastics practice very 3 days and tennis practice every 8 days. How many days will it take for me to have BOTH gymnastics and tennis practice on the same day?
Answer:
24 days
Step-by-step explanation:
Find LCM, or least common multiple
List out all the multiples of 3:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ...
Now 8:
8, 16, 24, 32, 40, 48, 56, 63, 72, 80, 88...
LCM is 24, because they both hit 24 first.
If my answer is incorrect, pls correct me!
If you like my answer and explanation, mark me as brainliest!
-Chetan K
Explanation:
This is because 24 is the LCM (lowest common multiple) of 8 and 3.
We multiply the given values to get 8*3 = 24. This trick only works if the original two numbers have no common factors other than 1.
We can verify that 24 is indeed the LCM by listing out the multiples
multiples of 8 are: 8, 16, 24, 32, 40, ...multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...We see that 24 is found in both lists and it's the smallest such item. The next common item would be 48, which is a multiple of 24. In other words, every 24 days, your two practices will line up on the same day.
arshad's father bought x sweets .(x-4)were eaten by children and 20 were left.how many sweets did his father bring
Answer:
24
Step-by-step explanation:
20+4
simple
x-4=20
x=20+4
x=24
mark me as brainliest
Answer:
24 sweets
Step-by-step explanation:
Remaining sweets = 20
x - 4 = 20
Add 4 to both sides.
x = 20 +4
x = 24
I’m struggling, if someone can explain this please
Answer:
5
Step-by-step explanation:
6*3÷√9-1
=6*3÷3-1
=6*(3/3)-1
=6*1-1
=6-1
=5
Answer:
5
Step-by-step explanation:
We already know the square root of 9 is 3 because 3 times 3 equals 9
So now the equation is 6 * 3/3 - 1
3 divided by 3 is 1
6 * 1-1
6 times 1 is 6
6-1
You should get 5
Which polynomial correctly combines the like terms and puts the given polynomial in standard form?
one-thirdx + 8x4 – two-thirdsx2 – x
8x4 – one-thirdx2 – x
8x4 + two-thirdsx2 + two-thirdsx
8x4 – two-thirdsx2 – two-thirdsx
8x4 – two-thirdsx2 – x
Answer:
C
Step-by-step explanation:
How many people can Liam buy lunch for
Answer:
At most, Liam can only buy lunch for 6 people.
Step-by-step explanation:
It isn't going to be a decimal answer because there can't be half of a person
All you have to do is divide 50 by 8 and round down because you don't want to spend more than 50 dollars.
Writing it mathematically, it would be:
p [tex]\leq[/tex] 6
Can you help me out plz
Answer:
An isometry preserves all the following except orientation .
Please help me solve this problem
Answer:
-4
they wanted you to compute using x as 3
-2*3 + 2 = -4
Step-by-step explanation:
Correct gets 5 stars and brainliest
Answer:
13 mi
Step-by-step explanation:
Since this is a right triangle, we can use the Pythagorean theorem
a^2+b^2 = c^2 where a and b are the legs and c is the hypotenuse
5^2 +12^2 = c^2
25+144 = c^2
169 = c^2
Taking the square root of each side
sqrt(169) = sqrt(c^2)
13 = c
