We can prove that the quadratic equation x² + (1 - k)x + k - 3 = 0 has two real roots for all real values of k.
How can we prove the quadratic equation?The discriminant of the quadratic equation ax² + bx + c = 0 is given by Δ = b² - 4ac.
If Δ is positive, the equation has two real roots. If Δ is zero, the equation has one real root. If Δ is negative, the equation has two complex roots.
For the given equation, x² + (1 - k)x + k - 3 = 0, we can find the discriminant:
Δ = (1 - k)² - 4(k - 3)
= 1 - 2k + k² - 4k + 12
= k² - 6k + 13
To show that the roots are real for all real values of k, we need to show that Δ ≥ 0 for all k.
Δ = k² - 6k + 13
= (k - 3)² + 4
Since the square of any real number is non-negative, we have (k - 3)² ≥ 0. Therefore, Δ = (k - 3)² + 4 ≥ 4 for all real values of k.
Since Δ is always positive or zero, the quadratic equation x² + (1 - k)x + k - 3 = 0 has two real roots for all real values of k.
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Marcus jogged 1 2/5 miles on Saturday and 9/10 miles on Sunday. Rounding the
fractions to the nearest 1/2, approximately how many miles did he jog over the
weekend? Express your answer as an improper fraction.
Marcus jogged [tex]2\frac{1}{2}[/tex] miles over the weekend.
Marcus jogged [tex]1\frac{2}{5}[/tex] miles on Saturday and 9/10 miles on Sunday.
we first need to find a common denominator:
[tex]1\frac{2}{5}[/tex] = 7/5
9/10 = 0.9 miles
Now, we can round each of these to the nearest 1/2:
7/5 is closer to [tex]1\frac{1}{2}[/tex] than 1, so we round to [tex]1\frac{1}{2}[/tex]
0.9 is closer to 1 than 1/2, so we round to 1
Adding these rounded values, we get:
[tex]1\frac{1}{2}[/tex] + 1 =[tex]2\frac{1}{2}[/tex]
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Offering 100 points to the first correct answer. Need asap, please!
A rectangle has sides measuring (6x + 4) units and (2x + 11) units.
Part A: What is the expression that represents the area of the rectangle? Show your work. (4 points)
Part B: What are the degree and classification of the expression obtained in Part A? (3 points)
Part C: How does Part A demonstrate the closure property for polynomials? (3 points)
Expression used for area of a rectangle is 12x² + 74x + 44 .
Its degree and classification is 2 and quadratic
and demonstration of a closure property is given by product of a polynomial is a polynomial.
Length of the rectangle = (6x + 4) units
and Width of the rectangle = (2x + 11) units.
The area of the rectangle is given by the product of its length and width,
Area = (6x + 4) × (2x + 11)
Find the product,
⇒ Area = 6x × 2x + 6x × 11 + 4 × 2x + 4 × 11
Simplifying this expression, we get,
⇒ Area = 12x² + 74x + 44
The expression that represents the area of the rectangle is 12x² + 74x + 44.
The degree of the expression 12x² + 74x + 44 is 2,
since the highest power of x is 2.
The classification of this expression is quadratic.
Part A demonstrates the closure property for polynomials because the product of two polynomials is also a polynomial.
Here, the area of the rectangle is the product of two polynomials (6x + 4) and (2x + 11).
And the result is also a polynomial (12x² + 74x + 44).
This shows that the set of polynomials is closed under multiplication.
Which is one of the defining properties of a mathematical system.
Therefore, the expression for area of a rectangle is (12x² + 74x + 44) ,
Degree and classification is 2 and quadratic and closure property is demonstrate by product of a polynomial.
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What are the polar coordinates of the point P shown below?
Select the correct answer below:
(5,5π4)
(4,−5π4)
(5,7π4)
(4,5π4)
(4,7π4)
(5,−5π4)
Answer:(4,5π/4)
Step-by-step explanation:
Note that the point is on the circle whose radius is labeled 4, so r=4. The angle of the point is 5π4, so θ=5π4. Thus, the polar coordinates are (4,5π4).
What does 96 tell you about the volume of one small cube(1/4)
The larger cube is made up of 384 small cubes, each with a volume of 1/4.
We have,
If we assume that 96 is the total volume of a larger cube made up of smaller cubes, and each small cube has a volume of 1/4.
We can make the following calculation:
Let x be the number of small cubes that make up the larger cube.
The volume of the larger cube = Volume of each small cube x Number of small cubes
96 = (1/4) x
Solving for x.
Multiply 4 on both sides.
96 x 4 = x
x = 384
Therefore,
The larger cube is made up of 384 small cubes, each with a volume of 1/4.
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Stephan has a box in the shape of a hexagonal prism where the hexagonal bases are regular. A net of the box is below.
In the hexagon diagram, all the sides are drawn with a rectangle on every side and each is added and another hexagon is added to a rectangle.
Note: Figure is not drawn to scale.
He measured the height of the box to be 7 in. Then, Stephan drew a line from the center of one of the hexagons to each of its vertices and noticed that all the triangles he created have a height of 9 in and a base of 10 in.
The geometry of the hexagon is equally divided into six parts of the triangles are shaded inside. The top side is marked as 10 inches, and the length of the triangle is 9 inches.
Note: Figure is not drawn to scale.
What is the surface area of the hexagonal prism?
A.
750 sq in
B.
690 sq in
C.
960 sq in
D.
1,380 sq in
I’ll give brainliest Write an inequality that represents this situation below.
Your suitcase for vacation will fit at most 11 outfits you already have 5 packed
Answer:
An inequality that represents this situation is:
x + 5 ≤ 11
where x is the number of outfits you can pack in addition to the 5 you already have.
Step-by-step explanation:
i got a what if If you have 6 outfits packed already, then the inequality becomes:
x + 6 ≤ 11
where x is the number of outfits you can pack in addition to the 6 you already have. happy to help
Use the image to determine the direction and angle of rotation. Graph of triangle ABC in quadrant 1 with point A at 1 comma 3. A second polygon A prime B prime C prime in quadrant 4 with point A prime at 3 comma negative 1. 90° clockwise rotation 180° clockwise rotation 180° counterclockwise rotation 90° counterclockwise rotation
Which term of the sequence 1/4;-1;-21/4;...is equal to -131/2
11, - 6, -1, 4, 9, 14, 19, ... are mapped onto 4. ... A;-l = (-ltQn (mod N),. (A5.34) ... 131 2, 14, 34, 38, 42, 78, 90, 178, 778, 974(1000).
Step-by-step explanation:
the nearest .1/2 incp, we say the 'unit 131/2 incl. 1.4 a measUrement is-stated tp- be 546 inch, this UM= the
4/5 is a real number or imaginary
4/5 is a real number.
What are real numbers?A real number is a value in mathematics that expresses a quantity along a continuous line. Real numbers can be expressed as fractions or decimals and can be positive, negative, or zero.
4/5 is a rational number in this situation, which is a form of real number that may be written as a ratio of two integers. A real number can also be irrational, which means that it cannot be stated as a ratio of two integers and that its decimal representation goes on indefinitely without repeating.
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Can someone please help me with this? I'm struggling and it's already late :(
Answer:
(2, -5)
(4, -3)
(3, -1)
Step-by-step explanation:
When reflected across the X-axis, the sign of the y will change to the opposite.
(x, y) → (x, -y)
Let's solve
(2,5) → (2, -5)
(4,3) → (4, -3)
(3,1) → (3, -1)
So, the missing coordinates are: (2, -5)
(4, -3)
(3, -1)
How many liters each of a 50% acid solution and a 60% acid solution must be used to produce 60 liters of a 55% acid solution? (Round to two decimal places if necessary.)
To produce 60 liters of a 55% acid solution, we need 18 liters of a 50% acid solution and 42 liters of a 60% acid solution.
Let x be the number of liters of the 50% acid solution needed, and y be the number of liters of the 60% acid solution needed.
We can set up two equations based on the amount of acid and the total volume:
0.50x + 0.60y = 0.55(60)
x + y = 60
Simplifying the first equation:
0.50x + 0.60y = 33
5x + 6y = 330 (multiplying both sides by 10)
We can solve for one variable in terms of the other in the second equation, and substitute into the first equation:
y = 60 - x
5x + 6(60 - x) = 330
Simplifying:
x = 18
So we need 18 liters of the 50% acid solution, and:
y = 60 - x = 42
So we need 42 liters of the 60% acid solution.
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Find the values of the 6 trigonometric functions of angle A.
The values of the trigonometric functions are
1. sin A = 12/13
2. cos A = 5/13
3. Tan A = 12/5
4. cot A = 5/12
5. cosec A = 13/12
6. sec A = 13/5
What are trigonometric functions?The trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. Examples of these functions are; sin, cos , tan cot, cosec , sec e.t.c
Sin(tetha) = opp/hyp
cos(tetha) = adj/hyp
tan(tetha) = opp/adj
here the opposite side to angle A is 12 and the adjascent is 5 while the hypotenuse is 13.
Therefore,
1. sin A = 12/13
2. cos A = 5/13
3. Tan A = 12/5
4. cot A =1/tan A = 5/12
5. cosec A = 1/sin = 13/12
6. sec A = 1/cos = 13/5
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Determine if the question is a Statistical Question
Answer:
jordans van
Step-by-step explanation:
yes satiscal
[tex]x-\sqrt{x} +4-2\sqrt{x+3} =0\\[/tex]
The solution of the equation x - √x + 4 - 2√(x+3) =0 is given by x ≈ 0.449 or x = 2.
The expression is equal to,
x - √x + 4 - 2√(x+3) =0
By rearranging the terms and simplifying the expression as follows,
x - √x + 4 - 2√(x+3) = 0
⇒x + 4 = √x + 2√(x+3)
Square both sides to eliminate the square root,
⇒ (x + 4)² = (√x + 2√(x+3))²
⇒ x² + 8x + 16 = x + 4(x+3) + 4√x√x+3
⇒x² + 8x + 16 -x -4x -12 = 4√x√x+3
⇒ x² + 3x + 4 = 4√x√x+3
⇒x² + 3x + 4 = 4√x√(x+3)
⇒ x² + 3x + 4 = 4√(x²+3x)
Squaring both sides again gives,
⇒ (x² + 3x + 4)² = 16(x²+3x)
Expanding the left side gives,
x⁴ + 6x³ + 17x² + 24x + 16 = 16x² + 48x
Simplifying and rearranging the terms,
x⁴ + 6x³ + x² - 24x + 16 = 0
Factor this equation as follows,
(x² + 4x - 2)(x² + 2x - 8) = 0
The solutions are,
x²+ 4x - 2 = 0 or x² + 2x - 8 = 0
Solve each quadratic equation using the quadratic formula,
For x² + 4x - 2 = 0
x = (-4 ± √(4² - 4(1)(-2))) / (2(1))
⇒ x = (-4 ± √24) / 2
Simplifying the expression under the square root gives,
⇒ x = (-4 ± 2√6) / 2
⇒ x = -2 ± √6
For x² + 2x - 8 = 0
x = (-2 ± √(2² - 4(1)(-8))) / (2(1))
⇒x = (-2 ± √36) / 2
Simplifying the expression under the square root gives,
⇒ x = (-2 ± 6) / 2
⇒ x = -4 or x = 2
Discard the negative solution -2-√6 and -4
As it would lead to a negative value under the square root.
The final solutions are,
x ≈ 0.449 or x = 2
Therefore, the solution of the given equation is equal to x ≈ 0.449 or x = 2.
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The above question is incomplete , the complete question is:
Solve for x.
x - √x + 4 - 2√(x+3) =0
ASAP NEDDDD HELPPPPPP
By translation, the image of the triangle X is equivalent to the triangle E.
How to determine the image of a figure by rigid transformations
Herein we find the representation of a triangle, whose vertices are described below:
A(x, y) = (5, 5), B(x, y) = (5, 6), C(x, y) = (7, 5)
Whose image must be determined by a rigid transformation known as translation:
P'(x, y) = P(x, y) + T(x, y)
Where:
P(x, y) - Original point.P'(x, y) - Image of the point.T(x, y) - Translation vector.Now we proceed to determine the images of the vertices of the triangle:
A'(x, y) = (5, 5) + (4, - 3)
A'(x, y) = (9, 2)
B'(x, y) = (5, 6) + (4, - 3)
B'(x, y) = (9, 3)
C'(x, y) = (7, 5) + (4, - 3)
C'(x, y) = (11, 2)
Therefore, the triangle E is the image of the triangle X.
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L={d,e,g} J ={b,c,h} what is the union of L and J
Step-by-step explanation:
well, the combination of all set elements.
FYI : if some elements appear in both sets, we only use these once for the result set (this is not the case here, but it might be in other questions).
the combination of both sets is then
{b, c, d, e, g, h}
If you can buy one bunch of seedless green grapes for $2, then how many can you buy with $18?
If you can buy 1 bunch of seedless green grapes for $2, then $18 we can buy 9 bunches.
The concept used here is a simple division to find out how many bunches of seedless green grapes can be purchased with a given amount of money.
In this case, we know the cost of one bunch of seedless green-grapes ($2) and we are given a total amount of money ($18) that we can spend on grapes. We can use division to find out how many bunches we can buy with that amount.
⇒ Number of bunches = ($18)/($2) per bunch,
⇒ Number of bunches = 9 bunches;
Therefore, with $18, you can buy 9 bunches of seedless green grapes.
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The volume of a cylinder is 637 cm³. If the radius is 3 cm, what is the height
of the cylinder?
If the radius is 3 cm, the height of the cylinder is approximately 7.06 cm, which is closest to option B, 7 cm. The correct option is B.
The formula for the volume of a cylinder is V = πr²h, where V is the volume, r is the radius, and h is the height.
We are given that the volume of the cylinder is 637 cm³ and the radius is 3 cm. Substituting these values into the formula, we get:
637 = π(3²)h
Simplifying:
637 = 9πh
h = 637 / (9π)
h ≈ 7.06
Thus, the height of the cylinder is approximately 7.06 cm, which is closest to option B, 7 cm.
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Find the area of the green region in the circle to the nearest tenth
The area of the green region of the circle is 96.4 ft².
What is area?Area is the region bounded by a plane shape.
To calculate the area of the green region of the circle, we use the formula below
Formula:
A = πr²∅/360Where:
A = Area of the green regionr = Radius of the circle∅ = Angle of the sector at the center of the circleFrom the diagram,
Given:
r = 17/2 = 8.5 feet∅ = (180-27°) = 153°π = 3.14Substitute these values into equation 1
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The area of the green region in the circle given is expressed as approximately: a. 96.5 square feet.
How to Find the Area of a Sector in a Circle?The sector of a circle is calculated as:
A = ∅/360 * πr², where r is the radius and ∅ is the reference or central angle of the sector.
Given the following:
Central or reference angle (∅) = 180 - 27 = 153 degrees
Radius (r) = 17/2 = 8.5 feet
Plug in the values:
Area of the green region = 153/360 * π * 8.5²
Area of the green region ≈ 96.5 square feet.
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A police car is chasing a speeding car on a right-angled intersection such that former is approaching to the intersection, while the latter is moving away from it. At the point where the police car is at 80 mph and it is 44 yards from the intersection, the police radar detected that the distance between them is increasing at 45 mph, while the other car is 117 yards away from the intersection. What is the rate of the speeding car?
The rate of the speeding car is 0.493.
We have,
The police car is at 80 mph and it is 44 yards from the intersection, is increasing at 45 mph, while the other car is 117 yards away from the intersection.
So, Rate of speeding car
= (45- 80)/ (117- 44)
= -35/(-73)
= 35/73
= 0.493
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What side-angle relationship is given in the following triangle?
A. ASA
B. AAA
C. SSS
D. SAS
Without more information or a diagram, it's not possible to definitively determine which side-angle relationship (ASA, AAA, SSS, SAS) the triangle fits. These acronyms represent different combinations of sides and angles known in a triangle.
Explanation:Given the question, it's clear that we are discussing a concept in Geometry - the side-angle relationships in a triangle. However, without more information or an accompanying diagram, it isn't possible to definitively determine which of the side-angle relationships (ASA, AAA, SSS, SAS) the triangle fits. These acronyms represent different combinations of side and angle information known about a triangle. For example, ASA means that we know two Angles and the Side between them. SAS relates to knowledge of two Sides and the Angle between them, and so forth. More information or a visual representation would allow a more conclusive answer.
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An entrepreneur invests in a new play. The cost includes an overhead of $33,750 plus production costs of $1700 per performance. A sold-out performance brings in $2325 . Assume every performance is sold out, and let x represent the number of sold-out performances.
The entrepreneur's revenue is $2325x, entrepreneur's cost is $33,750 + $1700x and entrepreneur's profit from x sold-out performances is $625x - $33,750
The entrepreneur's revenue from x sold-out performances is given by the formula:
Revenue = (Price per Performance) x (Number of Performances)
Revenue = $2325 x x
Revenue = $2325x
The entrepreneur's cost from x sold-out performances is given by the formula:
Cost = Overhead + (Production Cost per Performance) x (Number of Performances)
Cost = $33,750 + $1700x
The entrepreneur's profit from x sold-out performances is the revenue minus the cost:
Profit = Revenue - Cost
Profit = $2325x - ($33,750 + $1700x)
Profit = $625x - $33,750
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=3sinθ/2cos2θ
Which of the following gives the Cartesian form of the equation above?
Select the correct answer below:
y=2/3x2
2y4(x2+y2)=3x2
2x4(x2+y2)=3y2
y=√3/2x
Answer: correct answer is A 2/3x^2
Step-by-step explanation:Multiply each side of the equation by 2cos2θ to get 2rcos2θ=3sinθ.
Multiply each side of the equation by r to get 2r2cos2θ=3rsinθ. Using the relationships x=rcosθ and y=rsinθ, the equation becomes 2x2=3y.
Divide each side of the equation by 3 and reverse the order of the equation to find that y=2/3x^2
Which of the polar coordinates given below correctly describes the point A on the graph shown here? Select all correct answers.
Select all that apply:
(5,5π4)
(5,7π4)
(5,−3π4)
(−5,π4)
(−5,5π4)
(−5,−7π4)
Answer:
(5,5π/4)
(5,−3π/4)
(−5,π/4)
(−5,−7π/4)
Step-by-step explanation:The correct answer is:
r = 5 units (distance from the pole)
θ = 5π/4 radians (angle measured counterclockwise from the ray θ = 0)
Note: The polar coordinates for point A on the given graph are r = 5 and θ = 5π/4. The r-value represents the distance from the pole to the point, and the θ-value represents the angle measured counterclockwise from the reference ray θ = 0.
Factorise fully 5a + 15ab
Answer:
Therefore, the fully factorised form is 5a(1 + 3b).
Step-by-step explanation:
5a + 15ab as 5a(1 + 3b) by taking out the common factor of 5a from both terms.
solve the equation -16=a- 19
Answer:
a = 3
Step-by-step explanation:
Collect like-terms:
[tex] - 16 = a - 19[/tex]
[tex]a = - 16 + 19[/tex]
[tex]a = 3[/tex]
Answer: 3
Step-by-step explanation: you take 19 from 3 giving you -16
Solve 19×6 using the distributive law.
Answer: (10 * 6) + (9 * 6) = 114
Step-by-step explanation:
Using the distributive property, we can write:
19 × 6 = (10 + 9) × 6
= (10 × 6) + (9 × 6) // Distributive property of multiplication over addition
= 60 + 54
= 114
Therefore, 19 × 6 = 114 when using the distributive law.
The point C(-3, 2) is rotated 270° clockwise around the origin. What are the coordinates of
the resulting point, C'?
-10 -8
-6
4
C
-2
104
8
6
4
1
2
4
9
-6
8
10
4
6
8 10
The coordinate of the resulting point C' after 270-degree rotation clockwise is (-2, -3).
Given that:
Point, C(-3, 2)
Rotation does not change the shape and size of the geometry. But changes the orientation of the geometry.
The coordinate of the resulting point C' after 270-degree rotation clockwise is given as,
C' = (-y, x)
C' = (-2, -3)
The coordinate of the resulting point C' after 270-degree rotation clockwise is (-2, -3).
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Given the definitions of f(x) and g(x) below,
find the value of (gof)(-7).
f(x)
13
g(x) = 2x² - 6x – 10
=
- 2x
The composition of functions is solved and gof ( -7 ) = 1,286
Given data ,
Let the first function be represented as f ( x )
Now , the value of f ( x ) is
f ( x ) = 13 - 2x
Let the second function be represented as g ( x )
Now , the value of g ( x ) is
g ( x ) = 2x² - 6x - 10
Now , g o f ( x ) = 2 ( 13 - 2x )² - 6 ( 13 - 2x ) - 10
On simplifying , we get
g o f ( x ) = 2 ( 169 + 4x² - 52x ) - 78 + 12x - 10
g o f ( x ) = 250 + 8x² - 92x
Now , the composition of function is
g o f ( -7 ) = 8 ( -7 )² - 92 ( -7 ) + 250
g o f ( -7 ) = 392 + 894
g o f ( -7 ) = 1,286
Hence , the function is g o f ( -7 ) = 1,286
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I don't know how to as well.